Proceedings of International Conference on Physics in ... - KEK
Proceedings of International Conference on Physics in ... - KEK Proceedings of International Conference on Physics in ... - KEK
- Page 2 and 3: High Energy Accelerator Research Or
- Page 5: iii Group photo taken in front <str
- Page 9 and 10: Contents Overview and Tutorial talk
- Page 12 and 13: energy and Ip the ionization potent
- Page 14 and 15: estoring frequency (or the shorter
- Page 17 and 18: The Heisenberg-Schwinger effect: No
- Page 19 and 20: take into account the fact that the
- Page 21 and 22: cult to extract the exponentially s
- Page 23 and 24: [35] M. B. Voloshin and K. G. Seliv
- Page 25 and 26: quency measured in the lab. By Maxw
- Page 27 and 28: positrons, each of
- Page 29 and 30: Abstract Strong-Field Effects in Be
- Page 31 and 32: At high field a is nolonger a const
- Page 33 and 34: Abstract Second order QED processes
- Page 35 and 36: p f p i Figure 4: The BIP vertex co
- Page 37 and 38: Figure 1: (Left) Time evolution <st
- Page 39 and 40: momenta of left an
- Page 41 and 42: many interesting phenomena were fou
- Page 43 and 44: ted photons could provide a window
- Page 45 and 46: various sources [28]. These include
- Page 47 and 48: where A(x) denotes a vector potenti
- Page 49 and 50: The above equation gives a microsco
- Page 51 and 52: confinement-deconfinement transitio
<str<strong>on</strong>g>Proceed<strong>in</strong>gs</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<str<strong>on</strong>g>Internati<strong>on</strong>al</str<strong>on</strong>g> <str<strong>on</strong>g>C<strong>on</strong>ference</str<strong>on</strong>g> <strong>on</strong><br />
<strong>Physics</strong> <strong>in</strong> Intense Fields<br />
PIF2010<br />
24-26 November 2010, <strong>KEK</strong>, Tsukuba, Japan<br />
Edited by K. Itakura, S. Iso and T. Takahashi<br />
High Energy Accelerator Research Organizati<strong>on</strong><br />
<strong>KEK</strong> <str<strong>on</strong>g>Proceed<strong>in</strong>gs</str<strong>on</strong>g> 2010-13<br />
February 2011<br />
A/H
High Energy Accelerator Research Organizati<strong>on</strong> (<strong>KEK</strong>), 2011<br />
<strong>KEK</strong> Reports are available from:<br />
High Energy Accelerator Research Organizati<strong>on</strong> (<strong>KEK</strong>)<br />
1-1 Oho, Tsukuba-shi<br />
Ibaraki-ken, 305-0801<br />
JAPAN<br />
Ph<strong>on</strong>e: +81-29-864-5137<br />
Fax: +81-29-864-4604<br />
E-mail: irdpub@mail.kek.jp<br />
Internet: http://www.kek.jp
<str<strong>on</strong>g>C<strong>on</strong>ference</str<strong>on</strong>g> poster<br />
i
iii<br />
Group photo taken <strong>in</strong> fr<strong>on</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> Kenkyu H<strong>on</strong>kan <strong>on</strong> 25 th November 2010
Preface<br />
The <strong>in</strong>ternati<strong>on</strong>al c<strong>on</strong>ference <strong>on</strong> <strong>Physics</strong> <strong>in</strong> Intense Fields (PIF 2010) was<br />
held from November 24 to 26, 2010, <strong>in</strong> <strong>KEK</strong> (High Energy Accelerator Research<br />
Organizati<strong>on</strong>), Tsukuba, Japan. The purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>ference was to discuss<br />
prospects <str<strong>on</strong>g>of</str<strong>on</strong>g> the fundamental physics <strong>in</strong> str<strong>on</strong>g electromagnetic fields, and the<br />
emphasis was particularly put <strong>on</strong> its <strong>in</strong>terdiscipl<strong>in</strong>ary aspects.<br />
Recent developments <str<strong>on</strong>g>of</str<strong>on</strong>g> the high-<strong>in</strong>tensity lasers open a new w<strong>in</strong>dow to fundamental<br />
physics as well as applied researches. In particular, the ultra-high<br />
<strong>in</strong>tensity realm <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum electrodynamics (QED) is with<strong>in</strong> reach, and its<br />
n<strong>on</strong>-perturbative nature will be experimentally studied <strong>in</strong> near future. Investigati<strong>on</strong>s<br />
us<strong>in</strong>g the high-<strong>in</strong>tensity lasers are <strong>in</strong>timately tied up with the str<strong>on</strong>gfield<br />
dynamics <strong>in</strong> other areas <str<strong>on</strong>g>of</str<strong>on</strong>g> physics, such as the quark-glu<strong>on</strong> plasma <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
quantum chromodynamics (QCD), astrophysical phenomena <strong>in</strong> magnetars with<br />
critically str<strong>on</strong>g magnetic fields or dielectric breakdown <strong>in</strong> str<strong>on</strong>gly correlated<br />
systems. In such circumstances, collaborati<strong>on</strong>s and discussi<strong>on</strong>s over a wide range<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> physicists are extremely important and necessary towards understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
str<strong>on</strong>g-field dynamics.<br />
In the c<strong>on</strong>ference, more than a hundred participants gathered from various<br />
countries and from various areas <str<strong>on</strong>g>of</str<strong>on</strong>g> physics, <strong>in</strong>clud<strong>in</strong>g laser physics, plasma<br />
physics, particle physics, nuclear physics, c<strong>on</strong>densed matter physics, astrophysics<br />
and accelerator physics. In order to share <strong>in</strong>terests am<strong>on</strong>g the participants <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
such wide varieties, several tutorial talks <strong>on</strong> the basics <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tense fields were<br />
given, as well as many c<strong>on</strong>tributi<strong>on</strong>s <strong>on</strong> the recent hot topics. Thanks to big<br />
efforts <str<strong>on</strong>g>of</str<strong>on</strong>g> the participants, most <str<strong>on</strong>g>of</str<strong>on</strong>g> them could be <strong>in</strong>cluded <strong>in</strong> this proceed<strong>in</strong>gs.<br />
On behalf <str<strong>on</strong>g>of</str<strong>on</strong>g> the organiz<strong>in</strong>g committee, we would like to thank all the lectures,<br />
speakers and poster presenters for their c<strong>on</strong>tributi<strong>on</strong>s, participants from<br />
various countries, and the secretaries who worked very hard for the c<strong>on</strong>ference.<br />
The c<strong>on</strong>ference was f<strong>in</strong>ancially supported by <strong>KEK</strong> (directly by the director) and<br />
Sokendai (the Center for the Promoti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Integrated Sciences). We especially<br />
acknowledge the director <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>KEK</strong>, Atsuto Suzuki, who str<strong>on</strong>gly supported the<br />
c<strong>on</strong>ference.<br />
Satoshi Iso <strong>KEK</strong>, Japan<br />
Tohru Takahashi Hiroshima University, Japan<br />
v
C<strong>on</strong>tents<br />
Overview and Tutorial talks<br />
A Recent Development <strong>in</strong> High Field Science (T. Tajima and G. Mourou) 1<br />
The Heisenberg-Schw<strong>in</strong>ger Effect: N<strong>on</strong>perturbative Vacuum Pair Producti<strong>on</strong> (G. Dunne) 7<br />
N<strong>on</strong>l<strong>in</strong>ear QED<br />
QED <strong>in</strong> Ultra-Intense Laser Fields (T. He<strong>in</strong>zl) 14<br />
Str<strong>on</strong>g-Field Effects <strong>in</strong> Beam-Beam Interacti<strong>on</strong> <strong>in</strong> L<strong>in</strong>ear Colliders (K. Yokoya) 19<br />
Sec<strong>on</strong>d Order QED Processes and Their Radiative Correcti<strong>on</strong>s (A. Hart<strong>in</strong>) 23<br />
Heavy-i<strong>on</strong> collisi<strong>on</strong>s and Quark-Glu<strong>on</strong> Plasma<br />
Str<strong>on</strong>g Field Dynamics <strong>in</strong> Heavy I<strong>on</strong> Collisi<strong>on</strong>s (K. Itakura) 26<br />
Yoctosec<strong>on</strong>d phot<strong>on</strong> pulse generati<strong>on</strong> <strong>in</strong> heavy i<strong>on</strong> collisi<strong>on</strong>s (A. Ipp) 32<br />
Fields, Instant<strong>on</strong>s, and Currents (K. Fukushima) 36<br />
Critical Behavior <str<strong>on</strong>g>of</str<strong>on</strong>g> Charm<strong>on</strong>ium: QCD Sec<strong>on</strong>d Order Stark Effect (K. Morita and S.H. Lee) 40<br />
Unruh radia <strong>on</strong><br />
On the Unruh Effect (R. Schützhold) 44<br />
Can We Detect ″Unruh Radiati<strong>on</strong>″ <strong>in</strong> the High Intensity Lasers? (S. Zhang, et al.) 46<br />
Quantum Fields <strong>in</strong> Accelerated Frames (F. Lenz) 50<br />
Axi<strong>on</strong>-like par cle searches<br />
Sh<strong>in</strong><strong>in</strong>g Light through Walls: en Route towards a New Particle <strong>Physics</strong> Fr<strong>on</strong>tier (A. L<strong>in</strong>dner) 54<br />
Prob<strong>in</strong>g Extremely Light Fields via Res<strong>on</strong>ance Scatter<strong>in</strong>g by Focus<strong>in</strong>g Intense Laser (K. Homma) 59<br />
Schw<strong>in</strong>ger mechanism <strong>in</strong> QGP and c<strong>on</strong>densed ma er<br />
Dynamical View <str<strong>on</strong>g>of</str<strong>on</strong>g> Pair Creati<strong>on</strong> via the Schw<strong>in</strong>ger Mechanism (N. Tanji) 63<br />
Exact Soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> Pair Producti<strong>on</strong>s <strong>in</strong> Str<strong>on</strong>g Electric Field with F<strong>in</strong>ite Width (A. Iwazaki) 67<br />
Str<strong>on</strong>g Field <strong>Physics</strong> <strong>in</strong> C<strong>on</strong>densed Matter (T. Oka) 70<br />
N<strong>on</strong>-L<strong>in</strong>ear Charge Transport <strong>in</strong> Plasma under Str<strong>on</strong>g Field (S. Nakamura) 74<br />
Recent developments <strong>in</strong> Schw<strong>in</strong>ger mechanism<br />
Brilliant Hard γ-Producti<strong>on</strong> and e + e – -Creati<strong>on</strong> <strong>in</strong> Vacuum with Ultra-High Laser Fields:<br />
Test<strong>in</strong>g Theoretical Predicti<strong>on</strong>s at ELI-NP (D. Habs, et al.) 78<br />
Numerical Simulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QED Cascades <strong>in</strong> Intense Laser Fields (N. Elk<strong>in</strong>a and H. Ruhl) 83<br />
Schw<strong>in</strong>ger Limit Atta<strong>in</strong>ability with Extreme Light (S.V. Bulanov, et al.) 88<br />
Pair Creati<strong>on</strong> <strong>in</strong> QED-Str<strong>on</strong>g Pulsed Laser Fields (N. Naumova, et al.) 93<br />
Laser Accelerati<strong>on</strong> up to Black Holes and B-mes<strong>on</strong> Decay (H. Hora, et al.) 97<br />
vii
Recent progress <str<strong>on</strong>g>of</str<strong>on</strong>g> ultra-<strong>in</strong>tense lasers<br />
Present Status <str<strong>on</strong>g>of</str<strong>on</strong>g> Ultra-Intense Lasers and High-Field <strong>Physics</strong> <strong>in</strong> the World (H. Takabe) 101<br />
Reach<strong>in</strong>g the Schw<strong>in</strong>ger Limit with X-Rays (C. K. Rhodes, et al.) 107<br />
Magnetars<br />
N<strong>on</strong>l<strong>in</strong>ear QED Effects by Str<strong>on</strong>g Magnetic Field <strong>in</strong> Astrophysics (K. Kohri) 111<br />
Wide-Band X-ray Observati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> Magnetars (K. Makishima) 116<br />
QCD Orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Str<strong>on</strong>g Magnetic Fields <strong>in</strong> Compact Stars (T. Tatsumi) 121<br />
New technologies<br />
Recent Progress and Prospects <strong>on</strong> Laser-Plasma Accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Charged Particles (K. Nakajima) 125<br />
Fly<strong>in</strong>g Mirror as a Tool to Access Ultra-High Fields (M. Kando, et al.) 130<br />
Poster presenta <strong>on</strong>s<br />
4-Mirror Laser Stack<strong>in</strong>g Cavity for High Intensity Polarized Phot<strong>on</strong> Generati<strong>on</strong> (T. Akagi, et al.)<br />
Current Status <str<strong>on</strong>g>of</str<strong>on</strong>g> LFEX Laser and Exa-watt Laser C<strong>on</strong>cept at ILE/Osaka (J. Kawanaka, LFEX-Team,<br />
134<br />
EXA-Team, and H. Azechi) 137<br />
X-ray Emissi<strong>on</strong> from Magnetars and Its Physical Interpretati<strong>on</strong> (T. Enoto) 141<br />
The Nielsen-Olesen Instabilities <strong>in</strong> the Glasma (H. Fujii, et al.) 144<br />
First Order Quantum Correcti<strong>on</strong> to the Larmor Radiati<strong>on</strong> (G. Nakamura) 147<br />
Fast Vacuum Decay <strong>in</strong>to Particle Pairs <strong>in</strong> Str<strong>on</strong>g Electric and Magnetic Fields (Y. Hidaka, et al.) 150<br />
N<strong>on</strong>can<strong>on</strong>ical Lie Perturbati<strong>on</strong> Analysis for the Relativistic P<strong>on</strong>deromotive Force (N. Iwata, et al.) 153<br />
Particle Based Integrated Code EPIC3D for Laser-Matter Interacti<strong>on</strong> (Y. Kishimoto) 156<br />
X-Ray Generati<strong>on</strong> via Laser Compt<strong>on</strong> Scatter<strong>in</strong>g by Laser-Accelerated Electr<strong>on</strong> Beam (E.Miura, et al.) 159<br />
Measurement <str<strong>on</strong>g>of</str<strong>on</strong>g> Nanometer Scale Beam Size by the Sh<strong>in</strong>take M<strong>on</strong>itor (M. Oroku, et al.) 162<br />
Investigat<strong>in</strong>g the One-Phot<strong>on</strong> Annihilati<strong>on</strong> Channel <strong>in</strong> an e – e + Plasma Created from Vacuum <strong>in</strong><br />
Str<strong>on</strong>g Laser Fields (A.V. Tarakanov, et al.) 165<br />
Accelerator Test Facility (ATF) and Future Prospect (T. Tauchi) 168<br />
Unruh radiati<strong>on</strong> and Interference effect (Y. Yamamoto et al.) 171<br />
Program <str<strong>on</strong>g>of</str<strong>on</strong>g> PIF2010 174<br />
List <str<strong>on</strong>g>of</str<strong>on</strong>g> participants 178<br />
viii
energy and Ip the i<strong>on</strong>izati<strong>on</strong> potential. A fracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
harm<strong>on</strong>ic spectrum is selected to produce pulse durati<strong>on</strong>s<br />
down to 100 as [17, 18] pulses, the shortest be<strong>in</strong>g at 80 as<br />
[19].<br />
If we want to go even shorter, we need to resort to even<br />
higher <strong>in</strong>tensities and leave the n<strong>on</strong>l<strong>in</strong>ear bound electr<strong>on</strong><br />
regime to go <strong>in</strong>to the relativistic regime, which is, for 1 µm<br />
wavelength, greater than 10 18 W/cm 2 . Such <strong>in</strong>tensitesy is<br />
are, today, comm<strong>on</strong>ly available us<strong>in</strong>g Chirped Pulse Amplificati<strong>on</strong><br />
[20] and Optical Parametric Chirped Pulse Amplificati<strong>on</strong><br />
[21] systems.<br />
In the relativistic regime, electr<strong>on</strong>s oscillat<strong>in</strong>g <strong>in</strong> the<br />
laser field become relativistic and change their “mass” dur<strong>in</strong>g<br />
their oscillati<strong>on</strong>s by a factor proporti<strong>on</strong>al to the Lorentz<br />
factor γ, which <strong>in</strong> turn is also proporti<strong>on</strong>al to the normalized<br />
vector potential a0. If a laser pulse can produce this<br />
<strong>in</strong>tensity at a target’s surface, the enormous p<strong>on</strong>deromotive<br />
laser pressure makes the electr<strong>on</strong> critical surface oscillate<br />
<strong>in</strong> and out at relativistic velocity. As a c<strong>on</strong>sequence, the<br />
light imp<strong>in</strong>g<strong>in</strong>g <strong>on</strong> this oscillat<strong>in</strong>g mirror is modulated periodically,<br />
result<strong>in</strong>g <strong>in</strong> high harm<strong>on</strong>ics [22, 23]. Relativistic<br />
High Harm<strong>on</strong>ic Generati<strong>on</strong> gives the prospect <str<strong>on</strong>g>of</str<strong>on</strong>g> a much<br />
broader harm<strong>on</strong>ic spectrum, higher efficiency with no cut<str<strong>on</strong>g>of</str<strong>on</strong>g>f<br />
def<strong>in</strong>ed by the plasma frequency [22, 24]. This has been<br />
experimentally verified [25] us<strong>in</strong>g the l<strong>on</strong>g pulse durati<strong>on</strong><br />
(300 fs) <str<strong>on</strong>g>of</str<strong>on</strong>g> the Vulcan laser and observ<strong>in</strong>g the 3200th harm<strong>on</strong>ic<br />
order.<br />
A related scheme was shown based <strong>on</strong> a few-cycle pulse,<br />
focused <strong>on</strong> <strong>on</strong>e λ 2 –this is the so called λ 3 -regime [26]–<br />
the relativistic mirror ceases to be planar and deforms due<br />
to the <strong>in</strong>dentati<strong>on</strong> created by the focused gaussian beam.<br />
As it moves, PIC simulati<strong>on</strong> shows, it simultaneously compresses<br />
the sub-cycle pulses and broadcasts them <strong>in</strong> specific<br />
directi<strong>on</strong>s. This technique provides an elegant possibility<br />
to both compress but also isolate <strong>in</strong>dividual attosec<strong>on</strong>d<br />
pulses. The predicted pulse durati<strong>on</strong> scales like<br />
T = 600(attosec<strong>on</strong>d)/a0. Here a0 is aga<strong>in</strong> the normalized<br />
vector potential, which is about unity at 10 18 W/cm 2<br />
and scales as the square root <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tensity. For <strong>in</strong>tensity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 22 W/cm 2 the compressed pulse could<br />
be <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>on</strong>ly a few attosec<strong>on</strong>ds. The same authors<br />
have simulated the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> th<strong>in</strong> sheets <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s<br />
with γ <str<strong>on</strong>g>of</str<strong>on</strong>g> few tens and with attosec<strong>on</strong>d durati<strong>on</strong> [27].<br />
These electr<strong>on</strong> bunches could provide a way to produce,<br />
by coherent Thoms<strong>on</strong> scatter<strong>in</strong>g, efficient beams <str<strong>on</strong>g>of</str<strong>on</strong>g> X-rays<br />
or even γ-rays. A similar c<strong>on</strong>cept called ‘relativistic fly<strong>in</strong>g<br />
mirror’ has been advocated and dem<strong>on</strong>strated [28], us<strong>in</strong>g<br />
a th<strong>in</strong> sheet <str<strong>on</strong>g>of</str<strong>on</strong>g> accelerated electr<strong>on</strong>s. Reflecti<strong>on</strong> from this<br />
relativistic mirror is highly efficient and <strong>in</strong>stills pulse compressi<strong>on</strong>.<br />
COMPRESSION IN THE ULTRA<br />
RELATIVISTIC REGIME<br />
Can we go further <strong>in</strong> time compressi<strong>on</strong>? When <strong>on</strong>e<br />
wishes to go bey<strong>on</strong>d coherent X-rays to gamma rays, the<br />
‘mirror’ that compresses the laser <strong>in</strong>to gamma rays has to<br />
be <str<strong>on</strong>g>of</str<strong>on</strong>g> extremely high density (∼ 10 27 cm −3 ) so that the<br />
laser may be coherently reflected <strong>in</strong>to gamma phot<strong>on</strong>s. We<br />
suggest here that this may be achieved by a comb<strong>in</strong>ati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the relativistically fly<strong>in</strong>g mirror just menti<strong>on</strong>ed above with<br />
the implosi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this fly<strong>in</strong>g mirror so that its density may be<br />
enhanced by ten times <strong>in</strong> each dimensi<strong>on</strong> (thus thousandfold<br />
<strong>in</strong> its density). We surmise that this may be achieved<br />
by a large energy pulse (∼ MJ) at the ultra-relativistic (even<br />
i<strong>on</strong>s become relativistically mov<strong>in</strong>g <strong>in</strong> the optical fields) <strong>in</strong>tensity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> 10 24 W/cm 2 <strong>on</strong> a partial shell <str<strong>on</strong>g>of</str<strong>on</strong>g> a c<strong>on</strong>cave spherical<br />
target. This ultra-relativistic fly<strong>in</strong>g mirror [29] with<br />
the implod<strong>in</strong>g shell is c<strong>on</strong>ceptually capable <str<strong>on</strong>g>of</str<strong>on</strong>g> coherently<br />
backscatter<strong>in</strong>g an <strong>in</strong>jected 10 keV coherent X-ray pulse like<br />
the <strong>on</strong>e menti<strong>on</strong>ed above [26], produc<strong>in</strong>g a possibility <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
coherent gamma rays <str<strong>on</strong>g>of</str<strong>on</strong>g> 100 ys durati<strong>on</strong>.<br />
In relativistic compressi<strong>on</strong>, the c<strong>on</strong>cept described above<br />
relies <strong>on</strong>ly <strong>on</strong> electr<strong>on</strong>s <strong>in</strong> a th<strong>in</strong> fly<strong>in</strong>g sheet. Unfortunately,<br />
when we try further compressi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the pulse<br />
length, the frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s enters the doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
gamma rays. C<strong>on</strong>sequently, such a fly<strong>in</strong>g or oscillat<strong>in</strong>g<br />
electr<strong>on</strong> mirror at the solid density cannot <strong>in</strong>teract<br />
with gamma rays. However, at or near the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
10 24 W/cm 2 , i<strong>on</strong>s <strong>in</strong> the laser field can become relativistic.<br />
This characterizes the <strong>on</strong>set <str<strong>on</strong>g>of</str<strong>on</strong>g> the ultra-relativistic regime.<br />
Below this regime i<strong>on</strong>s resp<strong>on</strong>d to the laser transverse electric<br />
field with harm<strong>on</strong>ic moti<strong>on</strong>. Thus the i<strong>on</strong> moti<strong>on</strong> is<br />
l<strong>in</strong>ear and it cannot directly extract the laser energy. However,<br />
near or above this threshold, i<strong>on</strong>s beg<strong>in</strong> to resp<strong>on</strong>d to<br />
the v × B force, which produces the n<strong>on</strong>l<strong>in</strong>ear thrust force<br />
forward, the ultra-relativistic n<strong>on</strong>l<strong>in</strong>earity. Here new opportunities<br />
for pulse shorten<strong>in</strong>g arise, based <strong>on</strong> the ability<br />
to accelerate real matter (both electr<strong>on</strong>s and i<strong>on</strong>s) to approach<br />
the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light. In turn, it takes significant energy<br />
to move and compress matter <strong>in</strong>to this regime.<br />
We have devised a c<strong>on</strong>figurati<strong>on</strong> <strong>in</strong> which the laser at<br />
the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 23 − 10 24 W/cm 2 irradiates a th<strong>in</strong> c<strong>on</strong>cave<br />
shell target (30 micr<strong>on</strong> × 30 micr<strong>on</strong> × 100 nm). This<br />
causes a relativistic matter flow (not <strong>on</strong>ly electr<strong>on</strong>s but i<strong>on</strong>s<br />
as well). In this regime electr<strong>on</strong>s are held to the same speed<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong>s. The relativistically mov<strong>in</strong>g matter driven by a MJ<br />
laser pulse (red) rapidly c<strong>on</strong>verges due to the geometry <strong>in</strong><br />
its transverse dimensi<strong>on</strong>s. The l<strong>on</strong>gitud<strong>in</strong>al compressi<strong>on</strong><br />
occurs typically <strong>in</strong> this ultra-relativistic drive (Esirkepov,<br />
2004). Thus we can achieve typically 10×10×10 compressi<strong>on</strong>.<br />
This is the ‘ultra-relativistic implod<strong>in</strong>g mirror’. The<br />
relativistic Lorentz factor is estimated to be γi ∼ γe ∼ 30.<br />
Simultaneously, the relativistic mirror (<strong>in</strong> the usual relativistic<br />
regime, menti<strong>on</strong>ed earlier) driven by a 10 kJ laser<br />
pulse generates coherent X-rays. We let these X-rays collide<br />
<strong>in</strong>to the collaps<strong>in</strong>g shell. The collaps<strong>in</strong>g shell with a<br />
density <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s <strong>on</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 27 /cm 3 backscatters<br />
X-rays <strong>in</strong>to gamma rays coherently. If the X-ray energy<br />
is 10 4 eV, we obta<strong>in</strong> coherent gamma rays <strong>on</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
4 × 10 7 eV. Such a high energy gamma beam pulse is may<br />
be <strong>on</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> several times 10 ys to 100 ys. In Fig. 1<br />
we plot all these data po<strong>in</strong>ts, start<strong>in</strong>g from the world first<br />
laser by Maiman [3] to what we c<strong>on</strong>sider <strong>in</strong> this secti<strong>on</strong>.
Figure 1: The Pulse Intensity-Durati<strong>on</strong> C<strong>on</strong>jecture is shown. An <strong>in</strong>verse l<strong>in</strong>ear dependence exists between the pulse durati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> coherent light emissi<strong>on</strong> and its <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser driver <strong>in</strong> the generati<strong>on</strong> volume over 18 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude.<br />
These entries encompass different underly<strong>in</strong>g physical regimes, whose n<strong>on</strong>l<strong>in</strong>earities are aris<strong>in</strong>g from molecular, bound<br />
atomic electr<strong>on</strong>, relativistic plasma, and ultra-relativistic, and further eventually from vacuum nature. The blue patches<br />
are from the experiments, while the red from the simulati<strong>on</strong> or theory. This figure is an excerpt from [30].<br />
Remarkably, all these po<strong>in</strong>ts l<strong>in</strong>e up <strong>in</strong> a s<strong>in</strong>gle l<strong>in</strong>e over<br />
some 18 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude.[30]<br />
As the pulse is compressed to this extremely short durati<strong>on</strong><br />
<strong>in</strong> the forego<strong>in</strong>g scenario, a modest efficiency could<br />
produce sizable n<strong>on</strong>l<strong>in</strong>earities <strong>in</strong> vacuum, although its n<strong>on</strong>l<strong>in</strong>ear<br />
coefficient, n2, is 18 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude smaller<br />
than that <str<strong>on</strong>g>of</str<strong>on</strong>g> a typical optical transparent medium like glass.<br />
If the critical power <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum is 10 24 W at 1.0 µm, it<br />
will be 6 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude less for a zeptosec<strong>on</strong>d pulse,<br />
or 10 18 W. Under this c<strong>on</strong>diti<strong>on</strong> the vacuum critical power<br />
could be atta<strong>in</strong>ed with a mere millijoule. It is quite fasc<strong>in</strong>at<strong>in</strong>g<br />
to imag<strong>in</strong>e that a filament <strong>in</strong> vacuum analogous to<br />
those produced <strong>in</strong> air could be produced with an appropriate<br />
setup (such as counter stream<strong>in</strong>g c<strong>on</strong>figurati<strong>on</strong>). As <strong>in</strong><br />
air, the filament size would be limited by “vacuum breakdown”<br />
or pair creati<strong>on</strong>, when the <strong>in</strong>tensity would reach<br />
10 29 W/cm 2 corresp<strong>on</strong>d<strong>in</strong>g to a filament <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 −5 cm diameter.<br />
Further compressi<strong>on</strong> could be obta<strong>in</strong>ed with zeptosec<strong>on</strong>d<br />
pulses by self phase modulati<strong>on</strong> <strong>in</strong> vacuum. If we c<strong>on</strong>sider<br />
that the self phase modulati<strong>on</strong> scales like<br />
∆ω<br />
ω<br />
≃ Ln2<br />
c<br />
dI<br />
dt<br />
En<br />
∝ n2<br />
T 2<br />
where ω is the carrier frequency, ∆ω is the self-phase modulati<strong>on</strong><br />
broaden<strong>in</strong>g, n2 the n<strong>on</strong>l<strong>in</strong>ear <strong>in</strong>dex <str<strong>on</strong>g>of</str<strong>on</strong>g> refracti<strong>on</strong>, L<br />
the propagati<strong>on</strong> length and En the pulse energy and T the<br />
<strong>in</strong>put pulse durati<strong>on</strong>. It is clear that <strong>in</strong> spite <str<strong>on</strong>g>of</str<strong>on</strong>g> a very small<br />
n2 coefficient, extremely short pulse can produce a sizable<br />
n<strong>on</strong>l<strong>in</strong>ear effect. This c<strong>on</strong>firms what we know already <strong>in</strong><br />
the visible range where we had to wait for picosec<strong>on</strong>d- femtosec<strong>on</strong>d<br />
pulses to produce sizable n<strong>on</strong>l<strong>in</strong>ear effects <strong>in</strong> the<br />
visible like self-phase modulati<strong>on</strong> or Kerr lens mode lock<strong>in</strong>g.<br />
VACUUM NONLINEARITIES<br />
We have learned that: matter exhibits n<strong>on</strong>l<strong>in</strong>earities<br />
when exposed to str<strong>on</strong>g enough laser radiati<strong>on</strong>; manifestly<br />
n<strong>on</strong>l<strong>in</strong>earities vary depend<strong>in</strong>g <strong>on</strong> the strength <str<strong>on</strong>g>of</str<strong>on</strong>g> the ‘bend<strong>in</strong>g’<br />
field (and thus the <strong>in</strong>tensity). The str<strong>on</strong>ger we ‘bend’<br />
the c<strong>on</strong>stituent matter, the more rigid the ‘bend<strong>in</strong>g’ force<br />
we need to exert; the more rigid the force is, the higher the<br />
(1)
estor<strong>in</strong>g frequency (or the shorter the time scale) is. The<br />
n<strong>on</strong>l<strong>in</strong>earities <str<strong>on</strong>g>of</str<strong>on</strong>g> matter may vary, but this resp<strong>on</strong>se is universal,<br />
rang<strong>in</strong>g over molecular, atomic, plasma electr<strong>on</strong>ic<br />
and i<strong>on</strong>ic, and even the stiffest <str<strong>on</strong>g>of</str<strong>on</strong>g> all vacuum, n<strong>on</strong>l<strong>in</strong>earities.<br />
Thus, we have traversed nature’s display <str<strong>on</strong>g>of</str<strong>on</strong>g> the universal<br />
behavior <str<strong>on</strong>g>of</str<strong>on</strong>g> the direct correlati<strong>on</strong> between the brevity <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
pulse be<strong>in</strong>g generated and the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> its driv<strong>in</strong>g laser<br />
over the widest <strong>in</strong>tensity range our laboratory has to ever to<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g>fer.<br />
For example, we know that the laser self-focuses above<br />
the critical power with:<br />
χ3 n<strong>on</strong>l<strong>in</strong>earity<br />
Pcr = λ2/(2πn0n2) ∼ GW (2)<br />
relativistic plasma n<strong>on</strong>l<strong>in</strong>earity<br />
Pcr = mc 5 /e 2 (ω/ωp) 2 ∼ 17(ω/ωp) 2 GW, (3)<br />
vacuum n<strong>on</strong>l<strong>in</strong>earity [2]<br />
Pcr = (90/28)cE 2 Sλ 2 /α ∼ 10 15 (λ/λ1µ) 2 GW, (4)<br />
e.g. X-ray <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 keV, Pcr ∼ 10 PW,<br />
where ES is the Schw<strong>in</strong>ger field, above which the vacuum<br />
becomes sufficiently str<strong>on</strong>gly polarized and divulges electr<strong>on</strong><br />
and positr<strong>on</strong> pairs [31, 32, 33, 34, 35]. If we compare<br />
the critical power <str<strong>on</strong>g>of</str<strong>on</strong>g> self-focus<strong>in</strong>g <strong>in</strong> a gas with the χ3<br />
n<strong>on</strong>l<strong>in</strong>earity with that <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum, we f<strong>in</strong>d that the ratio is<br />
nearly α 6 with no other parameters <strong>in</strong>volved. On the other<br />
hand, the rati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Keldysh power to the Schw<strong>in</strong>ger<br />
power is aga<strong>in</strong> α 6 (i.e. EK/ES = α 3 ). We know that the<br />
Keldysh field is the field to create the potential energy <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Rydberg energy WB over the Bohr radius aB. On the other<br />
hand, the Schw<strong>in</strong>ger field is to create the potential energy <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
2mc 2 = α −2 WB over the distance <str<strong>on</strong>g>of</str<strong>on</strong>g> the Compt<strong>on</strong> length<br />
αaB. See Fig. 2.<br />
Figure 2: The explorati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum may be learned from<br />
that <str<strong>on</strong>g>of</str<strong>on</strong>g> an atom. While the size <str<strong>on</strong>g>of</str<strong>on</strong>g> the atom is <str<strong>on</strong>g>of</str<strong>on</strong>g> aB (the<br />
Bohr radius) and the potential depth is <str<strong>on</strong>g>of</str<strong>on</strong>g> the Rydberg energy<br />
WB, the size and the potential depth that we wish to<br />
explore <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum is <strong>in</strong>dicated here <strong>in</strong> this figure, <strong>on</strong>ly by<br />
a factor <str<strong>on</strong>g>of</str<strong>on</strong>g> the f<strong>in</strong>e structure c<strong>on</strong>stant α to its some power.<br />
We also know that <strong>in</strong> atomic physics, there is the Keldysh<br />
parameter [36] γK whereas the equivalent <strong>in</strong> vacuum is<br />
γV σ = mσωc/eE = 1/a0.(σ = e, or q, electr<strong>on</strong> or quark)<br />
(5)<br />
When the Keldysh parameter is smaller than unity, the<br />
process <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong>izati<strong>on</strong> is n<strong>on</strong>-perturbative;, while greater<br />
than 1, it is multi-phot<strong>on</strong> process [36]. Similarly when<br />
γV σ <strong>in</strong> vacuum is smaller than 1, the vacuum breakdown<br />
is n<strong>on</strong>-pertubative QED, while greater than 1, it is perturbative<br />
[32, 33, 34]. When we <strong>in</strong>ject an XUV phot<strong>on</strong><br />
to an atom to i<strong>on</strong>ize it and we apply a sufficiently <strong>in</strong>tense<br />
CEP-locked laser to accelerate the ejected electr<strong>on</strong>,<br />
we can make attosec<strong>on</strong>d resoluti<strong>on</strong> streak<strong>in</strong>g by the time<str<strong>on</strong>g>of</str<strong>on</strong>g>-flight<br />
detecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> [19, 37]. An equivalent<br />
process <strong>in</strong> vacuum is the above menti<strong>on</strong>ed Nikishov/Ritus<br />
process [32], <strong>in</strong> which a gamma phot<strong>on</strong> is <strong>in</strong>jected <strong>in</strong>to vacuum<br />
while a sufficiently str<strong>on</strong>g enough laser (or XUV) EM<br />
fields are applied. Nikishov et al.[32, 33, 34] showed that<br />
the Schw<strong>in</strong>ger vacuum breakdown is many orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude<br />
reduced.<br />
If we take advantage <str<strong>on</strong>g>of</str<strong>on</strong>g> this process, it is c<strong>on</strong>ceivable<br />
to accomplish electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong> from vacuum<br />
with a very str<strong>on</strong>g laser field, but still <strong>on</strong>e that is <strong>in</strong> a realistically<br />
achievable <strong>in</strong>tensity regime. With the adopti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> this process, we see the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> streak<strong>in</strong>g vacuum<br />
with laser (or XUV coherent phot<strong>on</strong>s) with zeptosec<strong>on</strong>d<br />
time resoluti<strong>on</strong>. This should open up a possibility to start<br />
the explorati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the time-dependent dynamics (<strong>in</strong> c<strong>on</strong>trast<br />
to the spectroscopy) <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum <strong>in</strong> that regime. If we<br />
become more ambitious, perhaps with higher <strong>in</strong>tensity and<br />
shorter time-scale, we might even be able to resolve the<br />
dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> quarks and glu<strong>on</strong>s out <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum.<br />
PROBING THE TEXTURE OF QUANTUM<br />
VACUUM WITH ATTOSECOND<br />
METROLOGY IN THE PEV ENERGY<br />
FRONTIER<br />
As discussed above, we can take advantage <str<strong>on</strong>g>of</str<strong>on</strong>g> the process<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum breakdown with the assistance <str<strong>on</strong>g>of</str<strong>on</strong>g> a highenergy<br />
phot<strong>on</strong> under str<strong>on</strong>g coherent fields, as <strong>in</strong>vestigated<br />
by Nikishov and Ritus some 40 years ago [32]. This is<br />
to use the process that an ultrahigh energy gamma-particle<br />
can assist to break down the vacuum with a substantially<br />
suppressed electric-field threshold compared with the wellknown<br />
Schw<strong>in</strong>ger value. This is the n<strong>on</strong>l<strong>in</strong>ear QED effect.<br />
The probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum breakdown is derived as<br />
[<br />
P (E) ∝ exp − 8<br />
3<br />
ES<br />
E<br />
· mc2<br />
ω<br />
where ES the Schw<strong>in</strong>ger field, ω is the gamma energy,<br />
E is the applied electric field <strong>in</strong> vacuum such as a laser.<br />
With a PeV gamma-ray particle, the exp<strong>on</strong>ent factor <str<strong>on</strong>g>of</str<strong>on</strong>g> (6)<br />
is reduced by the ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> MeV to PeV (mc 2 /ω) over<br />
the expressi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Schw<strong>in</strong>ger’s without the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
]<br />
(6)
Figure 3: The attosec<strong>on</strong>d metrology to detect the arrival<br />
time <str<strong>on</strong>g>of</str<strong>on</strong>g> gamma phot<strong>on</strong>s with<strong>in</strong> such a time scale is suggested.<br />
It utilizes the property <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum explored by the<br />
pi<strong>on</strong>eer<strong>in</strong>g works by Schw<strong>in</strong>ger, Nikishov et al. This approach<br />
closely parallels with that <str<strong>on</strong>g>of</str<strong>on</strong>g> the atom streak<strong>in</strong>g with<br />
a CEP laser with an XUV phot<strong>on</strong> i<strong>on</strong>izati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong><br />
from the atom. Here <strong>in</strong>stead <str<strong>on</strong>g>of</str<strong>on</strong>g> an atom, we simply use the<br />
vacuum electr<strong>on</strong>-positr<strong>on</strong> pair.<br />
gamma particle. This means that the vacuum breakdown<br />
field plummets from the value <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 16 V/cm to 10 10 V/cm.<br />
We suggest that by employ<strong>in</strong>g time-synchr<strong>on</strong>ized somewhat<br />
<strong>in</strong>tense laser field (at 10 10 W/cm 2 ) at the “goal l<strong>in</strong>e”<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the gamma-phot<strong>on</strong> arrival (see Fig. 3), we cause sudden<br />
breakdown <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum and its avalanched particles <str<strong>on</strong>g>of</str<strong>on</strong>g> e − e +<br />
as so<strong>on</strong> as <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the high- energy gamma particles arrives.<br />
The PeV gamma particle triggers the vacuum breakdown.<br />
The time scale <str<strong>on</strong>g>of</str<strong>on</strong>g> breakdown is far faster than fs. The<br />
exploitati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this phenomen<strong>on</strong> should allow an ultrafast<br />
signal <str<strong>on</strong>g>of</str<strong>on</strong>g> the PeV gamma-phot<strong>on</strong> arrival. S<strong>in</strong>ce the trigger<br />
phenomen<strong>on</strong> is exp<strong>on</strong>entially sensitive, we could play<br />
a game <str<strong>on</strong>g>of</str<strong>on</strong>g> adjust<strong>in</strong>g the value <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field to see and<br />
differentiate different types <str<strong>on</strong>g>of</str<strong>on</strong>g> trigger phenomenology and<br />
parameters, depend<strong>in</strong>g up<strong>on</strong> the gamma particle energies.<br />
The creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> PeV energy gamma particles is a tremendous<br />
challenge. For example, Pr<str<strong>on</strong>g>of</str<strong>on</strong>g>. Suzuki [38] has challenged<br />
us if this may be accomplished with<strong>in</strong> our life time.<br />
Tajima et al. (2011) has ventured and showed to use the<br />
Laser Wakefield Accelerati<strong>on</strong> (LWFA) based <strong>on</strong> a MJ class<br />
laser to reach PeV energy electr<strong>on</strong>s. For the prob<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
quantum vacuum texture such as Ellis has been spearhead<strong>in</strong>g<br />
can be d<strong>on</strong>e with a small number <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s generated<br />
gamma particles. Thus we are relieved from the tough lum<strong>in</strong>osity<br />
requirements. However, we need to test a precise<br />
arrival <str<strong>on</strong>g>of</str<strong>on</strong>g> gamma phot<strong>on</strong>s at the ’goal l<strong>in</strong>e’ [39]. Here we<br />
suggested the exploitati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger-Nikishov process<br />
to see this time resoluti<strong>on</strong>.<br />
CONCLUSIONS<br />
In c<strong>on</strong>clusi<strong>on</strong>, evidences over more than 18 orders <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the Pulse Intensity-Durati<strong>on</strong> C<strong>on</strong>jecture has<br />
been accumulated experimentally and through simulati<strong>on</strong>.<br />
It shows that the pulse durati<strong>on</strong> goes <strong>in</strong>versely with the<br />
<strong>in</strong>tensity from the millisec<strong>on</strong>d to the attosec<strong>on</strong>d and zeptosec<strong>on</strong>d<br />
regimes. Most notably it predicts that the short-<br />
est coherent pulse <strong>in</strong> the zeptosec<strong>on</strong>d-yoctosec<strong>on</strong>d regime<br />
should be produced by the largest laser, like ELI or NIF<br />
and the Megajoule, if they are rec<strong>on</strong>figurated [40] <strong>in</strong>to femtosec<strong>on</strong>d<br />
pulse systems.<br />
This C<strong>on</strong>jecture may prove to be an <strong>in</strong>valuable guide<br />
for future ultra-<strong>in</strong>tense and ultrashort pulse experiments. It<br />
fosters the hope that zeptosec<strong>on</strong>d and perhaps yocto sec<strong>on</strong>d<br />
pulses could be produced us<strong>in</strong>g kJ-MJ systems. It opens up<br />
the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> tak<strong>in</strong>g snap shots <str<strong>on</strong>g>of</str<strong>on</strong>g> nuclear reacti<strong>on</strong>s and<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> peek<strong>in</strong>g <strong>in</strong>to the nuclear <strong>in</strong>terior <strong>in</strong> the same way that<br />
Zewail [41] exam<strong>in</strong>ed chemical reacti<strong>on</strong>s or Corkum and<br />
Krausz [42] probed atoms. The other excit<strong>in</strong>g prospect is<br />
the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> study<strong>in</strong>g the n<strong>on</strong>l<strong>in</strong>ear optical properties<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum. This C<strong>on</strong>jecture <strong>in</strong> short implies:<br />
1. that the shortest coherent pulse will be produced by<br />
the largest laser, like the ELI or NIF and The Megajoule<br />
pump<strong>in</strong>g a femtosec<strong>on</strong>d Ti: Sapphire or an<br />
OPCPA system.<br />
2. that pulses <strong>in</strong> the zeptosec<strong>on</strong>d-yoctosec<strong>on</strong>d may be<br />
produced.<br />
3. that even with modest efficiency, extremely high<br />
power rival<strong>in</strong>g the critical power <strong>in</strong> vacuum may be<br />
produced. It opens up the regime <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum n<strong>on</strong>l<strong>in</strong>earity.<br />
It ties the three dist<strong>in</strong>ct discipl<strong>in</strong>es <str<strong>on</strong>g>of</str<strong>on</strong>g> science, i.e. ultrafast<br />
science, high- field science, and large-energy laser science<br />
together with a s<strong>in</strong>gle stroke.<br />
We have also discussed an example <str<strong>on</strong>g>of</str<strong>on</strong>g> the PeV energy<br />
fr<strong>on</strong>tier us<strong>in</strong>g the largest energy laser (e.g. NIF) relax<strong>in</strong>g<br />
the requirement <str<strong>on</strong>g>of</str<strong>on</strong>g> lum<strong>in</strong>osity. In this example, LWFA enables<br />
us to reach PeV <strong>in</strong> a manageable, albeit still large<br />
scale, experimental realizati<strong>on</strong> [39]. Here aga<strong>in</strong> we are utiliz<strong>in</strong>g<br />
the str<strong>on</strong>g- field vacuum n<strong>on</strong>l<strong>in</strong>earity to help achieve<br />
ultrafast metrology.<br />
ACKNOWLEDGMENTS<br />
We would like to acknowledge the fruitful discussi<strong>on</strong>s<br />
with S. Bulanov, E. Goulielmakis, T. Esirkepov, M. Kando,<br />
F. Krausz, A. Suzuki, J. Nees, N. Naumova, E. Moses, S.<br />
Iso, K. Itakura and N. Artemiev. T. Tajima was supported<br />
<strong>in</strong> part by the Blaise Pascal Foundati<strong>on</strong> and by DFG Cluster<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Excellence MAP (Munich Centre for Advanced Phot<strong>on</strong>ics).<br />
REFERENCES<br />
[1] http://www.extreme-light-<strong>in</strong>frastructure.eu<br />
[2] G. A. Mourou, T. Tajima, S. V. Bulanov, Optics <strong>in</strong> the relativistic<br />
regime. Rev. Mod. Phys. 78, 309-371 (2006).<br />
[3] T. H. Maiman, Stimulated Optical Radiati<strong>on</strong> <strong>in</strong> Ruby. Nature<br />
187, 493-494 (1960).<br />
[4] R. W. Hellwarth, Advances <strong>in</strong> Quantum Electr<strong>on</strong>ics<br />
(Columbia University Press, New York, 1961).
The Heisenberg-Schw<strong>in</strong>ger effect: N<strong>on</strong>perturbative Vacuum Pair Producti<strong>on</strong><br />
Abstract<br />
Gerald V. Dunne<br />
Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, University <str<strong>on</strong>g>of</str<strong>on</strong>g> C<strong>on</strong>necticut, Storrs, CT 06269, USA<br />
The Heisenberg-Schw<strong>in</strong>ger effect is the n<strong>on</strong>-perturbative<br />
producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>-positr<strong>on</strong> pairs when an external<br />
electric field is applied to the quantum electrodynamical<br />
(QED) vacuum. The <strong>in</strong>herent <strong>in</strong>stability <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum<br />
<strong>in</strong> an electric field was <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the first n<strong>on</strong>-trivial predicti<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> QED, but the effect is so weak that it has not<br />
yet been directly observed. However, there are excit<strong>in</strong>g<br />
new developments <strong>in</strong> ultra-high <strong>in</strong>tensity lasers, which may<br />
so<strong>on</strong> br<strong>in</strong>g us to the verge <str<strong>on</strong>g>of</str<strong>on</strong>g> this extreme ultra-relativistic<br />
regime. This necessitates a fresh look at both experimental<br />
and theoretical aspects <str<strong>on</strong>g>of</str<strong>on</strong>g> the Heisenberg-Schw<strong>in</strong>ger effect.<br />
I describe some new theoretical ideas aimed at mak<strong>in</strong>g this<br />
elusive effect observable, by careful shap<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser<br />
pulses.<br />
INTRODUCTION<br />
The experimental observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Heisenberg-<br />
Schw<strong>in</strong>ger effect, the n<strong>on</strong>-perturbative producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
electr<strong>on</strong>-positr<strong>on</strong> pairs from vacuum subjected to an electric<br />
field, would open a new w<strong>in</strong>dow <strong>in</strong>to the largely unexplored<br />
regime <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>perturbative quantum field theory, a<br />
regime <strong>in</strong> which we can study matter <strong>in</strong> extreme envir<strong>on</strong>ments<br />
<strong>in</strong> a c<strong>on</strong>trollable way. This has significant implicati<strong>on</strong>s<br />
bey<strong>on</strong>d QED, for example <strong>in</strong> particle physics, plasma<br />
physics and gravitati<strong>on</strong>al physics, as is discussed <strong>in</strong> various<br />
talks at this c<strong>on</strong>ference. It is well known that quantum<br />
vacuum fluctuati<strong>on</strong>s mean that the QED vacuum behaves<br />
like a polarizable medium that modifies classical behavior,<br />
lead<strong>in</strong>g to novel quantum effects [1, 2, 3, 4, 5, 6, 7, 10, 11,<br />
12, 8, 9]. Some <str<strong>on</strong>g>of</str<strong>on</strong>g> these effects, such as the Casimir effect<br />
or the vacuum birefr<strong>in</strong>gence effect, are perturbative and<br />
can be well described by perturbative quantum field theory.<br />
The Heisenberg-Schw<strong>in</strong>ger effect is a n<strong>on</strong>-perturbative effect<br />
that cannot be described by any s<strong>in</strong>gle Feynman diagram;<br />
its essence is a truly n<strong>on</strong>-perturbative process, which<br />
makes it both fasc<strong>in</strong>at<strong>in</strong>g and difficult. The process can be<br />
viewed pictorially as <strong>in</strong> Figure 1: a virtual electr<strong>on</strong>-positr<strong>on</strong><br />
pair <strong>in</strong> vacuum is accelerated apart by an external electric<br />
field, becom<strong>in</strong>g a real asymptotic e + e − pair if they ga<strong>in</strong><br />
the b<strong>in</strong>d<strong>in</strong>g energy <str<strong>on</strong>g>of</str<strong>on</strong>g> 2mc 2 from the external field. This<br />
sets the basic scale at which we might expect this process<br />
to become significant: when the work d<strong>on</strong>e separat<strong>in</strong>g the<br />
pair by a Compt<strong>on</strong> wavelength matches 2mc 2 :<br />
Ec = m2 c 3<br />
e ¯h ≈ 1016 V/cm<br />
Figure 1: Pair producti<strong>on</strong> as the separati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a virtual vacuum<br />
dipole pair under the <strong>in</strong>fluence <str<strong>on</strong>g>of</str<strong>on</strong>g> an external electric<br />
field.<br />
Ic = c<br />
8π E 2 c ≈ 4 × 10 29 W/cm 2<br />
Corresp<strong>on</strong>d<strong>in</strong>gly, the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> pair producti<strong>on</strong> is exp<strong>on</strong>entially<br />
suppressed by the Heisenberg-Euler factor<br />
[<br />
PHE ∼ exp − π m2 c3 ]<br />
, (2)<br />
e E ¯h<br />
An analogous estimate for atomic i<strong>on</strong>izati<strong>on</strong> [e.g., for H],<br />
aga<strong>in</strong> us<strong>in</strong>g the approximati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a c<strong>on</strong>stant electric field,<br />
leads to<br />
P hydrogen [<br />
∼ exp − 2<br />
3<br />
m 2 e 5<br />
E ¯h 4<br />
]<br />
(1)<br />
, (3)<br />
sett<strong>in</strong>g the basic scale <str<strong>on</strong>g>of</str<strong>on</strong>g> field strength and <strong>in</strong>tensity near<br />
which we expect to observe n<strong>on</strong>perturbative i<strong>on</strong>izati<strong>on</strong> effects<br />
<strong>in</strong> atomic systems:<br />
E i<strong>on</strong>izati<strong>on</strong><br />
c = m2 e 5<br />
¯h 4<br />
= α3 Ec ≈ 4 × 10 9 V/cm<br />
I i<strong>on</strong>izati<strong>on</strong><br />
c = α 6 Ic ≈ 6 × 10 16 W/cm 2<br />
Indeed, this is close to the scale <str<strong>on</strong>g>of</str<strong>on</strong>g> atomic i<strong>on</strong>izati<strong>on</strong> experiments,<br />
but <strong>in</strong> fact <strong>in</strong>tensities three orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude<br />
lower are rout<strong>in</strong>ely used. This is because the electric<br />
field <strong>in</strong> a laser is not c<strong>on</strong>stant, and careful shap<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the laser pulses makes i<strong>on</strong>izati<strong>on</strong> experiments possible at<br />
much lower <strong>in</strong>tensities. This simple observati<strong>on</strong>, together<br />
with the fact [see, for example, the talks by G. Mourou<br />
and T. Tajima at this c<strong>on</strong>ference] that there are plans at<br />
large laser facilities such as ELI [13], HiPER at Rutherford<br />
Laboratory, the NIF at Livermore, and the XFEL projects<br />
at SLAC and DESY, to approach the 10 25 − 10 26 W/cm 2<br />
<strong>in</strong>tensity regime, motivates us to ask: how critical is the<br />
Schw<strong>in</strong>ger critical field (1)? To answer this questi<strong>on</strong>, we<br />
need to review briefly some QFT formalism.<br />
(4)
THE QED EFFECTIVE ACTION<br />
In quantum field theory, the quantum correcti<strong>on</strong>s to classical<br />
Maxwell electrodynamics are encoded <strong>in</strong> the ”effective<br />
acti<strong>on</strong>” Γ[A] [14, 15]. For example, the polarizati<strong>on</strong><br />
tensor Πµν = δ2Γ c<strong>on</strong>ta<strong>in</strong>s the electric permittivity ϵij<br />
δAµδAν<br />
and the magnetic permeability µij <str<strong>on</strong>g>of</str<strong>on</strong>g> the quantum vacuum,<br />
and is obta<strong>in</strong>ed by vary<strong>in</strong>g the effective acti<strong>on</strong> Γ[A] with respect<br />
to the external probe Aµ(x). Γ[A] is def<strong>in</strong>ed <strong>in</strong> terms<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum-vacuum persistence amplitude<br />
[ ]<br />
i<br />
⟨0out | 0<strong>in</strong>⟩ = exp {Re(Γ) + i Im(Γ)} (5)<br />
¯h<br />
Re(Γ[A]) describes dispersive effects, such as vacuum birefr<strong>in</strong>gence,<br />
while Im(Γ[A]) describes absorptive effects,<br />
such as vacuum pair producti<strong>on</strong>. The imag<strong>in</strong>ary part encodes<br />
the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum par producti<strong>on</strong> as [14]<br />
Pproducti<strong>on</strong> = 1 − |⟨0out | 0<strong>in</strong>⟩| 2<br />
[<br />
= 1 − exp − 2<br />
]<br />
Im Γ<br />
¯h<br />
≈ 2<br />
Im Γ (6)<br />
¯h<br />
From a computati<strong>on</strong>al perspective, the effective acti<strong>on</strong> is<br />
def<strong>in</strong>ed as [14, 15]<br />
Γ[A] = ¯h ln det [iD/ − m] = ¯h tr ln [iD/ − m] . (7)<br />
Here, D/ ≡ γ µ Dµ, where the covariant derivative operator,<br />
Dµ = ∂µ − i e<br />
¯hc Aµ, def<strong>in</strong>es the coupl<strong>in</strong>g between electr<strong>on</strong>s<br />
and the electromagnetic field Aµ. When the gauge field<br />
Aµ is such that the field strength, Fµν = ∂µAν − ∂νAµ, is<br />
c<strong>on</strong>stant, this effective acti<strong>on</strong> was computed exactly [and<br />
n<strong>on</strong>-perturbatively] by Heisenberg and Euler [1]. For ex-<br />
ample, for a c<strong>on</strong>stant electric field E:<br />
Γ HE [E]<br />
Vol4<br />
= −¯h e2 E 2<br />
8π 2<br />
∫ ∞<br />
0<br />
ds m2<br />
e− eE<br />
s2 s<br />
(<br />
cot(s) − 1 s<br />
+<br />
s 3<br />
(8)<br />
The lead<strong>in</strong>g imag<strong>in</strong>ary part comes from the first pole <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
cot(s) functi<strong>on</strong>:<br />
Im Γ HE<br />
Vol4<br />
∼ ¯h e2 E 2 [ ]<br />
π m2<br />
exp −<br />
8π3 e E<br />
THE EFFECTIVE ACTION IN<br />
INHOMOGENEOUS BACKGROUND<br />
FIELDS<br />
It is essential to understand how this c<strong>on</strong>stant field result<br />
(9) is modified for more realistic <strong>in</strong>homogeneous fields,<br />
such as those describ<strong>in</strong>g ultra-short pulse focussed lasers.<br />
This is a difficult task, as standard perturbative effective<br />
field theory techniques do not apply. The first step <strong>in</strong><br />
this directi<strong>on</strong> is motivated by a sem<strong>in</strong>al result <str<strong>on</strong>g>of</str<strong>on</strong>g> Keldysh<br />
[16, 17] for the i<strong>on</strong>izati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> atoms <strong>in</strong> a m<strong>on</strong>ochromatic<br />
time dependent electric field E(t) = E cos(ωt). This <strong>in</strong>troduces<br />
a new scale to the problem, and Keldysh was able<br />
to compute the i<strong>on</strong>izati<strong>on</strong> probability as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
)<br />
(9)<br />
dimensi<strong>on</strong>less adiabaticity parameter γK, that characterized<br />
the fast [γK ≫ 1] and slow [γK ≪ 1] regimes.<br />
[The Keldysh parameter is related to the standard laser<br />
field strength parameter a0 as a0 = 1/γK.] Remarkably,<br />
Keldysh’s WKB result <strong>in</strong>terpolates smoothly between the<br />
n<strong>on</strong>-perturbative tunnel-i<strong>on</strong>izati<strong>on</strong> regime where γK ≪ 1,<br />
and the perturbative multi-phot<strong>on</strong> regime where γK ≫<br />
1. This formalism was generalized to the Heisenberg-<br />
Schw<strong>in</strong>ger effect <strong>in</strong> QED [18, 19, 20], with an analogous<br />
”adiabaticity parameter”<br />
Ppair prod. ∼<br />
γK ≡ mcω<br />
. (10)<br />
eE<br />
⎧ [<br />
⎨ exp −<br />
⎩<br />
πm2c 3<br />
]<br />
eE¯h , γK ≪ 1<br />
) 2 (11)<br />
2mc /¯hω<br />
, γK ≫ 1<br />
( eE<br />
mω<br />
The γK ≪ 1 regime corresp<strong>on</strong>ds to n<strong>on</strong>perturbative tunnel<strong>in</strong>g,<br />
while γK ≫ 1 is the perturbative multiphot<strong>on</strong><br />
regime. In the perturbative multi-phot<strong>on</strong> regime, this QED<br />
pair producti<strong>on</strong> effect has been observed <strong>in</strong> a beautiful experiment<br />
(E-144) at SLAC [21], <strong>in</strong> which a laser pulse collided<br />
with the (highly relativistic) SLAC electr<strong>on</strong> beam,<br />
lead<strong>in</strong>g to n<strong>on</strong>l<strong>in</strong>ear Compt<strong>on</strong> scatter<strong>in</strong>g <strong>in</strong>volv<strong>in</strong>g 4-5 phot<strong>on</strong>s,<br />
produc<strong>in</strong>g a high energy gamma phot<strong>on</strong> that decays<br />
<strong>in</strong>to an electr<strong>on</strong>-positr<strong>on</strong> pair. By c<strong>on</strong>trast, it is hoped that<br />
<strong>in</strong> future laser facilities it will be possible to probe deep<br />
<strong>in</strong>to the n<strong>on</strong>perturbative regime where γK ≪ 1, to see the<br />
truly n<strong>on</strong>perturbative Heisenberg-Schw<strong>in</strong>ger effect <str<strong>on</strong>g>of</str<strong>on</strong>g> pair<br />
producti<strong>on</strong> directly from vacuum.<br />
The Keldysh approach captures an enormous amount<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> important physical <strong>in</strong>formati<strong>on</strong>. Various methods have<br />
been developed which can be used to compute the pair producti<strong>on</strong><br />
probability when the background electric field depends<br />
<strong>on</strong> just <strong>on</strong>e coord<strong>in</strong>ate. The problem can be understood<br />
as a <strong>on</strong>e-dimensi<strong>on</strong>al scatter<strong>in</strong>g problem, based <strong>on</strong><br />
Feynman’s <strong>in</strong>terpretati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> positr<strong>on</strong>s as electr<strong>on</strong>s propagat<strong>in</strong>g<br />
backwards <strong>in</strong> time [22]. Then the probability<br />
can be extracted from the reflecti<strong>on</strong> coefficient for<br />
a ”Schröd<strong>in</strong>ger” problem <str<strong>on</strong>g>of</str<strong>on</strong>g> scatter<strong>in</strong>g <strong>in</strong> the time doma<strong>in</strong>.<br />
The reflecti<strong>on</strong> probability can be computed exactly<br />
[numerically], as <strong>in</strong> the quantum k<strong>in</strong>etic approach<br />
[23, 24, 25, 26], or estimated us<strong>in</strong>g semiclassical WKB<br />
arguments [18, 19, 20, 27]. A natural ”<strong>in</strong>verse questi<strong>on</strong>”<br />
arises: can we shape the laser pulses <strong>in</strong> order to enhance<br />
the pair producti<strong>on</strong> effect, or to make it more dist<strong>in</strong>ctive?<br />
PULSE SHAPING EFFECTS FOR<br />
TIME-DEPENDENT FIELDS<br />
C<strong>on</strong>t<strong>in</strong>u<strong>in</strong>g the approximati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>sider<strong>in</strong>g a timedependent<br />
electric field, several recent results suggest that<br />
the peak laser <strong>in</strong>tensity at which appreciable vacuum pair<br />
producti<strong>on</strong> could be observe is <strong>in</strong> the 10 25 − 10 26 W/cm 2<br />
<strong>in</strong>tensity range, which is significant s<strong>in</strong>ce this is the targeted<br />
goal <str<strong>on</strong>g>of</str<strong>on</strong>g> the ELI project, and with<strong>in</strong> range <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
HiPER facility. An important set <str<strong>on</strong>g>of</str<strong>on</strong>g> ideas [28, 29] is to<br />
comb<strong>in</strong>e multiple pulses, each <str<strong>on</strong>g>of</str<strong>on</strong>g> a lower <strong>in</strong>tensity, and to
take <strong>in</strong>to account the fact that the spatial focuss<strong>in</strong>g regi<strong>on</strong> is<br />
much larger than the scale <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> Compt<strong>on</strong> wavelength.<br />
Quantitative analyses then suggest a critical <strong>in</strong>tensity<br />
3 or 4 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude below the Schw<strong>in</strong>ger limit<br />
(1). It has also been shown recently [30] that plane wave<br />
fields <str<strong>on</strong>g>of</str<strong>on</strong>g> f<strong>in</strong>ite extent, and therefore hav<strong>in</strong>g n<strong>on</strong>trivial shape<br />
dependence, produce <strong>in</strong>terest<strong>in</strong>g and novel effects for vacuum<br />
pair producti<strong>on</strong>.<br />
Dynamically Assisted Schw<strong>in</strong>ger Mechanism<br />
Another related but dist<strong>in</strong>ct idea is the ”dynamically assisted<br />
Schw<strong>in</strong>ger mechanism” [31], <strong>in</strong> which a simple superpositi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> two time-dependent pulses, <strong>on</strong>e str<strong>on</strong>g but<br />
slow, and the other weak but fast, can lead to a significant<br />
enhancement <str<strong>on</strong>g>of</str<strong>on</strong>g> the tunnel<strong>in</strong>g process associated with the<br />
Heisenberg-Schw<strong>in</strong>ger effect. This ”dynamically assisted<br />
Schw<strong>in</strong>ger mechanism” was <strong>in</strong>troduced <strong>in</strong> [31] with fields:<br />
Eslow(t) = E sech 2 (Ωt) ; Efast(t) = ϵ sech 2 (ωt) (12)<br />
with parametric hierarchies: 0 < ϵ ≪ E ≪ Ec, 0 <<br />
Ω ≪ ω ≪ m. Surpris<strong>in</strong>gly, even though both field amplitudes,<br />
E and ϵ, are below the critical field Ec <strong>in</strong> (1), there is<br />
significant enhancement <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair producti<strong>on</strong> rate when<br />
the frequencies follow this hierarchy <str<strong>on</strong>g>of</str<strong>on</strong>g> scales. The n<strong>on</strong>perturbative<br />
pair producti<strong>on</strong> process that we would associate<br />
with the slow str<strong>on</strong>g field <strong>in</strong>teracts with the perturbative<br />
multiphot<strong>on</strong> pair producti<strong>on</strong> process that we would<br />
associate with the fast weaker field to produce a str<strong>on</strong>ger<br />
impact than each process separately. A specific realizati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> this idea was proposed recently [32], <strong>in</strong>volv<strong>in</strong>g a<br />
str<strong>on</strong>g, slow optical laser pulse and a weak, fast X-ray<br />
pulse. Particles <strong>in</strong> the Dirac sea can perturbatively absorb<br />
a high-frequency phot<strong>on</strong> from the weak, fast field,<br />
thereby lower<strong>in</strong>g the effective tunnel barrier for the n<strong>on</strong>perturbative<br />
process. This leads to predicti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> observable<br />
pair-producti<strong>on</strong> yields based <strong>on</strong> current technology.<br />
This dynamically assisted Schw<strong>in</strong>ger mechanism is<br />
closely related to a catalysis mechanism [33], <strong>in</strong> which <strong>on</strong>e<br />
can view the problem as the propagati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
X-ray probe pulse <strong>in</strong> an <strong>in</strong>tense and effectively c<strong>on</strong>stant<br />
electric field provided by the str<strong>on</strong>ger and slower optical<br />
laser pulse. There is a n<strong>on</strong>-zero absorpti<strong>on</strong> coefficient for<br />
phot<strong>on</strong> propagati<strong>on</strong> <strong>in</strong> such a str<strong>on</strong>g field, and from this <strong>on</strong>e<br />
can deduce the rate <str<strong>on</strong>g>of</str<strong>on</strong>g> pair producti<strong>on</strong>. Technically, this requires<br />
the computati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the imag<strong>in</strong>ary part <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong><br />
polarizati<strong>on</strong> tensor <strong>in</strong> an electric field. As the X-ray frequency<br />
approaches the threshold <str<strong>on</strong>g>of</str<strong>on</strong>g> 2m there is a dramatic<br />
exp<strong>on</strong>ential enhancement<br />
{<br />
α<br />
Im(Π) ≈ √ eE exp −<br />
π(π − 2) m2<br />
}<br />
(π − 2) (13)<br />
eE<br />
This exp<strong>on</strong>ential enhancement leads to a w<strong>in</strong>dow <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
opportunity [33] <strong>in</strong> the range <str<strong>on</strong>g>of</str<strong>on</strong>g> laser <strong>in</strong>tensity up to<br />
I ≈ 9 × 10 25 W/cm 2 <strong>in</strong> which this catalyzed Schw<strong>in</strong>ger<br />
mechanism is dramatically enhanced relative to the pure<br />
Heisenberg-Schw<strong>in</strong>ger effect with just the str<strong>on</strong>g optical<br />
laser pulse, without the catalyz<strong>in</strong>g X-ray pulse. This catalysis<br />
mechanism can also be viewed as phot<strong>on</strong>-stimulated<br />
pair-producti<strong>on</strong> [34], realiz<strong>in</strong>g the more general mechanism<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> an <strong>in</strong>duced metastable decay process [35].<br />
Interference effects from sub-cycle temporal<br />
pulse structure<br />
Recently it has become clear that the WKB analysis <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
[18, 19, 20] must be extended to <strong>in</strong>corporate <strong>in</strong>terference<br />
effects when the temporal pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse has subcycle<br />
structure. This <strong>in</strong>terference phenomen<strong>on</strong> is extremely<br />
sensitive to the temporal pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse; for example,<br />
to the carrier phase <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse, and also to the<br />
presence <str<strong>on</strong>g>of</str<strong>on</strong>g> “chirp” <strong>in</strong> the pulse [see below]. The technical<br />
reas<strong>on</strong> is that for such fields, the over-the-barrier scatter<strong>in</strong>g<br />
problem typically has multiple semiclassical saddle po<strong>in</strong>ts<br />
[i.e., sets <str<strong>on</strong>g>of</str<strong>on</strong>g> turn<strong>in</strong>g po<strong>in</strong>ts], and the <strong>in</strong>terference between<br />
different saddle po<strong>in</strong>ts leads directly to oscillatory res<strong>on</strong>ance<br />
behavior <strong>in</strong> the l<strong>on</strong>gitud<strong>in</strong>al momentum spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the produced particles. Such phenomena are familiar from<br />
str<strong>on</strong>g-field atomic and molecular physics, discussed l<strong>on</strong>g<br />
ago <strong>in</strong> the theory <str<strong>on</strong>g>of</str<strong>on</strong>g> atomic i<strong>on</strong>izati<strong>on</strong> [36, 37, 17], and observed<br />
experimentally <strong>in</strong> photoi<strong>on</strong>izati<strong>on</strong> spectra [38, 39].<br />
In the c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g> QED vacuum pair producti<strong>on</strong>, these <strong>in</strong>terferences<br />
effects were first studied numerically <strong>in</strong> [40],<br />
and given a simple quantitative semiclassical explanati<strong>on</strong><br />
<strong>in</strong> [41] <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the Stokes phenomen<strong>on</strong>: the <strong>in</strong>terference<br />
produced by the patch<strong>in</strong>g together <str<strong>on</strong>g>of</str<strong>on</strong>g> semiclassical approximati<strong>on</strong>s<br />
<strong>in</strong> different regi<strong>on</strong>s. Specifically, for an electric<br />
field <str<strong>on</strong>g>of</str<strong>on</strong>g> the form<br />
E(t) = E0 cos(ωt + ϕ) exp<br />
(<br />
− t2<br />
2τ 2<br />
)<br />
(14)<br />
<strong>on</strong>e f<strong>in</strong>ds oscillatory behavior <str<strong>on</strong>g>of</str<strong>on</strong>g> the l<strong>on</strong>gitud<strong>in</strong>al electr<strong>on</strong>positr<strong>on</strong><br />
momentum spectrum, which becomes pr<strong>on</strong>ounced<br />
when ωτ ∼ 4 [i.e., when the number <str<strong>on</strong>g>of</str<strong>on</strong>g> oscillati<strong>on</strong>s under<br />
the envelope exceeds 4], and <strong>in</strong> particular when the<br />
carrier phase ϕ approaches π/2. Most dramatically, when<br />
ϕ = π/2 there are values <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>-positr<strong>on</strong> momentum<br />
at which the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> producti<strong>on</strong> vanishes. The<br />
analytic explanati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> these numerical results lies <strong>in</strong> the<br />
<strong>in</strong>terference between different semiclassical saddle po<strong>in</strong>ts.<br />
With just <strong>on</strong>e dom<strong>in</strong>ant saddle po<strong>in</strong>t [as occurs for a c<strong>on</strong>stant<br />
electric field E(t) = E0, or for an electric field with a<br />
simple s<strong>in</strong>gle-bump structure like E(t) = E0 sech 2 (ωt), or<br />
E(t) = E0/(1 + (ωt) 2 ) 2 ] we have the familiar probability<br />
expressi<strong>on</strong> with exp<strong>on</strong>ential factor <str<strong>on</strong>g>of</str<strong>on</strong>g> the form e −Sc , where<br />
Sc is the classical (Euclidean) acti<strong>on</strong> [27]. With two saddle<br />
po<strong>in</strong>ts <str<strong>on</strong>g>of</str<strong>on</strong>g> comparable amplitude, the expressi<strong>on</strong> generalizes<br />
to [41]<br />
P ≈ e −S(1)<br />
c + e −S(2)<br />
1<br />
c −<br />
± 2 cos(2α) e 2 S(1)<br />
1<br />
c − 2 S(2)<br />
c (15)<br />
where α is an <strong>in</strong>tegral c<strong>on</strong>nect<strong>in</strong>g different saddle po<strong>in</strong>ts,<br />
and characterizes the <strong>in</strong>terference. The ± <strong>in</strong> (15) refers
Figure 2: Turn<strong>in</strong>g po<strong>in</strong>ts <strong>in</strong> the complex t plane for the ”cos<strong>in</strong>e” [left frame] electric field with carrier phase ϕ = 0,<br />
E(t) = E0 cos(ωt) exp ( −t 2 /(2τ 2 ) ) , and for the ”s<strong>in</strong>e” [right frame] electric field with carrier phase ϕ = π/2, E(t) =<br />
E0 s<strong>in</strong>(ωt) exp ( −t 2 /(2τ 2 ) ) . Turn<strong>in</strong>g po<strong>in</strong>ts closest to the real axis tend to dom<strong>in</strong>ate. In the first case there is <strong>on</strong>e dom<strong>in</strong>ant<br />
saddle po<strong>in</strong>t, while <strong>in</strong> the sec<strong>on</strong>d case there are two, and the <strong>in</strong>terference between them leads to oscillatory behavior <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the momentum spectra for electr<strong>on</strong>s and positr<strong>on</strong>s produced by such a laser pulse [41, 42].<br />
to scalar/sp<strong>in</strong>or QED, reflect<strong>in</strong>g the expected opposite sign<br />
dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>terference effects <strong>on</strong> quantum statistics,<br />
and expla<strong>in</strong><strong>in</strong>g both qualitatively and quantitatively the numerical<br />
observati<strong>on</strong> <strong>in</strong> [43]. Includ<strong>in</strong>g also a chirp parameter<br />
b, by replac<strong>in</strong>g the cos<strong>in</strong>e factor by cos(b t 2 + ωt + ϕ)<br />
<strong>on</strong>e f<strong>in</strong>ds a dramatic effect <strong>on</strong> the form <str<strong>on</strong>g>of</str<strong>on</strong>g> the momentum<br />
spectrum [44]. Besides their theoretical <strong>in</strong>terest, the<br />
acute sensitivity <str<strong>on</strong>g>of</str<strong>on</strong>g> the momentum spectra to the laser<br />
pulse shape may provide dist<strong>in</strong>ctive experimental signatures<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Heisenberg-Schw<strong>in</strong>ger effect, and eventually<br />
could provide sub-cycle pulse resoluti<strong>on</strong> at extremely short<br />
time scales, us<strong>in</strong>g the quantum vacuum fluctuati<strong>on</strong>s. Another<br />
suggesti<strong>on</strong> [40] to use these vacuum <strong>in</strong>terference effects<br />
is an all-optical time-doma<strong>in</strong> analogue <str<strong>on</strong>g>of</str<strong>on</strong>g> the doubleslit<br />
experiment, a detailed experimental proposal for which<br />
is <strong>in</strong> [45].<br />
SPATIAL AND TEMPORAL PULSE<br />
SHAPE EFFECTS<br />
Ideally we would like to be able to compute the imag<strong>in</strong>ary<br />
part <str<strong>on</strong>g>of</str<strong>on</strong>g> the effective acti<strong>on</strong> Γ[A] for gauge fields<br />
Aµ(⃗x, t) that represent the full spatial and temporal structure<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a realistic laser pulse c<strong>on</strong>figurati<strong>on</strong>. This is a n<strong>on</strong>trivial<br />
questi<strong>on</strong>, s<strong>in</strong>ce temporal <strong>in</strong>homogeneities tend to enhance<br />
the rate [because it is easier to tunnel through an<br />
oscillat<strong>in</strong>g barrier], while spatial <strong>in</strong>homogeneities tend to<br />
suppress the rate [because the field falls <str<strong>on</strong>g>of</str<strong>on</strong>g>f as the particles<br />
accelerate away from the nucleati<strong>on</strong> po<strong>in</strong>t]. This<br />
raises an <strong>in</strong>terest<strong>in</strong>g questi<strong>on</strong>: how do these compet<strong>in</strong>g effects<br />
play out <strong>in</strong> an ultra-short laser pulse that is tightly<br />
spatially focussed? This is a technically difficult questi<strong>on</strong><br />
to answer, because the c<strong>on</strong>venti<strong>on</strong>al WKB and QKE approaches<br />
have not yet been generalized to higher dimensi<strong>on</strong>s<br />
<strong>in</strong> any computati<strong>on</strong>ally efficient manner. Nevertheless,<br />
several promis<strong>in</strong>g approaches have recently been developed.<br />
Worldl<strong>in</strong>e <strong>in</strong>stant<strong>on</strong> formalism<br />
A natural multi-dimensi<strong>on</strong>al semiclassical approach is<br />
provided by Feynman’s worldl<strong>in</strong>e formulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the QED<br />
effective acti<strong>on</strong> [46, 47]. Feynman formulated a firstquantized<br />
form <str<strong>on</strong>g>of</str<strong>on</strong>g> QED, which amounts to represent<strong>in</strong>g<br />
the effective acti<strong>on</strong> as a quantum mechanical path <strong>in</strong>tegral<br />
over closed loops xµ(τ) <strong>in</strong> four dimensi<strong>on</strong>al spacetime,<br />
with the closed loops be<strong>in</strong>g parametrized by the proper<br />
time τ. The propertime parametrizati<strong>on</strong> had been developed<br />
earlier by Fock and Nambu [48], and was also used<br />
by Schw<strong>in</strong>ger, <strong>in</strong> operator form rather than <strong>in</strong> path <strong>in</strong>tegral<br />
form, <strong>in</strong> his landmark QED computati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum<br />
pair producti<strong>on</strong> [14]. Feynman’s worldl<strong>in</strong>e path <strong>in</strong>tegral<br />
formalism has s<strong>in</strong>ce been extended significantly, primarily<br />
for applicati<strong>on</strong>s <strong>in</strong> perturbative quantum field theory [47],<br />
build<strong>in</strong>g <strong>on</strong> analogies and motivati<strong>on</strong> from the Polyakov<br />
formulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>in</strong>g theory. This rebirth has led to many<br />
beautiful advances <strong>in</strong> our understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> perturbative<br />
scatter<strong>in</strong>g amplitudes, but here I describe an applicati<strong>on</strong> to<br />
n<strong>on</strong>-perturbative processes. For simplicity, c<strong>on</strong>sider scalar<br />
QED. The effective acti<strong>on</strong> for a scalar charged particle<br />
(charge e, mass m) <strong>in</strong> a Euclidean classical gauge background<br />
Aµ(x) is the functi<strong>on</strong>al:<br />
Γ[A] =<br />
∫ ∞<br />
0<br />
× exp<br />
dT<br />
T e−m2 ∫<br />
T<br />
[<br />
−<br />
∫ T<br />
0<br />
dτ<br />
d 4 x (0)<br />
∫<br />
Dx<br />
x(T )=x(0)=x (0)<br />
(<br />
2 ˙x µ<br />
4 + e Aµ<br />
)]<br />
˙xµ<br />
The ma<strong>in</strong> technical difficulty is to compute the quantum<br />
mechanical path <strong>in</strong>tegral, a sum over closed trajectories<br />
<strong>in</strong> four-dimensi<strong>on</strong>al Euclidean space. One approach is a<br />
direct M<strong>on</strong>te Carlo analysis, as has been d<strong>on</strong>e for <strong>on</strong>edimensi<strong>on</strong>al<br />
<strong>in</strong>homogeneities [49]; this is a powerful approach<br />
s<strong>in</strong>ce it does not rely <strong>on</strong> any particular symmetry <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the background field, although it is computati<strong>on</strong>ally diffi-
cult to extract the exp<strong>on</strong>entially small pair producti<strong>on</strong> rate.<br />
A more analytic approach is to make a semiclassical approximati<strong>on</strong><br />
to the path <strong>in</strong>tegral, by solv<strong>in</strong>g the classical<br />
equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> [we set 2e = 1 to simplify notati<strong>on</strong>]:<br />
¨xµ = Fµν(x) ˙xν , (µ, ν = 1 . . . 4) . (16)<br />
In this semiclassical approximati<strong>on</strong>, the path <strong>in</strong>tegral<br />
is dom<strong>in</strong>ated by a classical loop called a ”worldl<strong>in</strong>e <strong>in</strong>stant<strong>on</strong>”<br />
[a closed-loop soluti<strong>on</strong> to the classical Euclidean<br />
equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong>], together with quantum fluctuati<strong>on</strong>s<br />
about this loop. The dom<strong>in</strong>ant exp<strong>on</strong>ential factor <strong>in</strong> the<br />
imag<strong>in</strong>ary part <str<strong>on</strong>g>of</str<strong>on</strong>g> the effective acti<strong>on</strong> is just<br />
exp (−S[xclassical]) (17)<br />
with the acti<strong>on</strong> evaluated <strong>on</strong> the worldl<strong>in</strong>e <strong>in</strong>stant<strong>on</strong> trajectory.<br />
This idea was first applied to the vacuum pair producti<strong>on</strong><br />
problem for a c<strong>on</strong>stant electric field <strong>in</strong> [50], and<br />
later extended to <strong>in</strong>homogeneous background field c<strong>on</strong>figurati<strong>on</strong>s<br />
[51]. The prefactor c<strong>on</strong>tributi<strong>on</strong>s, which can be<br />
physically significant, are most efficiently computed us<strong>in</strong>g<br />
an analogy [52] to the Gutzwiller trace formula <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>relativistic<br />
quantum mechanics, view<strong>in</strong>g the closed loop as<br />
a closed trajectory <strong>in</strong> phase space. This worldl<strong>in</strong>e <strong>in</strong>stant<strong>on</strong><br />
method is very general; the ma<strong>in</strong> technical challenge<br />
is f<strong>in</strong>d<strong>in</strong>g the closed classical trajectories <strong>in</strong> a given (Euclidean)<br />
background field.<br />
Wigner functi<strong>on</strong> formalism<br />
Given the str<strong>on</strong>g analogies between the problem <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum<br />
pair producti<strong>on</strong> and atomic physics <strong>in</strong> str<strong>on</strong>g laser<br />
fields, and also to various well-known c<strong>on</strong>densed matter<br />
problems such as Landau-Zener tunnel<strong>in</strong>g [53], it is quite<br />
natural to adapt the standard quantum optics approach <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the Wigner functi<strong>on</strong> [54] to the problem <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum pair<br />
producti<strong>on</strong>. This method has been developed recently <strong>in</strong><br />
both the real time and light-c<strong>on</strong>e formalisms [55, 56]. The<br />
essential idea is that the Wigner transform <str<strong>on</strong>g>of</str<strong>on</strong>g> the field twopo<strong>in</strong>t<br />
functi<strong>on</strong> c<strong>on</strong>ta<strong>in</strong>s <strong>in</strong>formati<strong>on</strong> about the pair producti<strong>on</strong><br />
process, and the Wigner functi<strong>on</strong> satisfies an equati<strong>on</strong><br />
that generalizes the quantum k<strong>in</strong>etic equati<strong>on</strong> bey<strong>on</strong>d the<br />
<strong>on</strong>e-dimensi<strong>on</strong>al evoluti<strong>on</strong>. For vacuum pair producti<strong>on</strong>,<br />
c<strong>on</strong>sider the vacuum matrix element <str<strong>on</strong>g>of</str<strong>on</strong>g> the equal time density<br />
[<strong>in</strong>clud<strong>in</strong>g the gauge field parallel transport operator<br />
for gauge <strong>in</strong>variance]<br />
C(⃗x, ⃗s, t) = ⟨0|e −ie<br />
∫ 1/2<br />
⃗A(⃗x+λ⃗s,t)·⃗s dλ<br />
−1/2<br />
×[Ψ(⃗x + ⃗s<br />
2 , t), ¯ Ψ(⃗x − ⃗s<br />
, t)] |0⟩ (18)<br />
2<br />
and its Wigner transform<br />
W(⃗x, ⃗p, t) = − 1<br />
∫<br />
2<br />
d 3 s e −i⃗p·⃗s C(⃗x, ⃗s, t) (19)<br />
This Wigner functi<strong>on</strong> approach maps the problem to phase<br />
space, and leads to coupled equati<strong>on</strong>s for the Wigner functi<strong>on</strong>,<br />
provid<strong>in</strong>g a formalism <strong>in</strong> which various semiclassical<br />
approximati<strong>on</strong>s can be explored. It also opens the possibility<br />
for study<strong>in</strong>g n<strong>on</strong>-equilibrium aspects <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair producti<strong>on</strong><br />
process.<br />
CONCLUSIONS<br />
The observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Heisenberg-Schw<strong>in</strong>ger effect<br />
presents a fundamental challenge both theoretically and experimentally.<br />
Theoretically we need new n<strong>on</strong>-perturbative<br />
techniques to provide efficient and precise calculati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the expected pair producti<strong>on</strong> rate <strong>in</strong> realistic short-pulse focussed<br />
laser fields. The approach must be sufficiently flexible<br />
and powerful to be able to optimize the pulse shape<br />
to maximize the pair producti<strong>on</strong> rate. Other important<br />
theoretical techniques <strong>in</strong>volve <strong>in</strong>tense numerical model<strong>in</strong>g<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> soluti<strong>on</strong>s and trajectories for particles <strong>in</strong> laser fields<br />
[57, 58]. An important recent idea c<strong>on</strong>cerns the possibility<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> QED cascade effects, and the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> an upper<br />
limit <strong>on</strong> an atta<strong>in</strong>able electric field. This is a fasc<strong>in</strong>at<strong>in</strong>g,<br />
difficult and fundamental questi<strong>on</strong> that has not yet<br />
been resolved [58, 59, 60, 61]. On the experimental side,<br />
the ma<strong>in</strong> challenges are to obta<strong>in</strong> higher laser <strong>in</strong>tensity, as<br />
close as possible to the critical field limit (1), and to be<br />
able to focus and shape the laser pulse(s) <strong>in</strong> both the space<br />
and time doma<strong>in</strong>. Recent progress at ELI, HiPER and with<br />
XFEL lasers suggests that we may be very close to enter<strong>in</strong>g<br />
this new physical regime <str<strong>on</strong>g>of</str<strong>on</strong>g> ultra-relativistic physics. Bey<strong>on</strong>d<br />
QED, there are fundamental questi<strong>on</strong>s to be answered<br />
c<strong>on</strong>cern<strong>in</strong>g back-reacti<strong>on</strong> effects, the physics bey<strong>on</strong>d the<br />
Schw<strong>in</strong>ger critical field, and c<strong>on</strong>cern<strong>in</strong>g the possible simulati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> gravitati<strong>on</strong>al effects such as Unruh and Hawk<strong>in</strong>g<br />
radiati<strong>on</strong>, us<strong>in</strong>g the very large electric field accelerati<strong>on</strong> to<br />
mimic str<strong>on</strong>g gravitati<strong>on</strong>al fields. Many <str<strong>on</strong>g>of</str<strong>on</strong>g> these questi<strong>on</strong>s<br />
are directly addressed <strong>in</strong> talks at this c<strong>on</strong>ference.<br />
Acknowledgements: I thank the organizers <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
PIF2010 c<strong>on</strong>ference, especially Satoshi Iso, for organiz<strong>in</strong>g<br />
an excellent meet<strong>in</strong>g. I also thank R. Alk<str<strong>on</strong>g>of</str<strong>on</strong>g>er, C.<br />
Dumlu, H. Gies, F. Hebenstreit, G. Mourou, C. Schubert,<br />
R. Schützhold and T. Tajima for collaborati<strong>on</strong>s and discussi<strong>on</strong>s,<br />
and I acknowledge support from the DOE through<br />
the grant DE-FG02-92ER40716.<br />
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QED IN ULTRA-INTENSE LASER FIELDS ∗<br />
T. He<strong>in</strong>zl † , School <str<strong>on</strong>g>of</str<strong>on</strong>g> Comput<strong>in</strong>g & Mathematics, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Plymouth, UK<br />
C. Harvey, A. Ildert<strong>on</strong> and M. Marklund, Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Ume˚a University, Sweden<br />
Abstract<br />
We present an overview <str<strong>on</strong>g>of</str<strong>on</strong>g> basic QED processes <strong>in</strong> the<br />
presence <str<strong>on</strong>g>of</str<strong>on</strong>g> an ultra-<strong>in</strong>tense laser background.<br />
INTRODUCTION<br />
The year 2010 has seen the 50th anniversary <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser.<br />
S<strong>in</strong>ce its <strong>in</strong>cepti<strong>on</strong> it has underg<strong>on</strong>e a very dynamic development<br />
culm<strong>in</strong>at<strong>in</strong>g <strong>in</strong> a multitude <str<strong>on</strong>g>of</str<strong>on</strong>g> everyday applicati<strong>on</strong>s.<br />
From the physics po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> view specificati<strong>on</strong> parameters<br />
have evolved <strong>in</strong> many directi<strong>on</strong>s, for <strong>in</strong>stance towards<br />
the X-ray regime <str<strong>on</strong>g>of</str<strong>on</strong>g> frequency. For the purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> this c<strong>on</strong>ference<br />
and this talk we are particularly <strong>in</strong>terested <strong>in</strong> ultrahigh<br />
<strong>in</strong>tensities. The historical development <str<strong>on</strong>g>of</str<strong>on</strong>g> these is pictured<br />
<strong>in</strong> Fig. 1 (adapted from [1]) with a notable breakthrough<br />
<strong>in</strong> 1985 due to the implementati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> chirped pulse<br />
amplificati<strong>on</strong> (CPA) [2].<br />
Figure 1: Time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> laser <strong>in</strong>tensity.<br />
The vertical axis <strong>on</strong> the right-hand side measures <strong>in</strong>tensity<br />
I <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the dimensi<strong>on</strong>less laser amplitude<br />
a0 = eEλ<br />
mc 2 ∼ I1/2 , (1)<br />
which is the energy ga<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong> (charge e, mass m)<br />
across a laser wavelength λ <strong>in</strong> the r.m.s. field E, <strong>in</strong> units<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> its rest energy, mc 2 . Hence, when a0 exceeds unity an<br />
∗ Work supported <strong>in</strong> part by ERC, C<strong>on</strong>tract No. 204059-QPQV.<br />
† the<strong>in</strong>zl@plymouth.ac.uk<br />
electr<strong>on</strong> prob<strong>in</strong>g the laser field will beg<strong>in</strong> to move relativistically.<br />
It is worth po<strong>in</strong>t<strong>in</strong>g out that ultra-<strong>in</strong>tense lasers produce<br />
the largest electromagnetic fields that are currently available<br />
<strong>in</strong> the lab. Of course, the downside is that the fields<br />
are pulsed (i.e. “short-lived”) and alternat<strong>in</strong>g. An overview<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the current magnitudes is given <strong>in</strong> Table 1.<br />
Table 1: Some typical current magnitudes.<br />
Quantity Magnitude<br />
Power P 10 15 W ≡ 1 PW<br />
Intensity I 10 15 W/cm 2<br />
Electric Field E 10 14 V/m<br />
Magnetic Field B 10 10 G<br />
Planned facilities where these magnitudes will be <strong>in</strong>creased<br />
further <strong>in</strong>clude the Vulcan 10 PW project at the<br />
Central Laser Facility <str<strong>on</strong>g>of</str<strong>on</strong>g> Rutherford Lab, UK and the European<br />
Extreme Light Infrastructure where up to 100 PW<br />
are envisaged.<br />
STRONG FIELDS: THEORY<br />
We are <strong>in</strong>terested <strong>in</strong> elementary processes occurr<strong>in</strong>g <strong>in</strong><br />
the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> an ultra-<strong>in</strong>tense laser. The appropriate theory<br />
is (a variant <str<strong>on</strong>g>of</str<strong>on</strong>g>) str<strong>on</strong>g-field quantum electrodynamics<br />
(QED) with the laser field be<strong>in</strong>g <strong>in</strong>cluded as an external<br />
background field. The extent to which this theory is under<br />
analytical c<strong>on</strong>trol depends sensitively <strong>on</strong> the model chosen<br />
for the laser beam. The simplest model is an <strong>in</strong>f<strong>in</strong>ite,<br />
m<strong>on</strong>ochromatic plane wave for which transiti<strong>on</strong> amplitudes<br />
can be calculated <strong>in</strong>clud<strong>in</strong>g an analytic evaluati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
appear<strong>in</strong>g oscillatory <strong>in</strong>tegrals [5]. The latter becomes difficult<br />
for pulsed plane waves such that this case presents<br />
more challeng<strong>in</strong>g technical difficulties. While pulsed plane<br />
waves have f<strong>in</strong>ite extent <strong>in</strong> time and l<strong>on</strong>gitud<strong>in</strong>al distance<br />
they are still <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>f<strong>in</strong>ite transverse size. Introduc<strong>in</strong>g a transverse<br />
pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile such as for a Gaussian beam certa<strong>in</strong>ly represents<br />
a more realistic model but turns out to be difficult to<br />
implement <strong>in</strong> str<strong>on</strong>g-field QED, the ma<strong>in</strong> reas<strong>on</strong> be<strong>in</strong>g the<br />
loss <str<strong>on</strong>g>of</str<strong>on</strong>g> too many c<strong>on</strong>servati<strong>on</strong> laws al<strong>on</strong>g with translati<strong>on</strong>al<br />
<strong>in</strong>variance. So, for the purposes <str<strong>on</strong>g>of</str<strong>on</strong>g> this talk we will exclusively<br />
be deal<strong>in</strong>g with (<strong>in</strong>f<strong>in</strong>ite or pulsed) plane waves.<br />
From a relativistic field theory po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> view, which we<br />
have to adopt for a0 > 1, plane electromagnetic waves are<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a quite peculiar nature. They are described by a wave<br />
4-vector k that is lightlike or null, i.e. k 2 = 0. The electromagnetic<br />
field strength, F = (E, B), <strong>on</strong>ly depends <strong>on</strong> the<br />
<strong>in</strong>variant phase, k·x = ωt/c−k·x, where ω is the laser fre-
quency measured <strong>in</strong> the lab. By Maxwell’s equati<strong>on</strong>s, fields<br />
are transverse, k ·F = 0, and, most importantly, <strong>in</strong>herits the<br />
null properties from k, E 2 − B 2 = E · B = F 3 = 0. This<br />
implies that there is no <strong>in</strong>tr<strong>in</strong>sic <strong>in</strong>variant scale characteris<strong>in</strong>g<br />
an electromagnetic plane wave field. For this reas<strong>on</strong><br />
<strong>on</strong>e needs an external probe momentum, p, to build <strong>in</strong>variants<br />
such that, for <strong>in</strong>stance, a 2 0 ∼ (p · F) 2 [6].<br />
The ma<strong>in</strong> effect <strong>on</strong> such a probe, say an electr<strong>on</strong>, may<br />
actually be understood <strong>in</strong> classical language. Due to the<br />
Lorentz force the electr<strong>on</strong> will undergo rapid quiver moti<strong>on</strong><br />
as may be seen by solv<strong>in</strong>g for its momentum p(τ) as a functi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> proper time τ. Averag<strong>in</strong>g over the latter the electr<strong>on</strong><br />
acquires a quasi-momentum, q ≡ 〈p(τ)〉 = p<strong>in</strong> + κ(a 2 0) k,<br />
which displays a l<strong>on</strong>gitud<strong>in</strong>al additi<strong>on</strong> to the <strong>in</strong>itial momentum<br />
weighted with an a0 dependent prefactor. Squar<strong>in</strong>g<br />
this expressi<strong>on</strong> and us<strong>in</strong>g k 2 = 0 <strong>on</strong>e f<strong>in</strong>ds that the electr<strong>on</strong><br />
has become heavier with an effective mass given by<br />
m 2 ∗ = m 2 (1 + a 2 0) . (2)<br />
This fundamental <strong>in</strong>tensity effect has been predicted l<strong>on</strong>g<br />
ago [7, 8], but has apparently never been measured. While<br />
the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> the mass shift may be understood quantum<br />
mechanically it nevertheless has c<strong>on</strong>sequences for the<br />
quantum theory. These are best analysed <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g<br />
field QED. Its <strong>in</strong>gredients are the usual particle c<strong>on</strong>tent<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> QED namely phot<strong>on</strong>s and electr<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> arbitrary energy<br />
serv<strong>in</strong>g e.g. as probes for “quantum diagnostics” when they<br />
are coupled to the external laser field. The basic quantum<br />
effect is then a dress<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s via c<strong>on</strong>t<strong>in</strong>uous<br />
emissi<strong>on</strong> and absorpti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> laser phot<strong>on</strong>s. For plane waves<br />
this can be taken <strong>in</strong>to account exactly employ<strong>in</strong>g the celebrated<br />
Volkov soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Dirac equati<strong>on</strong> [9]. In terms<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Feynman diagrams the situati<strong>on</strong> is depicted <strong>in</strong> Fig. 2.<br />
Figure 2: The dressed (Volkov) propagator <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>.<br />
Transiti<strong>on</strong> amplitudes are then c<strong>on</strong>structed <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Volkov electr<strong>on</strong> l<strong>in</strong>es coupled to probe phot<strong>on</strong>s. Laser phot<strong>on</strong>s<br />
are no l<strong>on</strong>ger explicit but hidden <strong>in</strong> the dressed propagator<br />
given by the left-hand side Fig. 2. The ma<strong>in</strong> issues<br />
to be discussed below will be the dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> these amplitudes<br />
<strong>on</strong> <strong>in</strong>tensity (a0) and f<strong>in</strong>ite pulse durati<strong>on</strong>, all <strong>in</strong><br />
a plane wave c<strong>on</strong>text. To give an idea <str<strong>on</strong>g>of</str<strong>on</strong>g> the parameter<br />
range <strong>in</strong>volved we present <strong>in</strong> Fig. 3 a bird’s eye view <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
probe energy vs. <strong>in</strong>tensity regimes. C<strong>on</strong>venti<strong>on</strong>al high energy<br />
physics takes place close to the vertical axis, above the<br />
electr<strong>on</strong> pair creati<strong>on</strong> threshold <str<strong>on</strong>g>of</str<strong>on</strong>g> 2mc 2 . In this regime, a<br />
first excursi<strong>on</strong> <strong>in</strong>to high-field physics has been made by the<br />
SLAC experiment E-144 which utilised 30 GeV phot<strong>on</strong>s<br />
(obta<strong>in</strong>ed via backscatter<strong>in</strong>g from the 50 GeV SLAC electr<strong>on</strong><br />
beam) to produce electr<strong>on</strong> positr<strong>on</strong> pairs <strong>in</strong> collisi<strong>on</strong>s<br />
with a 10 TW (a0 0.4) laser beam [10]. New facilities<br />
such as Vulcan 10 PW or ELI would venture deeply <strong>in</strong> the<br />
high-<strong>in</strong>tensity regi<strong>on</strong> (a0 ≫ 1) and may also come close to<br />
the pair threshold us<strong>in</strong>g Compt<strong>on</strong> backscatter<strong>in</strong>g from sufficiently<br />
energetic electr<strong>on</strong>s. The latter could <strong>in</strong> pr<strong>in</strong>ciple<br />
be the result <str<strong>on</strong>g>of</str<strong>on</strong>g> laser wake field accelerati<strong>on</strong>.<br />
Figure 3: Parameter range <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g field QED.<br />
A basic example <str<strong>on</strong>g>of</str<strong>on</strong>g> f<strong>in</strong>ite pulse durati<strong>on</strong> effects is displayed<br />
<strong>in</strong> Fig. 4 which shows the mass shift, ∆m 2 ≡<br />
m 2 ∗ − m 2 , <strong>in</strong> a pulse [11] as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the number N <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
cycles per pulse. There is obviously a “switch<strong>in</strong>g <strong>on</strong>” effect<br />
with a sudden <strong>in</strong>crease up<strong>on</strong> c<strong>on</strong>clud<strong>in</strong>g the first cycle after<br />
which the <strong>in</strong>f<strong>in</strong>ite plane wave result (2) is approached. It<br />
should be emphasised that for current ultra-short pulses <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
a few fs durati<strong>on</strong> the number n is <strong>in</strong>deed <str<strong>on</strong>g>of</str<strong>on</strong>g> order unity.<br />
Figure 4: The mass shift <strong>in</strong> a pulsed plane wave.<br />
STRONG FIELDS: EXAMPLES<br />
N<strong>on</strong>l<strong>in</strong>ear Compt<strong>on</strong> Scatter<strong>in</strong>g (NLC)<br />
The Feynman diagrams for NLC are shown <strong>in</strong> Fig. 5.<br />
Expand<strong>in</strong>g the Volkov l<strong>in</strong>es accord<strong>in</strong>g to Fig. 2 we f<strong>in</strong>d<br />
that <strong>on</strong>e is essentially summ<strong>in</strong>g over all processes <str<strong>on</strong>g>of</str<strong>on</strong>g> the
form e + nγL → e ′ + γ where a phot<strong>on</strong> γ is emitted up<strong>on</strong><br />
absorpti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> n laser phot<strong>on</strong>s γL by the <strong>in</strong>com<strong>in</strong>g electr<strong>on</strong>.<br />
Figure 5: Feynman diagrams for NLC.<br />
The process has been analysed <strong>in</strong> [5, 8] where formulae<br />
for cross secti<strong>on</strong>s or emissi<strong>on</strong> rates can be found. The ma<strong>in</strong><br />
features <str<strong>on</strong>g>of</str<strong>on</strong>g> NLC may be summarised as follows. There is<br />
no threshold to overcome which implies that there is a classical<br />
limit (n<strong>on</strong>l<strong>in</strong>ear Thoms<strong>on</strong> scatter<strong>in</strong>g) corresp<strong>on</strong>d<strong>in</strong>g to<br />
ω ≪ mc2 . In the l<strong>in</strong>ear regime (a0 ≪ 1) <strong>on</strong>e f<strong>in</strong>ds the<br />
usual Compt<strong>on</strong> upshift for the emitted phot<strong>on</strong> frequency,<br />
ω ′ 4γ2 eω where γe is the electr<strong>on</strong> gamma factor. This is<br />
nowadays be<strong>in</strong>g exploited for the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> m<strong>on</strong>oenergetic<br />
gamma rays <str<strong>on</strong>g>of</str<strong>on</strong>g> high peak brillance [12]. The n<strong>on</strong>l<strong>in</strong>ear<br />
regime (a0 > 1) is characterised by an <strong>in</strong>tensity dependent<br />
cross secti<strong>on</strong>, σ(a0), determ<strong>in</strong><strong>in</strong>g the number <str<strong>on</strong>g>of</str<strong>on</strong>g> produced<br />
phot<strong>on</strong>s, Nγ ∼ σ(a0)NeNγL . This very fact alters<br />
the Compt<strong>on</strong> upshift the maximum <str<strong>on</strong>g>of</str<strong>on</strong>g> which now becomes<br />
ω ′ n,max 4γ 2 enω/(1 + a 2 0) , n = 1, 2, . . . . (3)<br />
In particular, we note the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> higher harm<strong>on</strong>ics<br />
(n > 1) and the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> a 2 0 <strong>in</strong> the denom<strong>in</strong>ator. Thus,<br />
there is a reducti<strong>on</strong> (red-shift) <str<strong>on</strong>g>of</str<strong>on</strong>g> the k<strong>in</strong>ematic Compt<strong>on</strong><br />
edge which for the first harm<strong>on</strong>ic amounts to<br />
ω ′ max 4γ 2 eω −→ 4γ 2 eω/a 2 0 , (a0 ≫ 1) . (4)<br />
Figure 6: Emissi<strong>on</strong> spectrum for NLC.<br />
This redshift w.r.t. l<strong>in</strong>ear Compt<strong>on</strong> scatter<strong>in</strong>g should be a<br />
clear experimental signal and is highlighted <strong>in</strong> Fig. 6 (dis-<br />
play<strong>in</strong>g the γ emissi<strong>on</strong> spectrum) by a red arrow. The<br />
higher harm<strong>on</strong>ics are visible as side bands to the right <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the ma<strong>in</strong> (n = 1) spectral peak [13].<br />
Effects due to f<strong>in</strong>ite transverse pulse extensi<strong>on</strong> are easily<br />
understood qualitatively. The previous situati<strong>on</strong> is typical<br />
for laser beams that are not too str<strong>on</strong>gly focussed such that<br />
the electr<strong>on</strong> beam (radius rb) does not feel the decrease<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tensity <strong>in</strong> transverse directi<strong>on</strong>. Clearly, this requires<br />
rb ≪ w0 with w0 denot<strong>in</strong>g the laser beam waist size (see<br />
Fig. 7, right panel). On the other hand, when the laser beam<br />
is tightly focussed (w0 < rb, Fig. 7, left panel) the electr<strong>on</strong>s<br />
will also probe the boundaries <str<strong>on</strong>g>of</str<strong>on</strong>g> the beam which <strong>in</strong><br />
turn will modify the spectra <str<strong>on</strong>g>of</str<strong>on</strong>g> Fig. 6. It is somewhat unfortunate<br />
that the highly n<strong>on</strong>l<strong>in</strong>ear situati<strong>on</strong> (a0 ≫ 1) corresp<strong>on</strong>ds<br />
to a tight focus. This suggests that the experimental<br />
detecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the redshift will require a f<strong>in</strong>e tun<strong>in</strong>g compromise<br />
and <strong>in</strong> particular a very narrow electr<strong>on</strong> beam. For a<br />
detailed study the reader is referred to [14].<br />
Figure 7: Left: Tight laser focus. Right: Wide laser focus.<br />
Laser pair producti<strong>on</strong> (PP)<br />
The str<strong>on</strong>g-field QED Feynman diagram for laser <strong>in</strong>duced<br />
PP is obta<strong>in</strong>ed from the NLC diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> Fig. 5 via<br />
cross<strong>in</strong>g, i.e. by swapp<strong>in</strong>g the outgo<strong>in</strong>g gamma with the<br />
<strong>in</strong>com<strong>in</strong>g electr<strong>on</strong> which turns <strong>in</strong>to an outgo<strong>in</strong>g positr<strong>on</strong><br />
(Fig. 8).<br />
Figure 8: Feynman diagram for laser PP obta<strong>in</strong>ed from<br />
NLC via cross<strong>in</strong>g.<br />
Expand<strong>in</strong>g the diagram <strong>on</strong> the right-hand side corresp<strong>on</strong>ds<br />
to pair creati<strong>on</strong> stimulated by n laser phot<strong>on</strong>s,<br />
γ + nγL → e + e − . Both processes <str<strong>on</strong>g>of</str<strong>on</strong>g> Fig. 8 have been<br />
employed <strong>in</strong> the experiment SLAC E-144. First, high energy<br />
gammas (30 GeV) have been obta<strong>in</strong>ed through n<strong>on</strong>l<strong>in</strong>ear<br />
Compt<strong>on</strong> upshift<strong>in</strong>g whereup<strong>on</strong> these were brought<br />
<strong>in</strong>to collisi<strong>on</strong> with the laser aga<strong>in</strong>. The centre-<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass energy<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the collid<strong>in</strong>g phot<strong>on</strong>s (for the sec<strong>on</strong>d harm<strong>on</strong>ic)<br />
was just about enough to produce pairs <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and
positr<strong>on</strong>s, each <str<strong>on</strong>g>of</str<strong>on</strong>g> effective mass m∗ 1.2 m. A number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> about 10 2 positr<strong>on</strong>s has been reported <strong>in</strong> [15] (see [10]<br />
for a comprehensive overview). In a new theoretical development<br />
this two-step process (NLC + laser PP) has been<br />
reanalysed and led to the predicti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pair cascades where<br />
the produced charged particles are c<strong>on</strong>t<strong>in</strong>uously be<strong>in</strong>g reaccelerated<br />
emitt<strong>in</strong>g bremsstrahlung that <strong>in</strong> turn produces<br />
more pairs [16].<br />
Figure 9: Laser PP rates for a f<strong>in</strong>ite wave tra<strong>in</strong> (N = 1, 4, 8<br />
from top to bottom).<br />
Pursu<strong>in</strong>g a slightly complementary directi<strong>on</strong> we have recently<br />
<strong>in</strong>vestigated the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> f<strong>in</strong>ite pulse durati<strong>on</strong> <strong>on</strong><br />
laser PP us<strong>in</strong>g light-c<strong>on</strong>e field theory [17]. In an <strong>in</strong>f<strong>in</strong>ite<br />
plane wave the triple differential PP rate (w.r.t. l<strong>on</strong>gitud<strong>in</strong>al<br />
and transverse positr<strong>on</strong> momentum, say) is a delta comb<br />
above threshold, with sharp res<strong>on</strong>ance peaks ak<strong>in</strong> to an<br />
ideal <strong>in</strong>terference pattern. One expects this to get washed<br />
out as so<strong>on</strong> as f<strong>in</strong>ite pulse durati<strong>on</strong> becomes noticeable, i.e.<br />
when the number <str<strong>on</strong>g>of</str<strong>on</strong>g> cycle per pulse N = O(1). This is<br />
<strong>in</strong>deed what <strong>on</strong>e f<strong>in</strong>ds as is shown <strong>in</strong> Fig.s 9 and 10. For a<br />
f<strong>in</strong>ite wave tra<strong>in</strong> (sharply cut <str<strong>on</strong>g>of</str<strong>on</strong>g>f <strong>in</strong> k·x) the maxima rema<strong>in</strong><br />
at the delta comb positi<strong>on</strong>s (vertical l<strong>in</strong>es <strong>in</strong> Fig. 9) while<br />
for a more realistic pulse (smoothly decay<strong>in</strong>g envelope) this<br />
is no l<strong>on</strong>ger true (Fig. 10). Notably, <strong>in</strong> both cases, there is a<br />
signal below the n<strong>on</strong>l<strong>in</strong>ear (m∗) threshold which seems to<br />
be c<strong>on</strong>sistent with Fig. 4 as is the fact that the <strong>in</strong>f<strong>in</strong>ite plane<br />
wave results are recovered when N → ∞. The important<br />
bottom l<strong>in</strong>e here is that the spectra are f<strong>in</strong>gerpr<strong>in</strong>ts <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
pulse shapes which might even be turned <strong>in</strong>to a diagnostic<br />
tool.<br />
Figure 10: Laser PP rates for a smooth pulse (N = 1, 4, 8<br />
from top to bottom).<br />
Vacuum birefr<strong>in</strong>gence<br />
Our f<strong>in</strong>al example addresses an <strong>in</strong>terest<strong>in</strong>g effect due to<br />
external fields below the PP threshold. This possibility<br />
has already been po<strong>in</strong>ted out <strong>in</strong> the early days <str<strong>on</strong>g>of</str<strong>on</strong>g> QED by<br />
Heisenberg and Euler [18]: “...even <strong>in</strong> situati<strong>on</strong>s where the<br />
[phot<strong>on</strong>] energy is not sufficient for matter producti<strong>on</strong>, its<br />
virtual possibility will result <strong>in</strong> a ‘polarisati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum’<br />
and hence <strong>in</strong> an alterati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Maxwells equati<strong>on</strong>s”<br />
(our translati<strong>on</strong>).<br />
In terms <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g-field QED Feynman diagrams the<br />
physics <strong>in</strong>volved may be understood via the optical theorem<br />
(Fig. 11). For the case at hand this states that the total<br />
PP rate is given by the imag<strong>in</strong>ary part <str<strong>on</strong>g>of</str<strong>on</strong>g> the polarisati<strong>on</strong><br />
tensor which, <str<strong>on</strong>g>of</str<strong>on</strong>g> course, is <strong>on</strong>ly n<strong>on</strong>vanish<strong>in</strong>g above threshold.<br />
This appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> an imag<strong>in</strong>ary part corresp<strong>on</strong>ds to<br />
an absorptive process (phot<strong>on</strong>s ‘decay’). Effects below<br />
threshold, <strong>on</strong> the other hand, are dispersive, they modify,<br />
<strong>in</strong> particular, the propagati<strong>on</strong> properties <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s prob<strong>in</strong>g<br />
the external field. One natural expectati<strong>on</strong> is that the probe<br />
phot<strong>on</strong> polarisati<strong>on</strong> might be affected by the preferred di-
ecti<strong>on</strong>(s) associated with the external field. This is <strong>in</strong>deed<br />
what happens.<br />
Figure 11: The optical theorem for laser PP.<br />
Obviously, to study these effects <strong>on</strong>e needs to calculate<br />
the vacuum polarisati<strong>on</strong> tensor <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> the relevant<br />
external field (Fig. 11, right-hand side). For slowly<br />
vary<strong>in</strong>g fields this has been achieved l<strong>on</strong>g ago <strong>in</strong> Toll’s thesis<br />
[19] based <strong>on</strong> the Heisenberg-Euler lagrangian [18]. He<br />
has found that there is birefr<strong>in</strong>gence <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum due to<br />
a phase retardati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the probe electric field comp<strong>on</strong>ent<br />
parallel to the background magnetic field comp<strong>on</strong>ent. The<br />
effect is t<strong>in</strong>y (as are the Heisenberg-Euler correcti<strong>on</strong>s to<br />
Maxwell’s equati<strong>on</strong>s) and was l<strong>on</strong>g thought to be out <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
reach experimentally [20]. This situati<strong>on</strong> has changed with<br />
the <strong>in</strong>crease <strong>in</strong> laser <strong>in</strong>tensity as discussed <strong>in</strong> [21]. Let us<br />
discuss the physics <strong>in</strong>volved <strong>in</strong> some more detail.<br />
The propagati<strong>on</strong> properties <str<strong>on</strong>g>of</str<strong>on</strong>g> probe phot<strong>on</strong>s are determ<strong>in</strong>ed<br />
by the eigenvalues <str<strong>on</strong>g>of</str<strong>on</strong>g> the polarisati<strong>on</strong> tensor. An<br />
explicit calculati<strong>on</strong> shows that two <str<strong>on</strong>g>of</str<strong>on</strong>g> these are n<strong>on</strong>trivial<br />
imply<strong>in</strong>g birefr<strong>in</strong>gence. The eigenvalues translate <strong>in</strong>to two<br />
different <strong>in</strong>dices <str<strong>on</strong>g>of</str<strong>on</strong>g> refracti<strong>on</strong> which, follow<strong>in</strong>g Toll, may<br />
be written as<br />
n± = 1+ αɛ2 2 2 2<br />
11 ± 3 + O(ɛ ν ) 1 + O(αɛ ) . (5)<br />
45π<br />
There are three small dimensi<strong>on</strong>less parameters <strong>in</strong>volved,<br />
namely (i) the field strength, ɛ ≡ E/ES <strong>in</strong> units <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
Sauter-Schw<strong>in</strong>ger critical field, ES = m 2 c 3 /e 1.3 ×<br />
10 18 V/m, (ii) the probe frequency, ν = ω/mc 2 and the<br />
f<strong>in</strong>e structure c<strong>on</strong>stant, α = e 2 /4πc 1/137. The experimental<br />
challenge is to measure the ellipticity acquired<br />
by a l<strong>in</strong>early polarised probe beam travers<strong>in</strong>g the focus <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
an ultra-<strong>in</strong>tense beam. As shown <strong>in</strong> [21] the ellipticity <strong>in</strong>creases<br />
with focus size d, <strong>in</strong>tensity ɛ 2 and probe frequency<br />
ν, the latter suggest<strong>in</strong>g the use <str<strong>on</strong>g>of</str<strong>on</strong>g> probe X-rays. Recent advances<br />
<strong>in</strong> X-ray polarimetry [22] imply that it may become<br />
possible to observe the effect with <strong>in</strong>tensities <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 23 ...10 24<br />
W/cm 2 which seem <strong>in</strong> reach for the near future [3, 4].<br />
C<strong>on</strong>clusi<strong>on</strong><br />
The examples analysed <strong>in</strong> this secti<strong>on</strong> should have<br />
shown that an experimental programme dedicated to their<br />
study is feasible. NLC is unique <strong>in</strong> that there are no thresholds<br />
<strong>in</strong> energy or <strong>in</strong>tensity to be overcome, so its study is<br />
just a questi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> hav<strong>in</strong>g the resources (electr<strong>on</strong> and laser<br />
beams) <strong>in</strong> place. Laser PP and vacuum birefr<strong>in</strong>gence are<br />
more demand<strong>in</strong>g <strong>in</strong> <strong>in</strong>tensity but new facilities are under<br />
way. Hence, it seems we are <strong>on</strong> the verge <str<strong>on</strong>g>of</str<strong>on</strong>g> open<strong>in</strong>g up a<br />
new and excit<strong>in</strong>g <strong>in</strong>terdiscipl<strong>in</strong>ary field <str<strong>on</strong>g>of</str<strong>on</strong>g> physics that may<br />
be called laser particle physics.<br />
It is a pleasure to thank the organisers <str<strong>on</strong>g>of</str<strong>on</strong>g> PIF2010 for<br />
the excellent job they did, <strong>in</strong> particular for creat<strong>in</strong>g such an<br />
excit<strong>in</strong>g c<strong>on</strong>ference atmosphere.<br />
REFERENCES<br />
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Abstract<br />
Str<strong>on</strong>g-Field Effects <strong>in</strong> Beam-Beam Interacti<strong>on</strong> <strong>in</strong> L<strong>in</strong>ear Colliders<br />
Various types <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>-positr<strong>on</strong> l<strong>in</strong>ear colliders are be<strong>in</strong>g<br />
studied <strong>in</strong> the world. In all cases the <strong>in</strong>tense field effects<br />
associated with the beam-beam <strong>in</strong>teracti<strong>on</strong> due to the<br />
tightly focused beams are expected. We summarize the results<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the studies.<br />
INTRODUCTION<br />
It was obvious s<strong>in</strong>ce l<strong>on</strong>g ago that the electr<strong>on</strong>-positr<strong>on</strong><br />
collider after LEP had to be a l<strong>in</strong>ear collider rather than<br />
a r<strong>in</strong>g collider. Serious design studies <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear colliders<br />
started <strong>in</strong> mid 1980’s <strong>in</strong> several laboratories <strong>in</strong> the world.<br />
To reach the required lum<strong>in</strong>osity the beam must be much<br />
more tightly focused than <strong>in</strong> r<strong>in</strong>g colliders. It was so<strong>on</strong> recognized<br />
that the electromagnetic field due to the focused<br />
beam causes various phenomena related to quantum electrodynamics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tense field.<br />
The first obvious fact was that the phot<strong>on</strong> energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
synchrotr<strong>on</strong> radiati<strong>on</strong> (now called ‘beamstrahlung’) from<br />
the beam field can exceed the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the primary electr<strong>on</strong>s<br />
if the classical formula <str<strong>on</strong>g>of</str<strong>on</strong>g> synchrotr<strong>on</strong> radiati<strong>on</strong> is<br />
used. The full formula <str<strong>on</strong>g>of</str<strong>on</strong>g> Sokolov and Ternov[2] must be<br />
used.<br />
It was also po<strong>in</strong>ted out the beamstrahlung phot<strong>on</strong> can<br />
create electr<strong>on</strong>-positr<strong>on</strong> pairs <strong>in</strong> the str<strong>on</strong>g electromagnetic<br />
field. In the l<strong>in</strong>ear collider community this is ‘called coherent<br />
pair creati<strong>on</strong>’.<br />
The str<strong>on</strong>g field can cause depolarizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong><br />
(positr<strong>on</strong>) through the process <str<strong>on</strong>g>of</str<strong>on</strong>g> precessi<strong>on</strong> <strong>in</strong> a field and<br />
sp<strong>in</strong>-flip synchrotr<strong>on</strong> radiati<strong>on</strong>.<br />
Moreover, the idea <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>vert<strong>in</strong>g an electr<strong>on</strong>-electr<strong>on</strong><br />
collider <strong>in</strong>to a phot<strong>on</strong>-phot<strong>on</strong> (more <str<strong>on</strong>g>of</str<strong>on</strong>g>ten called gammagamma)<br />
collider was proposed. In this scheme a str<strong>on</strong>g<br />
laser beam is irradiated <strong>on</strong> the electr<strong>on</strong>s to create highenergy<br />
back-scattered phot<strong>on</strong>s. This process also requires<br />
a knowledge <str<strong>on</strong>g>of</str<strong>on</strong>g> high-field quantum electro-dynamics. It<br />
turned out that the n<strong>on</strong>l<strong>in</strong>ear effects impose str<strong>on</strong>g limitati<strong>on</strong>s<br />
<strong>on</strong> the electr<strong>on</strong>-phot<strong>on</strong> c<strong>on</strong>versi<strong>on</strong>.<br />
All these are unwanted effects for the performance <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear<br />
colliders. The theory and simulati<strong>on</strong> had almost been<br />
established by around 1995, although experimental verificati<strong>on</strong>s<br />
are very poor still now. In this report we briefly<br />
summarize what we have d<strong>on</strong>e <strong>in</strong> the past.<br />
For numerical examples <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear colliders <strong>in</strong> this paper<br />
we quote three future colliders, namely, ILC (<str<strong>on</strong>g>Internati<strong>on</strong>al</str<strong>on</strong>g><br />
L<strong>in</strong>ear Collider), CLIC (Compact L<strong>in</strong>ear Collider)<br />
and plasma collider. The technologies for these colliders<br />
∗ kaoru.yokoya@kek.jp<br />
Kaoru Yokoya ∗ , <strong>KEK</strong>, Japan<br />
are somewhat different but here we <strong>on</strong>ly need the beam parameters<br />
at the collisi<strong>on</strong> po<strong>in</strong>t. Tab.1 shows typical parameters<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> these colliders (those related to beam-beam <strong>in</strong>teracti<strong>on</strong><br />
<strong>on</strong>ly). Colliders us<strong>in</strong>g the plasma accelerati<strong>on</strong> technology<br />
are be<strong>in</strong>g plannned as far future projects. Their parameters<br />
are still uncerta<strong>in</strong>. In this table two possible parameter<br />
sets ‘Plasma1’ and ‘Plasma2’ are shown[1].<br />
Table 1: Example Parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> L<strong>in</strong>ear Colliders<br />
ILC ILC CLIC Plasma1 Plasma2<br />
E CM 0.5 1 3 10 10 TeV<br />
N 2 2 0.37 0.1 0.4 ×10 10<br />
σz 300 300 44 1 1 µm<br />
σx 470 550 40 2 2 nm<br />
σy 3.8 2.7 1 2 2 nm<br />
〈Υ〉 0.063 0.109 5.5 2000 9000<br />
δ B 3.9 5 30 25 50 %<br />
nγ 1.71 1.43 2 2.4 2.1<br />
E CM : center-<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass energy, N : number <str<strong>on</strong>g>of</str<strong>on</strong>g> particles per bunch, σz :<br />
rms bunch length, σx : horiz<strong>on</strong>tal rms beam size, σy : vertical rms beam<br />
size, 〈Υ〉 : field strength parameter, δ B : energy loss by beamstrahlung,<br />
nγ : number <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s per electr<strong>on</strong>.<br />
BEAMSTRAHLUNG<br />
The field strength by the electr<strong>on</strong> (positr<strong>on</strong>) beam at the<br />
collisi<strong>on</strong> po<strong>in</strong>t is approximately given by<br />
Ne<br />
≈<br />
σz × max(σx, σy)<br />
where the symbols are def<strong>in</strong>ed <strong>in</strong> Tab.1. The field ranges<br />
from ∼ 500 Tesla <strong>in</strong> ILC, ∼ 10 4 Tesla <strong>in</strong> CLIC to ∼ 10 6<br />
<strong>in</strong> plasma colliders (better to use magnetic field because<br />
electric field is cancelled between electr<strong>on</strong> and positr<strong>on</strong>).<br />
This is still low compared with the Schw<strong>in</strong>ger field<br />
(1)<br />
BSch = ESch/c = 4.4 × 10 9 Tesla (2)<br />
However, another Lorentz <strong>in</strong>variant quantity can be O(1):<br />
Υ ≡ e<br />
m3 <br />
(p µ Fµν) 2 = 2<br />
3<br />
¯hω C<br />
E<br />
= λeγ 2<br />
ρ<br />
B<br />
= γ , (3)<br />
BSch where pµ is the electr<strong>on</strong> 4-momentum, m the electr<strong>on</strong><br />
rest mass, Fµν the electromagnetic tensor, λe the Compt<strong>on</strong><br />
wavelength, ¯hω C the critical energy <str<strong>on</strong>g>of</str<strong>on</strong>g> radiati<strong>on</strong>, and<br />
E = γmc 2 the electr<strong>on</strong> energy. This parameter varies<br />
al<strong>on</strong>g the bunch. Its average can be approximately expressed<br />
by the beam parameters as<br />
〈Υ〉 = 5 Nr<br />
6<br />
2 eγ<br />
. (4)<br />
ασz(σx + σy)
This is listed <strong>in</strong> Tab.1. The effect is sizable <strong>in</strong> ILC and is<br />
dom<strong>in</strong>ant <strong>in</strong> colliders above a few TeV.<br />
An example <str<strong>on</strong>g>of</str<strong>on</strong>g> the lum<strong>in</strong>osity spectrum under str<strong>on</strong>g<br />
beamstrahlung is shown <strong>in</strong> Fig.1. It shows an extreme<br />
case <str<strong>on</strong>g>of</str<strong>on</strong>g> a plasma collider. The peak at low energies comes<br />
from the energy loss by multiple beamstrahlung. The spectrum<br />
for ILC is by far clearly dom<strong>in</strong>ated by the high-energy<br />
peak.<br />
Figure 1: An extreme example <str<strong>on</strong>g>of</str<strong>on</strong>g> lum<strong>in</strong>osity spectrum under<br />
beamstrahlung and coherent pair creati<strong>on</strong>. The parameter<br />
Plasma1 is used.<br />
COHERENT PAIR CREATION<br />
When a high-energy phot<strong>on</strong> (beamstrahlung <strong>in</strong> our case)<br />
travels <strong>in</strong> an <strong>in</strong>tense electromagnetic field, it can decay <strong>in</strong>to<br />
electr<strong>on</strong>-positr<strong>on</strong> pairs. The <strong>on</strong>e that has the same sign <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
charge as the <strong>on</strong>com<strong>in</strong>g beam is defected by a large angle<br />
due to the Coulomb field and causes serious backgrounds<br />
to the detector. The relevant Lorentz <strong>in</strong>variant quantity is<br />
χ ≡ e<br />
m3 <br />
(k µ Fµν) 2 = ω<br />
m<br />
B<br />
B Sch<br />
where kµ is the 4-momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> and ω its energy.<br />
When Υ is O(1), χ can also be O(1). The beamstrahlung<br />
and coherent pair creati<strong>on</strong> come from the same<br />
diagram seen <strong>in</strong> different channels as shown <strong>in</strong> Fig.2.<br />
The spectrum (energy distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair particles)<br />
is given by[3]<br />
dW CP<br />
dE+<br />
= α m<br />
√<br />
3π<br />
2<br />
ω2 ∞<br />
η<br />
K1/3(η ′ )dη ′ <br />
E−<br />
+<br />
η = 2<br />
3χ<br />
ω 2<br />
E+E−<br />
+<br />
E+<br />
E+<br />
E−<br />
<br />
(5)<br />
K2/3(η)<br />
<br />
, E− = ω − E+, (6)<br />
where α is the f<strong>in</strong>e structure c<strong>on</strong>stant, E+(E−) the f<strong>in</strong>al<br />
positr<strong>on</strong> (electr<strong>on</strong>) energy, Kν the modified Bessel functi<strong>on</strong>.<br />
This spectrum (normalized to unity) is plotted <strong>in</strong> Fig.3<br />
for various values <str<strong>on</strong>g>of</str<strong>on</strong>g> χ.<br />
Figure 2: Beamstrahlung and Coherent Pair Creati<strong>on</strong>. The<br />
double solid l<strong>in</strong>e <strong>in</strong>dicates electr<strong>on</strong> <strong>in</strong> an external field.<br />
Figure 3: Spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the coherent pair creati<strong>on</strong>.<br />
There is another process, sometimes called ‘trident cascade’,<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> creat<strong>in</strong>g pairs. The virtual phot<strong>on</strong> associated with<br />
an electr<strong>on</strong> can create pairs under a str<strong>on</strong>g field. This process<br />
has been studied <strong>in</strong> early 1970’s[4]. When Υ is very<br />
large (e.g., > 1000), the c<strong>on</strong>tributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this process may<br />
be even larger than the comb<strong>in</strong>ati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> beamstrahlung and<br />
coherent pair creati<strong>on</strong>.<br />
Early studies <strong>on</strong> beamstrahlung and coherent pair creati<strong>on</strong><br />
are reviewed <strong>in</strong>[5].<br />
BEAM-BEAM DEPOLARIZATION<br />
It is relatively easy to obta<strong>in</strong> polarized beams <strong>in</strong> l<strong>in</strong>ear<br />
colliders than <strong>in</strong> r<strong>in</strong>g colliders. The most important<br />
source <str<strong>on</strong>g>of</str<strong>on</strong>g> depolarizati<strong>on</strong> comes from beam-beam <strong>in</strong>teracti<strong>on</strong>.<br />
There are two mechanisms that causes depolarizati<strong>on</strong>,<br />
namely the precessi<strong>on</strong> <strong>in</strong> magnetic field and the sp<strong>in</strong>-flip<br />
synchrotr<strong>on</strong> radiati<strong>on</strong>. Both <str<strong>on</strong>g>of</str<strong>on</strong>g> these processes are wellknown<br />
except the correcti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the precessi<strong>on</strong> formula under<br />
str<strong>on</strong>g field.<br />
The relevant terms <strong>in</strong> the Thomas-BMT equati<strong>on</strong> is<br />
dS<br />
dt<br />
= e<br />
mγ (γa + 1)B T × S (7)<br />
where S is the sp<strong>in</strong> vector (<strong>in</strong> the rest frame), B T the transverse<br />
comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field and a the coefficient<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the anomalous magnetic moment.
At high field a is nol<strong>on</strong>ger a c<strong>on</strong>stant (a ≈ α/2π) but is<br />
a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Υ [6]:<br />
∞<br />
a(Υ) 2 xdx<br />
=<br />
a(0) Υ 0 (1 + x) 3<br />
∞ <br />
x<br />
s<strong>in</strong> t +<br />
0 Υ<br />
t3<br />
<br />
dt (8)<br />
3<br />
= 1+12Υ 2<br />
<br />
log 1 1 37<br />
+ log 3−<br />
Υ 2 12 +γ <br />
, (Υ ≪ 1)<br />
E<br />
This is plotted <strong>in</strong> Fig.4. S<strong>in</strong>ce the depolarizati<strong>on</strong> is propor-<br />
Figure 4: Anomalous magnetic moment <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> as a<br />
functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Υ.<br />
ti<strong>on</strong>al to a 2 , this correcti<strong>on</strong> already gives a sizable (welcome)<br />
effect <strong>in</strong> ILC at 1TeV, and str<strong>on</strong>gly suppresses the<br />
depolarizati<strong>on</strong> by precessi<strong>on</strong> <strong>in</strong> CLIC at 3TeV. The total<br />
depolarizati<strong>on</strong> through beam-beam <strong>in</strong>teracti<strong>on</strong> is a fracti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> percent <strong>in</strong> ILC and several percent <strong>in</strong> CLIC.<br />
The equati<strong>on</strong> (8) has not been experimentally c<strong>on</strong>firmed<br />
<strong>in</strong> spite the magnetic moments <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> and mu<strong>on</strong> at low<br />
field are known extremely accurately. It is quite uncerta<strong>in</strong><br />
whether it can be measured <strong>in</strong> the l<strong>in</strong>ear collider envior<strong>on</strong>ment<br />
because <str<strong>on</strong>g>of</str<strong>on</strong>g> the complexity <str<strong>on</strong>g>of</str<strong>on</strong>g> beam-beam <strong>in</strong>teracti<strong>on</strong>.<br />
GAMMA-GAMMA COLLIDER<br />
As is shown <strong>in</strong> Fig.5, An electr<strong>on</strong>-positr<strong>on</strong> collider (<strong>in</strong><br />
electr<strong>on</strong>-electr<strong>on</strong> mode) can be c<strong>on</strong>verted to a gammagamma<br />
collider by irradiat<strong>in</strong>g a laser just before the collisi<strong>on</strong><br />
po<strong>in</strong>t (less than 1cm) to get high energy backscattered<br />
phot<strong>on</strong>s. The comb<strong>in</strong>ati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>on</strong>gitud<strong>in</strong>ally polarized electr<strong>on</strong><br />
and circularly polarized laser can create phot<strong>on</strong>s with<br />
small energy spread.<br />
§¨©<br />
¡ ¢£¤¥¦ ¡ ¢£¤¥¦<br />
<br />
Figure 5: Gamma-Gamma collider scheme<br />
<br />
There are three Lorentz <strong>in</strong>variant quantities made from<br />
the 4-momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> pµ and the field tensor Fµν <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
laser (assume plane wave, wave number kµ): 1<br />
am<strong>on</strong>g which there is a relati<strong>on</strong><br />
Υ = e<br />
m3 <br />
(p µ Fµν) 2 (9)<br />
Λ =<br />
2k · p<br />
m2 ξ =<br />
(10)<br />
e <br />
−A µ<br />
Aµ<br />
m<br />
(11)<br />
2Υ = ξΛ. (12)<br />
(Aµ is the vector potential but ξ can be written <strong>in</strong> gauge<br />
<strong>in</strong>variant form by us<strong>in</strong>g eq(12).)<br />
The 2-dimensi<strong>on</strong>al parameter plane <str<strong>on</strong>g>of</str<strong>on</strong>g> Υ, ξ and Λ is<br />
shown <strong>in</strong> Fig.6. 2 Λ is the parameter <str<strong>on</strong>g>of</str<strong>on</strong>g> the center-<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass<br />
energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the Compt<strong>on</strong> scatter<strong>in</strong>g and is a good parameter<br />
<strong>in</strong> the regi<strong>on</strong> ξ ≪ 1, whereas Υ is better <strong>in</strong> the regi<strong>on</strong><br />
ξ ≫ 1 to describe str<strong>on</strong>g fields.<br />
Figure 6: Parameter plane <str<strong>on</strong>g>of</str<strong>on</strong>g> laser-electr<strong>on</strong> <strong>in</strong>teracti<strong>on</strong><br />
For gamma-gamma colliders large values <str<strong>on</strong>g>of</str<strong>on</strong>g> Λ (short<br />
laser wavelength) is preferred for obta<strong>in</strong><strong>in</strong>g higher energy<br />
phot<strong>on</strong>s from the given electr<strong>on</strong> energy. However, if Λ<br />
is too large, the produced phot<strong>on</strong>s decay <strong>in</strong>to e + e − pairs<br />
<strong>in</strong> the same laser field. For this reas<strong>on</strong>, we usually adopt<br />
Λ < 2 + 2 √ 2 = 4.83.<br />
A str<strong>on</strong>ger laser (larger ξ) is preferred for obta<strong>in</strong><strong>in</strong>g more<br />
phot<strong>on</strong>s but, when ξ is too large, n<strong>on</strong>-l<strong>in</strong>ear QED effects<br />
degrades the energy spectrum and polarizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> produced<br />
phot<strong>on</strong>s. Normally we choose ξ 2 < 0.5. Thus, the regi<strong>on</strong><br />
near the center <str<strong>on</strong>g>of</str<strong>on</strong>g> diagram <strong>in</strong> Fig.6 is chosen.<br />
Fig.7 is an (old) example <str<strong>on</strong>g>of</str<strong>on</strong>g> the spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s for a<br />
gamma-gamma collider. The parameters are : the electr<strong>on</strong><br />
energy Ee = 250GeV, laser wavelength 1µm (corresp<strong>on</strong>d<strong>in</strong>g<br />
to Λ = 4.8), ξ 2 = 0.4. The lower (green) curve shows<br />
1 Laser physicists denote ξ by a and accelerator physicists by K. Λ is<br />
more <str<strong>on</strong>g>of</str<strong>on</strong>g>ten denoted by x.<br />
2 Beamstrahlung is not a radiati<strong>on</strong> <strong>in</strong> periodic field but is added<br />
<strong>in</strong> the diagram for comparis<strong>on</strong> by identify<strong>in</strong>g the bunch length as<br />
wavelength/(2π).
the spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s from primary electr<strong>on</strong>s and the upper<br />
(red) curve shows all the phot<strong>on</strong>s <strong>in</strong>clud<strong>in</strong>g those from<br />
repeated Compt<strong>on</strong> scatter<strong>in</strong>g. The parameters are chosen<br />
so that the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> Compt<strong>on</strong> scatter<strong>in</strong>g for an electr<strong>on</strong><br />
is about 1 to get high γ-γ lum<strong>in</strong>osity. Hence multiple<br />
scatter<strong>in</strong>g is <strong>in</strong>evitable, produc<strong>in</strong>g low energy electr<strong>on</strong>s<br />
which causes background.<br />
Figure 7: An example <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> spectrum created by laser-<br />
Compt<strong>on</strong> scatter<strong>in</strong>g for a gamma-gamma collider<br />
The simple Compt<strong>on</strong> scatter<strong>in</strong>g would produce the maximum<br />
phot<strong>on</strong> energy ω = ΛEe/(1 + Λ)=207GeV. But<br />
due to the large value <str<strong>on</strong>g>of</str<strong>on</strong>g> ξ, this energy is lowered to<br />
ω = ΛEe/(1+Λ+ξ 2 )=194GeV and higher harm<strong>on</strong>ic phot<strong>on</strong>s<br />
ω = nΛEe/(1 + nΛ + ξ 2 ) are also produced (n =2<br />
and 3 are visible). In the simulati<strong>on</strong> the formulas for <strong>in</strong>f<strong>in</strong>ite<br />
plane wave laser with the local laser <strong>in</strong>tensity (adiabatic approximati<strong>on</strong>)<br />
is used. This means the value <str<strong>on</strong>g>of</str<strong>on</strong>g> ξ is vary<strong>in</strong>g<br />
with<strong>in</strong> the laser beam so that the peaks <strong>in</strong> Fig.7 are blurred.<br />
Another str<strong>on</strong>g field phenomen<strong>on</strong> related to gammagamma<br />
colliders is the rotati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> polarizati<strong>on</strong> <strong>in</strong><br />
a str<strong>on</strong>g laser field. Some gamma-gamma experiments demand<br />
l<strong>in</strong>early polarized phot<strong>on</strong>. Us<strong>in</strong>g l<strong>in</strong>early polarized<br />
laser, however, would produce blurred phot<strong>on</strong> spectrum.<br />
One possible idea is to produce phot<strong>on</strong>s by circularly polarized<br />
laser and to rotate the polarizati<strong>on</strong> by another laser<br />
which is l<strong>in</strong>early polarized.[7]. Circularly polarized laser<br />
rotates the polarizati<strong>on</strong> plane <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>early polarized phot<strong>on</strong>,<br />
whereas l<strong>in</strong>early polarized laser <strong>in</strong>terchanges l<strong>in</strong>ear and circular<br />
polarizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s. This phenomen<strong>on</strong> is effective<br />
near the threshould <str<strong>on</strong>g>of</str<strong>on</strong>g> pair creati<strong>on</strong>. Although this idea<br />
does not seem to be practical for gamma-gamma applicati<strong>on</strong><br />
but it is theoretically <strong>in</strong>terest<strong>in</strong>g.<br />
SIMULATION TOOLS<br />
In early stages <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear collider study a computer code<br />
CAIN[8] was written to treat the beam deformati<strong>on</strong> by<br />
Coulomb field, beamstrahlung, coherent pair creati<strong>on</strong>, and<br />
polarizati<strong>on</strong> behavior (trident cascade has not been <strong>in</strong>cluded<br />
yet). The code GUINEA-PIG[9] has also been written<br />
for the same purpose. The laser-electr<strong>on</strong> <strong>in</strong>teracti<strong>on</strong> has<br />
been <strong>in</strong>cluded <strong>in</strong> CAIN <strong>in</strong> a later stage. The figures 1 and<br />
7 are produced by CAIN. These codes are still evolv<strong>in</strong>g<br />
accord<strong>in</strong>g to the demands but their essential features have<br />
been established l<strong>on</strong>g ago.<br />
SUMMARY<br />
The Beam-beam <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> l<strong>in</strong>ear colliders is a place<br />
where we expect various phenomena related to the quantum<br />
electro-dynamics <strong>in</strong> <strong>in</strong>tense fields. Its study started <strong>in</strong><br />
mid 1980’s and is thought to be well established theoretically<br />
by now. Necessary simulati<strong>on</strong> tools have also been<br />
developed. N<strong>on</strong>etheless we have to wait for the realizati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear colliders for the experimental verificati<strong>on</strong>s.<br />
Up to now these <strong>in</strong>tese field effects are mostly unwanted<br />
phenomena for the performance <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear colliders. We<br />
hope we may f<strong>in</strong>d useful applicati<strong>on</strong>s <strong>in</strong> the future such as<br />
the positr<strong>on</strong> producti<strong>on</strong> us<strong>in</strong>g the coherent pair creati<strong>on</strong>.<br />
REFERENCES<br />
[1] ICFA-ICUIL Jo<strong>in</strong>t Task Force <strong>on</strong> Laser Accelerati<strong>on</strong><br />
Meet<strong>in</strong>g, Apr.8-10, 2010 at GSI, Darmstadt, Germany. The<br />
task force report will be published so<strong>on</strong> (as <str<strong>on</strong>g>of</str<strong>on</strong>g> Jan.2011).<br />
[2] A. A. Sokolov and I. M. Ternov, ‘Radiati<strong>on</strong> from<br />
Relativistic Electr<strong>on</strong>s’, American Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>,<br />
Translati<strong>on</strong> Series, New York, 1986.<br />
[3] V. N. Baier and V. M. Katkov, Sov. Phys. JETP<br />
26(1968)854. W. Y. Tsai and T. Erber,<br />
Phys. Rev. D10(1974)492.<br />
[4] V. I. Ritus, Nucl. Phys. B44(1972)236. V. N. Baier,<br />
V. M. Katkov and V. M. Strakhovenko, Soviet<br />
J. Nucl. Phys. 14(1972)572.<br />
[5] ‘Beam-Beam Phenomena <strong>in</strong> L<strong>in</strong>ear Colliders’,<br />
K. Yokoya and P. Chen, <strong>in</strong> Fr<strong>on</strong>tiers <str<strong>on</strong>g>of</str<strong>on</strong>g> Particle Beams:<br />
Intensity Limitati<strong>on</strong>. Lecture Notes <strong>in</strong> <strong>Physics</strong> 400,<br />
Spr<strong>in</strong>ger Verlag, (1991) page 414-445.<br />
[6] V. N. Baier, private communicati<strong>on</strong>.<br />
[7] G. L. Kotk<strong>in</strong> and V. G. Serbo,<br />
Phys. Lett. B413(1997)122-129.<br />
[8] CAIN:C<strong>on</strong>glomérat d’ABEL et d’Interacti<strong>on</strong>s<br />
N<strong>on</strong>-L<strong>in</strong>éaires. P. Chen, G. Hort<strong>on</strong>-Smith, T. Ohgaki,<br />
A. W. Weidemann and K. Yokoya, Workshop <strong>on</strong><br />
Gamma-Gamma Colliders, Berkeley, CA, March 28-31,<br />
1994. SLAC-PUB-6583, July 1994.<br />
Nucl. Instr. Meth. A355(1995)107-110. C<strong>on</strong>tact the present<br />
author for the laset <strong>in</strong>formati<strong>on</strong>.<br />
[9] D. Schulte, c<strong>on</strong>sult the web page<br />
http://www-sldnt.slac.stanford.edu/nlc/programs/<br />
gu<strong>in</strong>ea_pig/Orig<strong>in</strong>al%20GP%20Home.html<br />
and<br />
http://flc-mdi.lal.<strong>in</strong>2p3.fr/spip.php?rubrique40
Abstract<br />
Sec<strong>on</strong>d order QED processes and their radiative correcti<strong>on</strong>s<br />
A. Hart<strong>in</strong>, DESY FLC, Notkestrasse 85, Hamburg, Germany<br />
The effect <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tense external fields <strong>on</strong> physics processes<br />
can be taken <strong>in</strong>to account exactly by perform<strong>in</strong>g the usual<br />
S-matrix expansi<strong>on</strong> <strong>in</strong> the bound <strong>in</strong>teracti<strong>on</strong> picture. The<br />
effect so calculated is to predict res<strong>on</strong>ant cross-secti<strong>on</strong>s<br />
<strong>in</strong> sec<strong>on</strong>d order processes. The Compt<strong>on</strong> scatter<strong>in</strong>g <strong>in</strong><br />
such a framework is outl<strong>in</strong>ed. Res<strong>on</strong>ant <strong>in</strong>f<strong>in</strong>ities <strong>in</strong> the<br />
tree level process are mitigated by the electr<strong>on</strong> self energy<br />
and the vertex correcti<strong>on</strong>. The state <str<strong>on</strong>g>of</str<strong>on</strong>g> the calculati<strong>on</strong> and<br />
<strong>in</strong>clusi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> these radiative correcti<strong>on</strong>s is discussed.<br />
INTRODUCTION<br />
The Dirac equati<strong>on</strong> for fermi<strong>on</strong>s embedded <strong>in</strong> <strong>in</strong>tense<br />
electromagnetic plane wave fields can be solved exactly.<br />
The wave functi<strong>on</strong> soluti<strong>on</strong>s <strong>in</strong>clude an <strong>in</strong>f<strong>in</strong>ite summati<strong>on</strong><br />
over <strong>in</strong>teracti<strong>on</strong>s with multiple external field phot<strong>on</strong>s and<br />
display a quasi-energy level structure. Res<strong>on</strong>ant transiti<strong>on</strong>s<br />
between these quasi-energy levels is predicted for quantum<br />
electrodynamical processes with a fermi<strong>on</strong> propagator<br />
and which therefore are at least sec<strong>on</strong>d order. Transiti<strong>on</strong><br />
probabilities for these higher order processes can be<br />
calculated <strong>in</strong> the bound <strong>in</strong>teracti<strong>on</strong> picture (BIP) <strong>in</strong> which<br />
the fermi<strong>on</strong>-external field states <strong>in</strong>teract with free phot<strong>on</strong><br />
states.<br />
I review here the BIP Compt<strong>on</strong> scatter<strong>in</strong>g, a sec<strong>on</strong>d order<br />
QED process <strong>in</strong> the Bound Interacti<strong>on</strong> Picture which is<br />
predicted to have multiple res<strong>on</strong>ances <strong>in</strong> its cross-secti<strong>on</strong>.<br />
The BIP Pair Producti<strong>on</strong> also displays similar behaviour<br />
be<strong>in</strong>g related to the BIP Compt<strong>on</strong> scatter<strong>in</strong>g via a cross<strong>in</strong>g<br />
symmetry [1]. The BIP radiative correcti<strong>on</strong>s required to<br />
render these res<strong>on</strong>ances f<strong>in</strong>ite are the BIP electr<strong>on</strong> self<br />
energy and the BIP vertex correcti<strong>on</strong>, the calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
which is outl<strong>in</strong>ed <strong>in</strong> this paper.<br />
These sec<strong>on</strong>d order processes and their predicted res<strong>on</strong>ances<br />
could be tested experimentally <strong>in</strong> the <strong>in</strong>teracti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> two lasers - <strong>on</strong>e at least be<strong>in</strong>g high <strong>in</strong>tensity - and an<br />
electr<strong>on</strong> beam such as that provided by ATF2 at <strong>KEK</strong>.<br />
Indeed the cross-secti<strong>on</strong> at res<strong>on</strong>ance for sec<strong>on</strong>d order<br />
processes <strong>in</strong> the BIP is c<strong>on</strong>siderably larger for the 1st order<br />
processes such as <strong>on</strong>e phot<strong>on</strong> pair producti<strong>on</strong> and phot<strong>on</strong><br />
radiati<strong>on</strong> [2].<br />
QED IN THE BOUND INTERACTION<br />
PICTURE<br />
For quantum electrodynamical physics processes that<br />
take place <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> an external potential A e the<br />
Lagrangian density is written<br />
LQED = − 1<br />
4 F µν Fµν + ¯ ψ(i/∂ + eA + eA e − m)ψ (1)<br />
If the external field is sufficiently str<strong>on</strong>g it is desirable<br />
to c<strong>on</strong>sider its effect exactly. This is achieved <strong>in</strong> the<br />
BIP which c<strong>on</strong>siders the <strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the bound fermi<strong>on</strong>external<br />
field states with free bos<strong>on</strong> states <strong>in</strong> the usual time<br />
evoluti<strong>on</strong> perturbati<strong>on</strong> theory. The bound fermi<strong>on</strong>-external<br />
field states are obta<strong>in</strong>ed by solv<strong>in</strong>g the Dirac equati<strong>on</strong> with<br />
an external potential,<br />
e ¯ ψ(i/∂ + A e − m)ψ = 0 (2)<br />
Equati<strong>on</strong> 2 can be solved exactly when the external potential<br />
is a plane wave electromagnetic field. The soluti<strong>on</strong>s,<br />
referred to as Volkov soluti<strong>on</strong>s [3] are a product <str<strong>on</strong>g>of</str<strong>on</strong>g> the normal<br />
free fermi<strong>on</strong> soluti<strong>on</strong> with an extra phase S(x) and a<br />
magnetic moment term Ep(x),<br />
Ψ V p (x) = Ep(x)u(p)<br />
where Ep(x) =<br />
<br />
1 − e /Ae <br />
/k<br />
e<br />
2(kp)<br />
iS(x)<br />
(k·x)<br />
and S(x) = −i<br />
0<br />
<br />
e(Aep) (kp) − e2Ae2 <br />
dφ<br />
2(kp)<br />
For sec<strong>on</strong>d order processes <strong>in</strong> the BIP we also need the<br />
fermi<strong>on</strong> propagator <strong>in</strong> the external field, which turns out<br />
to be the normal propagator sandwiched between Volkov<br />
Ep(x) functi<strong>on</strong>s,<br />
(3)<br />
G e (x,x ′ <br />
d4p 1<br />
) = Ep(x) Ēp(x<br />
(2π) 4<br />
/p−m+iǫ<br />
′ ) (4)<br />
These new elements can be <strong>in</strong>cluded <strong>in</strong> Feynman diagrams<br />
<strong>in</strong> the usual way (be<strong>in</strong>g represented by double
pf f<br />
f<br />
pi<br />
p<br />
x2<br />
x1<br />
k<br />
ki<br />
p<br />
pi<br />
Figure 1: The BIP Compt<strong>on</strong> scatter<strong>in</strong>g.<br />
straight l<strong>in</strong>es). In effect, the Volkov Ep(x) functi<strong>on</strong>s can<br />
be grouped together around diagram vertices (with <strong>in</strong>com<strong>in</strong>g<br />
momentum q and outgo<strong>in</strong>g momentum p) <strong>in</strong> order to<br />
”dress” them,<br />
p<br />
kf<br />
ki<br />
γ e µ(p,q) = Ēp(x)γµEq(x) (5)<br />
COMPTON SCATTERING IN THE BOUND<br />
INTERACTION PICTURE<br />
The Volkov soluti<strong>on</strong>s to a fermi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> momentum p embedded<br />
<strong>in</strong> an external periodic field A e reveal an energy<br />
level structure depend<strong>in</strong>g <strong>on</strong> the discrete number s <str<strong>on</strong>g>of</str<strong>on</strong>g> external<br />
field phot<strong>on</strong>s k that <strong>in</strong>teract with the fermi<strong>on</strong> and given<br />
by the dispersi<strong>on</strong> relati<strong>on</strong> [4],<br />
(p ± sk) 2 = m 2 + |eA e | 2<br />
For Compt<strong>on</strong> scatter<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> a phot<strong>on</strong> ki with such a bound<br />
fermi<strong>on</strong> (figure 1) res<strong>on</strong>ant transiti<strong>on</strong>s are possible whenever<br />
the <strong>in</strong>termediate virtual particle reaches the mass shell.<br />
The k<strong>in</strong>ematics for such transiti<strong>on</strong>s are given by the c<strong>on</strong>diti<strong>on</strong><br />
that the propagator denom<strong>in</strong>ator is zero,<br />
(6)<br />
(pi + ki − sk) 2 = 2(pi · ki) − 2sk · (pi + ki) = 0 (7)<br />
If the particles are coll<strong>in</strong>ear then the res<strong>on</strong>ant c<strong>on</strong>diti<strong>on</strong><br />
simplifies to the c<strong>on</strong>diti<strong>on</strong> that the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>com<strong>in</strong>g<br />
phot<strong>on</strong> be an <strong>in</strong>teger multiple <str<strong>on</strong>g>of</str<strong>on</strong>g> the external field phot<strong>on</strong><br />
energy.<br />
RADIATIVE CORRECTIONS<br />
The res<strong>on</strong>ant <strong>in</strong>f<strong>in</strong>ities <strong>in</strong> the tree level process must be<br />
mitigated by <strong>in</strong>clud<strong>in</strong>g radiative correcti<strong>on</strong>s. The schema<br />
is to calculate the same loop diagrams <strong>in</strong> the BIP and to<br />
<strong>in</strong>clude them to all orders <strong>in</strong> a geometric series as <strong>in</strong> the<br />
p<br />
+<br />
k’<br />
p<br />
p<br />
p’<br />
p<br />
k’ p’<br />
+ p + ...<br />
Figure 2: The Corrected Bound fermi<strong>on</strong> Propagator.<br />
case <str<strong>on</strong>g>of</str<strong>on</strong>g> the normal <strong>in</strong>teracti<strong>on</strong> picture (figure 2).<br />
The fermi<strong>on</strong> self energy Σ e p has been calculated for the<br />
case <str<strong>on</strong>g>of</str<strong>on</strong>g> a c<strong>on</strong>stant crossed electromagnetic field [5] and<br />
a circularly polarised field [6]. The sum to all orders is<br />
straightforward and the corrected propagator G e p(rc) is<br />
G e p(rc) = G e p + G e pΣ e pG e p + G e pΣ e pG e pΣ e pG e p + ...<br />
k’<br />
<br />
d4p 1<br />
= Ep(x)<br />
(2π) 4<br />
/p−Σ e Ēp(x<br />
p− m + iǫ<br />
′ ) (8)<br />
There still rema<strong>in</strong>s the questi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> UV divergences <strong>in</strong><br />
the BIP self energy. In the literature, a term equivalent<br />
to the normal <strong>in</strong>teracti<strong>on</strong> picture self energy is separated<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g>f and the usual regularizati<strong>on</strong> and renormalizati<strong>on</strong><br />
procedures are carried out. However it now appears that<br />
the calculati<strong>on</strong> can be carried out without the appearance<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> UV divergences 1 .<br />
We need also to <strong>in</strong>clude <strong>in</strong> the BIP Compt<strong>on</strong> scatter<strong>in</strong>g,<br />
all terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the same order <strong>in</strong> the coupl<strong>in</strong>g c<strong>on</strong>stant <strong>in</strong><br />
the S-matrix expansi<strong>on</strong>. One such term required is the<br />
<strong>in</strong>terference term between the BIP vertex correcti<strong>on</strong> and<br />
the BIP phot<strong>on</strong> radiati<strong>on</strong> (figure 3)<br />
The BIP vertex correcti<strong>on</strong> (figure 4) is a rather complicated<br />
expressi<strong>on</strong> <strong>in</strong>clud<strong>in</strong>g three dressed vertices and <strong>in</strong>f<strong>in</strong>ite<br />
<strong>in</strong>tegrati<strong>on</strong>s (c<strong>on</strong>stant crossed external field) or summati<strong>on</strong>s<br />
(periodic external field) over c<strong>on</strong>tributi<strong>on</strong>s l,r,s<br />
from external field phot<strong>on</strong>s at each vertex,<br />
1 This will be the subject <str<strong>on</strong>g>of</str<strong>on</strong>g> a forthcom<strong>in</strong>g paper by the author<br />
2<br />
↔ ×<br />
Figure 3: A S-matrix <strong>in</strong>terference term <str<strong>on</strong>g>of</str<strong>on</strong>g> the same order<br />
as the BIP Compt<strong>on</strong> scatter<strong>in</strong>g.<br />
p<br />
p’<br />
~
p f<br />
p<br />
i<br />
Figure 4: The BIP vertex correcti<strong>on</strong>.<br />
k f<br />
− ieΓ e µ = 2ie 2<br />
∞<br />
d<br />
−∞<br />
4p (2π) 4 dl dr ds δ4 (pi + lk − pf − kf)<br />
• γ eν (pf,p ′ )<br />
1<br />
/p ′ − m γe µ(p ′ 1<br />
,p)<br />
/p−m γe ν(p,pi) 1<br />
k ′2<br />
where p ′ → p − kf + rk , k ′ → pi − p + sk<br />
Equati<strong>on</strong> 9 has been calculated for special k<strong>in</strong>ematics <strong>in</strong><br />
which the radiated phot<strong>on</strong> is parallel to the external field<br />
wave vector. For such a case no UV divergence exists and<br />
the result<strong>in</strong>g expressi<strong>on</strong> c<strong>on</strong>ta<strong>in</strong>s a term c<strong>on</strong>sistent with the<br />
known expressi<strong>on</strong> for the anomalous magnetic moment <strong>in</strong><br />
a c<strong>on</strong>stant crossed field [7]. Work is c<strong>on</strong>t<strong>in</strong>u<strong>in</strong>g to extend<br />
the calculati<strong>on</strong> to the general case.<br />
CONCLUSION<br />
<strong>Physics</strong> with str<strong>on</strong>g external fields such as those present<br />
<strong>in</strong> <strong>in</strong>tense laser-matter <strong>in</strong>teracti<strong>on</strong>s, at the <strong>in</strong>teracti<strong>on</strong><br />
po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> modern colliders or <strong>in</strong> an astrophysical sett<strong>in</strong>g, is<br />
experimentally <strong>in</strong>terest<strong>in</strong>g and theoretically challeng<strong>in</strong>g.<br />
Calculati<strong>on</strong>s which take <strong>in</strong>to account the effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
external field exactly, are necessary. Such calculati<strong>on</strong>s are<br />
performed <strong>in</strong> the BIP which, <strong>in</strong> effect, dress the vertices <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the corresp<strong>on</strong>d<strong>in</strong>g Feynman diagram.<br />
The transiti<strong>on</strong> rates for BIP processes c<strong>on</strong>ta<strong>in</strong><strong>in</strong>g propagators<br />
<strong>in</strong>dicate the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> res<strong>on</strong>ances which must be<br />
corrected by the <strong>in</strong>clusi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> BIP radiative correcti<strong>on</strong>s. To<br />
date <strong>on</strong>ly the BIP electr<strong>on</strong> self energy (and BIP vacuum<br />
polarizati<strong>on</strong>) have been calculated. In order to correct<br />
higher order tree level processes <strong>in</strong> the BIP, and to prove<br />
that IR divergences cancel assum<strong>in</strong>g a Bloch-Nordiseck<br />
type pro<str<strong>on</strong>g>of</str<strong>on</strong>g>, the BIP vertex correcti<strong>on</strong> is also required.<br />
(9)<br />
These calculati<strong>on</strong>s and their correcti<strong>on</strong>s are underway<br />
and much progress has been made. The ultimate aim is<br />
a realistic calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> BIP Compt<strong>on</strong> scatter<strong>in</strong>g crosssecti<strong>on</strong><br />
at res<strong>on</strong>ance and proposals for the experimental<br />
detecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> such predicted res<strong>on</strong>ances.<br />
REFERENCES<br />
[1] Hart<strong>in</strong> A 2006 PhD thesis University <str<strong>on</strong>g>of</str<strong>on</strong>g> L<strong>on</strong>d<strong>on</strong><br />
[2] Bamber C et al 1999 Phys Rev D 60(9) 092004<br />
[3] Volkov D M 1935 Z Phys 94 250<br />
[4] Zeldovich Y B 1967 Sov Phys JETP 24 1006<br />
[5] Ritus V I 1972 Ann. Phys. D 69 555-582<br />
[6] Becker W, Mitter H 1976 J Phys A 9(12) 2171<br />
[7] Ritus V I 1970 Sov. Phys. JETP 30(6) 1181
STRONG FIELD DYNAMICS IN HEAVY-ION COLLISIONS<br />
Abstract<br />
In high-energy heavy-i<strong>on</strong> collisi<strong>on</strong>s, there appear two<br />
different k<strong>in</strong>ds <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g fields: Str<strong>on</strong>g electromagnetic<br />
fields and str<strong>on</strong>g Yang-Mills (glu<strong>on</strong>) fields. I expla<strong>in</strong> the<br />
mechanisms how they appear <strong>in</strong> the collisi<strong>on</strong>s, the <strong>in</strong>terest<strong>in</strong>g<br />
<strong>in</strong>terplay between QED and QCD, and f<strong>in</strong>ally the<br />
importance <str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>g field dynamics <strong>in</strong> understand<strong>in</strong>g<br />
time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the full collisi<strong>on</strong> events.<br />
INTRODUCTION<br />
The ma<strong>in</strong> goal <str<strong>on</strong>g>of</str<strong>on</strong>g> this talk is to c<strong>on</strong>v<strong>in</strong>ce you that the<br />
high-energy heavy-i<strong>on</strong> collisi<strong>on</strong> (HIC) is the ideal place for<br />
the study <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g-field physics. This is primarily because<br />
they provide the “str<strong>on</strong>gest” fields that human be<strong>in</strong>g can<br />
create. Let us first overview how str<strong>on</strong>g they are. Table 1<br />
shows the comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic fields realized <strong>in</strong> various<br />
situati<strong>on</strong>s from weak to str<strong>on</strong>g fields. The str<strong>on</strong>gest steady<br />
magnetic field <strong>on</strong> earth, B = 4.5 × 10 5 Gauss, is achieved<br />
at Nati<strong>on</strong>al High Magnetic Field Laboratory <strong>in</strong> Florida [1],<br />
but there are several situati<strong>on</strong>s <strong>in</strong> Nature which create much<br />
str<strong>on</strong>ger magnetic fields. Typical examples <strong>in</strong>clude a neutr<strong>on</strong><br />
star which is a remnant <str<strong>on</strong>g>of</str<strong>on</strong>g> supernova explosi<strong>on</strong>. The<br />
magnetic field <str<strong>on</strong>g>of</str<strong>on</strong>g> a huge star is squeezed to a small regi<strong>on</strong><br />
after explosi<strong>on</strong> to become str<strong>on</strong>g B ∼ 10 12 Gauss.<br />
However, it is still weaker than the so-called “critical”<br />
magnetic field Bc = m 2 e/e = 4 × 10 13 Gauss, bey<strong>on</strong>d<br />
which the n<strong>on</strong>l<strong>in</strong>ear QED effects must be fully taken <strong>in</strong>to<br />
account. The corresp<strong>on</strong>d<strong>in</strong>g electric field Ec = m 2 e/e is<br />
called the “Schw<strong>in</strong>ger field” bey<strong>on</strong>d which the e + e − pair<br />
creati<strong>on</strong> from the vacuum becomes possible [2]. Recently,<br />
it has been recognized that some <str<strong>on</strong>g>of</str<strong>on</strong>g> the neutr<strong>on</strong> stars, called<br />
“magnetars”, would have much str<strong>on</strong>ger magnetic fields.<br />
The surface magnetic fields are estimated as ∼ 10 15 Gauss,<br />
well bey<strong>on</strong>d the critical value [3, 4]. So far, this is the<br />
str<strong>on</strong>gest static magnetic fields <strong>in</strong> Nature.<br />
Table 1: Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic fields (Gauss)<br />
Strength Realized as<br />
0.6 Earth’s magnetic field<br />
100 A typical hand-held magnet<br />
8.3×10 4 Superc<strong>on</strong>duct<strong>in</strong>g magnets <strong>in</strong> LHC<br />
4.5×10 5 Str<strong>on</strong>gest steady magnetic field [1]<br />
∼ 10 12 Surface field <str<strong>on</strong>g>of</str<strong>on</strong>g> neutr<strong>on</strong> stars<br />
(4×10 13 Critical magnetic field <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s)<br />
∼ 10 15 Surface field <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars<br />
∼ 10 17 N<strong>on</strong>central heavy-i<strong>on</strong> coll. at RHIC<br />
∼ 10 18 N<strong>on</strong>central heavy-i<strong>on</strong> coll. at LHC<br />
∗ e-mail: kazunori.itakura@kek.jp<br />
K. Itakura ∗<br />
Theory Center, IPNS, <strong>KEK</strong>, Japan<br />
Surpris<strong>in</strong>gly, high-energy HIC’s provide much str<strong>on</strong>ger<br />
magnetic fields. They are created <strong>in</strong> n<strong>on</strong>central collisi<strong>on</strong>s<br />
and the maximum strength amounts to ∼ 10 17 Gauss at<br />
RHIC <strong>in</strong> BNL and ∼ 10 18 Gauss at LHC <strong>in</strong> CERN, far<br />
above the critical value 4 × 10 13 Gauss. S<strong>in</strong>ce the orig<strong>in</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic fields is the fast mov<strong>in</strong>g nuclei with electric<br />
charges, such str<strong>on</strong>g fields last <strong>on</strong>ly for a very short period<br />
(typically <strong>on</strong>ly dur<strong>in</strong>g the passage <str<strong>on</strong>g>of</str<strong>on</strong>g> two collid<strong>in</strong>g nuclei).<br />
However, with the str<strong>on</strong>gest magnetic fields far bey<strong>on</strong>d the<br />
critical value, we expect many <strong>in</strong>terest<strong>in</strong>g phenomena related<br />
to n<strong>on</strong>l<strong>in</strong>ear QED to occur, as I will expla<strong>in</strong> later. In<br />
particular, it is quite <strong>in</strong>terest<strong>in</strong>g to see how the str<strong>on</strong>g magnetic<br />
field affects the Quark Glu<strong>on</strong> Plasma (QGP) which<br />
has also a very short life time.<br />
On the other hand, HIC’s generate another str<strong>on</strong>g<br />
fields: “color” electromagnetic fields described by Quantum<br />
Chromodynamics (QCD). The ma<strong>in</strong> motivati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
HIC’s is to create the QGP by liberat<strong>in</strong>g subatomic degrees<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> freedom, quarks and glu<strong>on</strong>s, from <strong>in</strong>side <str<strong>on</strong>g>of</str<strong>on</strong>g> nucle<strong>on</strong>s.<br />
QCD is the fundamental theory for the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> quarks<br />
and glu<strong>on</strong>s and is the n<strong>on</strong>-Abelian gauge theory with<br />
“color” SU(3) symmetry. Therefore, there exist “color”<br />
electromagnetic fields which are n<strong>on</strong>-Abela<strong>in</strong> analogs <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the ord<strong>in</strong>ary electromagnetic fields. High-energy HIC’s can<br />
create very str<strong>on</strong>g color electromagnetic fields. S<strong>in</strong>ce <strong>on</strong>e<br />
cannot directly compare the strength <str<strong>on</strong>g>of</str<strong>on</strong>g> color fields with<br />
that <str<strong>on</strong>g>of</str<strong>on</strong>g> ord<strong>in</strong>ary electromagnetic fields, let us compare them<br />
<strong>in</strong> unit <str<strong>on</strong>g>of</str<strong>on</strong>g> energy. Table 2 is the comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the ord<strong>in</strong>ary<br />
and color electromagnetic fields. For the electromagnetic<br />
fields we show √ eB <strong>in</strong> unit <str<strong>on</strong>g>of</str<strong>on</strong>g> MeV, and for color electromagnetic<br />
fields √ gB <strong>in</strong> the same energy scale where g<br />
is the coupl<strong>in</strong>g strength and B is the color magnetic fields.<br />
Notice that the critical magnetic field <strong>in</strong> QED is determ<strong>in</strong>ed<br />
by the electr<strong>on</strong> mass √ eBc = me, but the corresp<strong>on</strong>d<strong>in</strong>g<br />
field <strong>in</strong> QCD will be determ<strong>in</strong>ed by quark mass mq. Still,<br />
the strength √ gB ∼ 1 GeV at RHIC is well bey<strong>on</strong>d the<br />
critical value mq ∼ a few MeV, suggest<strong>in</strong>g the importance<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Schw<strong>in</strong>ger mechanism <str<strong>on</strong>g>of</str<strong>on</strong>g> q¯q pairs. Also we expect that<br />
the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> such str<strong>on</strong>g color fields is crucial for the<br />
early time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the created matter to the QGP.<br />
Table 2: √ eB for electromagnetic fields (EM) and √ gB<br />
for Yang-Mills fields (YM) <strong>in</strong> unit <str<strong>on</strong>g>of</str<strong>on</strong>g> energy (MeV)<br />
Strength Realized as<br />
0.5 (= me) Critical mag. field <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s (EM)<br />
∼ 2 − 3 Surface field <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars (EM)<br />
∼ 10 2 (∼ mπ) N<strong>on</strong>central HIC at RHIC (EM)<br />
∼ 4 × 10 2 N<strong>on</strong>central HIC at LHC (EM)<br />
∼ 10 3 (∼ Qs) Color magnetic fields at RHIC (YM)<br />
(2 − 3)×10 3 Color magnetic fields at LHC (YM)
Figure 1: (Left) Time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the matter created <strong>in</strong> high-energy heavy-i<strong>on</strong> collisi<strong>on</strong>s. (Right) N<strong>on</strong>central collisi<strong>on</strong>s.<br />
HIGH-ENERGY HEAVY-ION<br />
COLLISIONS<br />
As already menti<strong>on</strong>ed, the ma<strong>in</strong> motivati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the HIC’s<br />
is to create the QGP, a local equilibrium matter made <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
quarks and glu<strong>on</strong>s. In order to throw gigantic k<strong>in</strong>etic energies<br />
<strong>in</strong>to a small but f<strong>in</strong>ite volume, we need to collide<br />
‘large’ objects, namely two heavy i<strong>on</strong>s (bare nuclei without<br />
electr<strong>on</strong>s). Thus, the spatial extent <str<strong>on</strong>g>of</str<strong>on</strong>g> the created state<br />
is, <strong>in</strong>itially, <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> that <str<strong>on</strong>g>of</str<strong>on</strong>g> collid<strong>in</strong>g nuclei (e.g., for<br />
Au nuclei, about 10 fm = 10 −14 m). If the energy density<br />
is high enough to reach a very high temperature bey<strong>on</strong>d<br />
the critical value Tc ∼ 170 MeV, <strong>on</strong>e can naively expect<br />
that the QGP will be formed. In reality, however, the created<br />
state with high-energy densities will show quite n<strong>on</strong>trivial<br />
time evoluti<strong>on</strong>: It will quickly change <strong>in</strong>to a hightemperature<br />
QGP through <strong>in</strong>teracti<strong>on</strong>s between quarks and<br />
glu<strong>on</strong>s. Then, the QGP will cool down to hadr<strong>on</strong>ic matter<br />
as it rapidly expands (see Fig. 1, left). All these processes<br />
take place dur<strong>in</strong>g a very short time, and the life time <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
QGP is, at most, <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 fm/c = 10 −22 sec.<br />
In order to study the properties <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP, we need to extract<br />
<strong>in</strong>formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP from the huge number <str<strong>on</strong>g>of</str<strong>on</strong>g> f<strong>in</strong>al<br />
hadr<strong>on</strong>s. S<strong>in</strong>ce particle producti<strong>on</strong> can, <strong>in</strong> pr<strong>in</strong>ciple, occur<br />
at every stage <str<strong>on</strong>g>of</str<strong>on</strong>g> evoluti<strong>on</strong>, it is quite important to understand<br />
the evoluti<strong>on</strong> history <str<strong>on</strong>g>of</str<strong>on</strong>g> the created matter <strong>in</strong> HIC’s.<br />
In particular, the formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP is <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the most difficult<br />
problems <strong>in</strong> QCD (or maybe <strong>in</strong> theoretical physics) because<br />
it requires n<strong>on</strong>-perturbative descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-l<strong>in</strong>ear<br />
phenomena <strong>in</strong> n<strong>on</strong>-equilibrium states. However, many people<br />
believe that the str<strong>on</strong>g field dynamics should be relevant<br />
at least for the early time evoluti<strong>on</strong>. In fact, as already menti<strong>on</strong>ed,<br />
there emerge two different k<strong>in</strong>ds <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g fields <strong>in</strong><br />
HIC’s, and the “color” electromagnetic fields are expected<br />
to play important roles for the transiti<strong>on</strong> towards QGP. The<br />
str<strong>on</strong>g color electr<strong>on</strong>ic and magnetic fields give rise to the<br />
quark-antiquark pair producti<strong>on</strong> (Schw<strong>in</strong>ger mechanism)<br />
and Nielsen-Olesen <strong>in</strong>stability, respectively. We expect<br />
both <str<strong>on</strong>g>of</str<strong>on</strong>g> them will c<strong>on</strong>tribute to drive the system towards<br />
thermalizati<strong>on</strong>. On the other hand, ord<strong>in</strong>ary magnetic fields<br />
<strong>in</strong>deed have a significant <strong>in</strong>fluence <strong>on</strong> the created matter,<br />
but would be irrelevant for thermalizati<strong>on</strong>, because it does<br />
not appear <strong>in</strong> the central collisi<strong>on</strong>.<br />
STRONG MAGNETIC FIELDS IN<br />
NONCENTRAL HEAVY-ION COLLISIONS<br />
How do they appear?<br />
Note that the atomic nuclei used <strong>in</strong> HIC’s have large<br />
electric charges 1 Ze and they are mov<strong>in</strong>g at very fast speed<br />
close to the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light. Then, the Coulomb field (electric<br />
field) <str<strong>on</strong>g>of</str<strong>on</strong>g> each nucleus is highly Lorentz c<strong>on</strong>tracted to a<br />
str<strong>on</strong>g spike. Time variati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric fields <strong>in</strong>duces<br />
magnetic fields <str<strong>on</strong>g>of</str<strong>on</strong>g> the same strength. For example, c<strong>on</strong>sider<br />
a po<strong>in</strong>t charge mov<strong>in</strong>g <strong>on</strong> the z axis at rapidity (velocity)<br />
Y . Then, the magnetic field at the positi<strong>on</strong> ⃗x is given by<br />
e ⃗ B(⃗x) = Zα EM<br />
−(⃗x⊥ × ⃗ez) s<strong>in</strong>h Y<br />
[(⃗x⊥) 2 + (t s<strong>in</strong>h Y − z cosh Y ) 2 ,<br />
3/2<br />
]<br />
which has a large enhancement factor s<strong>in</strong>h Y . One can<br />
roughly estimate the magnetic field <strong>in</strong> n<strong>on</strong>central HIC’s<br />
(b ̸= 0) by simply summ<strong>in</strong>g the magnetic fields given<br />
above assum<strong>in</strong>g the trajectories <str<strong>on</strong>g>of</str<strong>on</strong>g> two collid<strong>in</strong>g nuclei<br />
shown <strong>in</strong> Fig. 1, right. Then, <strong>on</strong>e f<strong>in</strong>ds the maximum<br />
strength √ eB ∼ 100 MeV <strong>in</strong> n<strong>on</strong>central Au-Au collisi<strong>on</strong>s<br />
at RHIC ( √ sNN = 200 GeV) 2 when the impact parameter<br />
b ∼ 10 fm [5]. In fact, such a crude estimate is numerically<br />
c<strong>on</strong>firmed by the UrQMD simulati<strong>on</strong> [6]. As already<br />
commented, this is quite a str<strong>on</strong>g magnetic field, bey<strong>on</strong>d 3<br />
the “critical” value √ eBc = me = 0.5 MeV. S<strong>in</strong>ce this<br />
magnetic field arises <strong>on</strong>ly from the k<strong>in</strong>ematical c<strong>on</strong>figurati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> two charges, it becomes str<strong>on</strong>gest when two charges<br />
come closest, and decays rapidly as they recede from each<br />
other. Typically, it lasts <strong>on</strong>ly for a few fm/c. However, it is<br />
argued that the life time becomes l<strong>on</strong>ger <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the QGP, because the QGP has a large electric c<strong>on</strong>ductivity<br />
[7]. If this is <strong>in</strong>deed the case, life time <str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>g magnetic<br />
field could be <str<strong>on</strong>g>of</str<strong>on</strong>g> the same order <str<strong>on</strong>g>of</str<strong>on</strong>g> that <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP,<br />
namely, about 10 fm/c.<br />
1 Z = 79 (Au) at RHIC, 82 (Pb) at LHC and we take e > 0.<br />
2 √ sNN is the center-<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass energy per nucle<strong>on</strong>-nucle<strong>on</strong> collisi<strong>on</strong>.<br />
The total energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the Au-Au collisi<strong>on</strong> is A × √ sNN with A = 197<br />
be<strong>in</strong>g the mass number <str<strong>on</strong>g>of</str<strong>on</strong>g> Au.<br />
3 It makes sense to discuss the magnetic field bey<strong>on</strong>d the critical value,<br />
because, unlike the electric field, the vacuum does not break down. However,<br />
perturbative calculati<strong>on</strong> ‘breaks down’ because the <strong>in</strong>serti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> n<br />
external magnetic fields <strong>in</strong> Feynman diagrams c<strong>on</strong>tributes as (eB/m 2 e )n<br />
which becomes order 1 for B > Bc = m 2 e/e [3].
Possible effects <str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>g magnetic fields<br />
Even if the magnetic field lasts <strong>on</strong>ly for a short period,<br />
it is extraord<strong>in</strong>arily str<strong>on</strong>g and should have a large impact<br />
<strong>on</strong> dynamics. What <strong>on</strong>e would immediately expect<br />
is the n<strong>on</strong>l<strong>in</strong>ear QED effect. This has been discussed for a<br />
l<strong>on</strong>g time <strong>in</strong> “very peripheral” collisi<strong>on</strong>s b > 2RA (RA<br />
is the nuclear radius) [8]. In this case, the impact parameter<br />
b is too large for two nuclei to touch each other,<br />
but multiphot<strong>on</strong> exchange will occur due to large electric<br />
charges <str<strong>on</strong>g>of</str<strong>on</strong>g> the nuclei that compensate the smallness <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
α EM = e 2 /4π = 1/137. On the other hand, <strong>in</strong> the “n<strong>on</strong>central”<br />
collisi<strong>on</strong>s 0 < b < 2RA, two nuclei <strong>in</strong>deed touch<br />
each other, and the matter <strong>in</strong> the reacti<strong>on</strong> regi<strong>on</strong> will turn<br />
<strong>in</strong>to a QGP if the collisi<strong>on</strong> energy is large enough. Thus,<br />
<strong>in</strong> this case, we can study properties <str<strong>on</strong>g>of</str<strong>on</strong>g> a QGP <strong>in</strong> a str<strong>on</strong>g<br />
magnetic field as shown <strong>in</strong> Fig. 1, right (the orange ellipsoid<br />
represents a QGP). Below we discuss possible observable<br />
effects <str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>g magnetic field <strong>on</strong> the QGP.<br />
Glu<strong>on</strong> synchrotr<strong>on</strong> radiati<strong>on</strong><br />
First <str<strong>on</strong>g>of</str<strong>on</strong>g> all, notice that quarks/antiquarks have electric<br />
charges (e.g., eu = (2/3)e and ed = −(1/3)e for up<br />
and down quarks), while glu<strong>on</strong>s d<strong>on</strong>’t. Thus, the str<strong>on</strong>g<br />
magnetic field primarily affects quarks and antiquarks and<br />
<strong>in</strong>duces synchrotr<strong>on</strong> radiati<strong>on</strong>s [7]. Because <str<strong>on</strong>g>of</str<strong>on</strong>g> the largeness<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the QCD coupl<strong>in</strong>g, αs ≡ g 2 /4π ≫ αEM, radiati<strong>on</strong>s<br />
are predom<strong>in</strong>antly due to glu<strong>on</strong>s, <strong>in</strong>stead <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s<br />
as usual. Therefore, this phenomen<strong>on</strong> is a typical <strong>in</strong>terplay<br />
between QED and QCD. One <str<strong>on</strong>g>of</str<strong>on</strong>g> the observable effects<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the glu<strong>on</strong> synchrotr<strong>on</strong> radiati<strong>on</strong>s is the energy loss <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
quarks and antiquarks. Usually, energy loss <str<strong>on</strong>g>of</str<strong>on</strong>g> fast mov<strong>in</strong>g<br />
quarks/antiquarks is assumed to be due to <strong>in</strong>teracti<strong>on</strong> with a<br />
hot medium. So, the synchrotr<strong>on</strong> radiati<strong>on</strong> is c<strong>on</strong>sidered as<br />
another (less familiar) source <str<strong>on</strong>g>of</str<strong>on</strong>g> the energy loss. However,<br />
this is not just a correcti<strong>on</strong> to the ord<strong>in</strong>ary c<strong>on</strong>tributi<strong>on</strong>: In<br />
fact, it turned out that it is large enough to be comparable<br />
with the experimentally measured value [7]. For example,<br />
the energy loss per unit length due to synchrotr<strong>on</strong><br />
radiati<strong>on</strong> is −∆E/ℓ ∼ 0.2 ÷ 0.35 GeV/fm at RHIC and<br />
−∆E/ℓ ∼ 1.5 ÷ 2 GeV/fm at LHC, both <str<strong>on</strong>g>of</str<strong>on</strong>g> which are for<br />
up quarks with transverse momentum p⊥ = 10 ÷ 20 GeV.<br />
Another possible observable effects is the angular distributi<strong>on</strong><br />
specific to the synchrotr<strong>on</strong> radiati<strong>on</strong>, which is also<br />
discussed <strong>in</strong> Ref. [7].<br />
Phot<strong>on</strong> decay and phot<strong>on</strong> splitt<strong>in</strong>g<br />
Other n<strong>on</strong>l<strong>in</strong>ear QED effects <strong>in</strong>clude phot<strong>on</strong> decay <strong>in</strong>to<br />
fermi<strong>on</strong> pairs and phot<strong>on</strong> splitt<strong>in</strong>g. Note that several different<br />
k<strong>in</strong>ds <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s are produced <strong>in</strong> HIC’s. It is quite<br />
important to identify the orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s to understand<br />
the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> HIC’s: Direct phot<strong>on</strong>s are produced by<br />
hard collisi<strong>on</strong>s at the moment <str<strong>on</strong>g>of</str<strong>on</strong>g> HIC’s and thus carry<br />
the earliest-time <strong>in</strong>formati<strong>on</strong> before the formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP.<br />
Thermal phot<strong>on</strong>s are produced from the QGP. F<strong>in</strong>ally, decay<br />
phot<strong>on</strong>s are produced by the decay <str<strong>on</strong>g>of</str<strong>on</strong>g>, say, π 0 at the last<br />
stage <str<strong>on</strong>g>of</str<strong>on</strong>g> HIC’s. So far, all the analyses <strong>in</strong> relati<strong>on</strong> to pho-<br />
t<strong>on</strong>s assume that the produced phot<strong>on</strong>s do not <strong>in</strong>teract with<br />
the QGP and simply carry the <strong>in</strong>formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the matter<br />
when they are emitted. However, <strong>in</strong> a str<strong>on</strong>g magnetic field,<br />
this assumpti<strong>on</strong> is no l<strong>on</strong>ger true. In order to correctly extract<br />
the <strong>in</strong>formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the evolv<strong>in</strong>g matter, we have to take<br />
<strong>in</strong>to account the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g magnetic fields which will<br />
be present until the end <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP.<br />
In the vacuum, a real phot<strong>on</strong> cannot decay <strong>in</strong>to either a<br />
lept<strong>on</strong> pair or two phot<strong>on</strong>s. However, <strong>in</strong> a str<strong>on</strong>g magnetic<br />
field, both become possible because charged fermi<strong>on</strong>s are<br />
‘dressed’ by the magnetic field. In the vacuum, it is known<br />
that a fermi<strong>on</strong> loop with odd number <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s attached<br />
is zero (Furry’s theorem). In the magnetic field, however,<br />
it is not zero because the fermi<strong>on</strong> is ‘dressed’ and <strong>in</strong>cludes<br />
many external l<strong>in</strong>es <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field. Hence these are<br />
both characteristic phenomena <str<strong>on</strong>g>of</str<strong>on</strong>g> the n<strong>on</strong>l<strong>in</strong>ear QED.<br />
The cross secti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> decay <strong>in</strong>to a fermi<strong>on</strong> pair<br />
is easily obta<strong>in</strong>ed by chang<strong>in</strong>g k<strong>in</strong>ematical variables <strong>in</strong> the<br />
results <str<strong>on</strong>g>of</str<strong>on</strong>g> the synchrotr<strong>on</strong> radiati<strong>on</strong>, but <strong>in</strong> the present case,<br />
the e + e − pair producti<strong>on</strong> (<strong>in</strong>stead <str<strong>on</strong>g>of</str<strong>on</strong>g> a q¯q pair) is the dom<strong>in</strong>ant<br />
process. Decay rate <str<strong>on</strong>g>of</str<strong>on</strong>g> a real phot<strong>on</strong> <strong>in</strong> HIC’s <strong>in</strong>clud<strong>in</strong>g<br />
several channels (γ → ℓ + ℓ − or q¯q) was computed <strong>in</strong><br />
Ref. [9]. It was also po<strong>in</strong>ted out that such effects would<br />
generate an asymmetry <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> emissi<strong>on</strong> with respect<br />
to the reacti<strong>on</strong> plane (def<strong>in</strong>ed by two collid<strong>in</strong>g nuclei, see<br />
Fig. 1, right). If a phot<strong>on</strong> is produced <strong>in</strong> the directi<strong>on</strong> close<br />
to the vertical axis <str<strong>on</strong>g>of</str<strong>on</strong>g> the reacti<strong>on</strong> plane, the decay rate is<br />
large. Thus, we will see less phot<strong>on</strong>s <strong>in</strong> the vertical directi<strong>on</strong>s,<br />
which generates asymmetry to be measured as elliptic<br />
flow. The calculati<strong>on</strong> was d<strong>on</strong>e <strong>in</strong> a c<strong>on</strong>stant homogeneous<br />
magnetic field, but the asymmetry will be enhanced<br />
if <strong>on</strong>e <strong>in</strong>cludes the <strong>in</strong>homogeneity <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field as<br />
shown <strong>in</strong> Fig. 1, right. On the other hand, so far, there<br />
is no calculati<strong>on</strong> about the phot<strong>on</strong> splitt<strong>in</strong>g <strong>in</strong> HIC’s. One<br />
naively expects that this can be measured as anomalous enhancement<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> spectrum <strong>in</strong> the regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> smaller<br />
energies, because the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the parent phot<strong>on</strong> is shared<br />
by two s<str<strong>on</strong>g>of</str<strong>on</strong>g>ter phot<strong>on</strong>s.<br />
Chiral magnetic effects<br />
The last example is the chiral magnetic effect [5, 10].<br />
This is not the n<strong>on</strong>l<strong>in</strong>ear QED effect, but shows an <strong>in</strong>terest<strong>in</strong>g<br />
<strong>in</strong>terplay between QED and QCD. First <str<strong>on</strong>g>of</str<strong>on</strong>g> all, let us<br />
recall that quarks/antiquarks have handedness. For example,<br />
the sp<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a right-handed quark is parallel to its momentum,<br />
while that <str<strong>on</strong>g>of</str<strong>on</strong>g> a left-handed quark is anti-parallel.<br />
The chirality N5 <str<strong>on</strong>g>of</str<strong>on</strong>g> a matter is def<strong>in</strong>ed by the difference<br />
between the numbers <str<strong>on</strong>g>of</str<strong>on</strong>g> right- and left-handed particles:<br />
(<br />
) (<br />
)<br />
N5 ≡ N(qR) + N(¯qR) − N(qL) + N(¯qL) ,<br />
where N(qR) represents the number <str<strong>on</strong>g>of</str<strong>on</strong>g> right-handed<br />
quarks, and similarly for the others. If a str<strong>on</strong>g magnetic<br />
field is present, sp<strong>in</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> quarks po<strong>in</strong>t to the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
magnetic field, irrespectively <str<strong>on</strong>g>of</str<strong>on</strong>g> their handedness. When<br />
the numbers <str<strong>on</strong>g>of</str<strong>on</strong>g> left- and right-handed quarks are the same,<br />
no net current is generated <strong>in</strong> that directi<strong>on</strong> because the
momenta <str<strong>on</strong>g>of</str<strong>on</strong>g> left and right quarks are <strong>in</strong> the opposite directi<strong>on</strong>s.<br />
However, if an excitati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Yang-Mills field<br />
with a topological number change occurs, the handedness<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quarks, and c<strong>on</strong>sequently the chirality <str<strong>on</strong>g>of</str<strong>on</strong>g> the matter will<br />
change ∆N5 = N5(t = ∞) − N5(t = −∞) = −2Q with<br />
Q be<strong>in</strong>g the topological number change. This means that a<br />
net charge current will flow <strong>in</strong> the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic<br />
field. In HIC’s, as I will discuss <strong>in</strong> the next secti<strong>on</strong>, there<br />
appears a n<strong>on</strong>trivial Yang-Mills c<strong>on</strong>figurati<strong>on</strong> with n<strong>on</strong>zero<br />
topological charge. Therefore, if the str<strong>on</strong>g magnetic field<br />
and the topological number chang<strong>in</strong>g Yang-Mills field are<br />
both present, then an electric charge current will flow perpendicular<br />
to the reacti<strong>on</strong> plane, generat<strong>in</strong>g asymmetry <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
charge distributi<strong>on</strong>. The str<strong>on</strong>g magnetic field works to effectively<br />
align the sp<strong>in</strong>. More details are discussed <strong>in</strong> the<br />
talk by K. Fukushima [10].<br />
STRONG COLOR-ELECTROMAGNETIC<br />
FIELDS: CGC AND GLASMA<br />
Now let us turn to the color electromagnetic field <strong>in</strong> HIC<br />
that orig<strong>in</strong>ates from a huge number <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s <strong>in</strong>herent <strong>in</strong><br />
the nuclei before collisi<strong>on</strong>. Below, I expla<strong>in</strong> first how such<br />
a state with many glu<strong>on</strong>s (called ‘Color Glass C<strong>on</strong>densate’,<br />
CGC) appears at high energies, then how the <strong>in</strong>formati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> CGC is transferred to the str<strong>on</strong>g color fields created <strong>in</strong><br />
HIC (called ‘glasma’), and f<strong>in</strong>ally how the glasma evolves<br />
towards the QGP.<br />
From CGC to glasma: How do they appear?<br />
What is CGC?<br />
C<strong>on</strong>sider <strong>on</strong>e nucleus that is mov<strong>in</strong>g very fast <strong>in</strong> the z<br />
directi<strong>on</strong>. A nucleus is made <str<strong>on</strong>g>of</str<strong>on</strong>g> nucle<strong>on</strong>s, and they are further<br />
made <str<strong>on</strong>g>of</str<strong>on</strong>g> three quarks. Such a ‘valence’ picture works<br />
well <strong>on</strong>ly at low energies or when we use ‘low resoluti<strong>on</strong>’<br />
probes. On the other hand, if <strong>on</strong>e <strong>in</strong>creases the scatter<strong>in</strong>g<br />
energy, <strong>on</strong>e will <strong>in</strong>stead see a state with a huge number <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
glu<strong>on</strong>s. These glu<strong>on</strong>s are emitted either directly by the valence<br />
quarks or successively by (already emitted) glu<strong>on</strong>s<br />
(see Fig. 2). Such a highly dense glu<strong>on</strong>ic state is called the<br />
CGC (see Ref. [11] for a review), and is <strong>in</strong>deed observed<br />
experimentally <strong>in</strong> the electr<strong>on</strong> deep <strong>in</strong>elastic scatter<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g>f<br />
a prot<strong>on</strong>. It is physically important to dist<strong>in</strong>guish the roles<br />
played by large and small x c<strong>on</strong>stituents (x is the fracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
momentum carried by a c<strong>on</strong>stituent). Large x c<strong>on</strong>stituents<br />
(valence quarks and glu<strong>on</strong>s hav<strong>in</strong>g large momentum fracti<strong>on</strong>)<br />
are distributed <strong>on</strong> a Lorentz-c<strong>on</strong>tracted nucleus and<br />
their moti<strong>on</strong> is very slow compared to the collisi<strong>on</strong> time<br />
scale. Thus we treat them altogether as a static color source<br />
<strong>on</strong> a transverse disk ρ a (x⊥). We also assume that ρ a can<br />
be taken as random reflect<strong>in</strong>g the unpredictable c<strong>on</strong>figurati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> colored c<strong>on</strong>stituents at the moment <str<strong>on</strong>g>of</str<strong>on</strong>g> collisi<strong>on</strong>. On<br />
the other hand, with <strong>in</strong>creas<strong>in</strong>g energy (corresp<strong>on</strong>d<strong>in</strong>g to<br />
go<strong>in</strong>g to smaller x), glu<strong>on</strong>s with smaller x are successively<br />
created. Then, small x c<strong>on</strong>stituents (mostly glu<strong>on</strong>s) are collectively<br />
described as a coherent radiati<strong>on</strong> field A a µ created<br />
Figure 2: Emergence <str<strong>on</strong>g>of</str<strong>on</strong>g> CGC at high energies. With <strong>in</strong>creas<strong>in</strong>g<br />
energies (from left to right), multi-glu<strong>on</strong> producti<strong>on</strong><br />
occurs, which eventually leads to high-density saturated<br />
glu<strong>on</strong> matter, CGC.<br />
by the color source ρ a . Namely, we treat<br />
(DνF νµ ) a = J µ , J µ = δ µ+ δ(x − )ρ a (x⊥) , (1)<br />
where J µ is the current <strong>in</strong>duced by a fast-mov<strong>in</strong>g color<br />
charge distributi<strong>on</strong> ρ a (x⊥) (its trajectory is taken as x − =<br />
(z −vt)/ √ 2 = 0 when v ≃ c) and F a µν = ∂µA a ν −∂νA a µ −<br />
gf abc A b µA b ν is the field strength. The coherent glu<strong>on</strong> field<br />
A a µ is a n<strong>on</strong>-Abelian analog <str<strong>on</strong>g>of</str<strong>on</strong>g> the Weizsäcker-Williams<br />
field (or the boosted Coulomb field) <str<strong>on</strong>g>of</str<strong>on</strong>g> a mov<strong>in</strong>g electric<br />
charge. Hence, the collid<strong>in</strong>g nuclei at high energies are<br />
necessarily accompanied by str<strong>on</strong>g glu<strong>on</strong> fields.<br />
Most <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s <strong>in</strong> CGC have relatively large transverse<br />
momentum called the saturati<strong>on</strong> momentum, Qs ≫ ΛQCD<br />
which corresp<strong>on</strong>ds to (the <strong>in</strong>verse <str<strong>on</strong>g>of</str<strong>on</strong>g>) a typical transverse<br />
‘size’ <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s when they fill up the transverse nuclear<br />
disk (see Fig. 2). One can compute <strong>in</strong> QCD the energy (or<br />
x) and atomic mass number A dependences <str<strong>on</strong>g>of</str<strong>on</strong>g> Qs as<br />
Q 2 s(x, A) ∝ A 1/3 (1/x) λ , λ ≃ 0.3 , (2)<br />
which is <strong>in</strong>deed c<strong>on</strong>sistent with the x and A dependences <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
experimentally determ<strong>in</strong>ed Qs. One expects Q 2 s ∼ 1 GeV<br />
at RHIC ( √ sNN = 200 GeV, Au), while it <strong>in</strong>creases by a<br />
factor 3 at LHC ( √ sNN = 5.5 TeV, Pb). Note also that<br />
1/Qs corresp<strong>on</strong>ds to the correlati<strong>on</strong> length <strong>on</strong> the trans-<br />
verse disk with the correlati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> two glu<strong>on</strong>s roughly given<br />
by ∼ e−Q2s r2 ⊥. This means that the color field is regarded<br />
as correlated/ordered with<strong>in</strong> this correlati<strong>on</strong> length.<br />
One can roughly estimate the field strength <str<strong>on</strong>g>of</str<strong>on</strong>g> the CGC.<br />
When the CGC is realized, n<strong>on</strong>l<strong>in</strong>earity <strong>in</strong> the Yang-Mills<br />
theory is crucial. For example, the field strength F a µν c<strong>on</strong>ta<strong>in</strong>s<br />
both ∂A and gAA terms, and these two become <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the same magnitude ∂A ∼ gAA <strong>in</strong> the CGC. S<strong>in</strong>ce the<br />
typical momentum scale <strong>in</strong> CGC is given by Qs, <strong>on</strong>e f<strong>in</strong>ds<br />
A ∼ Qs/g that yields very str<strong>on</strong>g color electric (E) and<br />
magnetic (B) fields: E, B ∼ Q2 s/g.
Figure 3: Flux tube structure <str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma<br />
What is glasma?<br />
It is now obvious that the high-energy HIC should be described<br />
as a collisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> two CGC’s. In the center-<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass<br />
frame (see Fig. 1, left), the current <strong>in</strong> eq. (1) is replaced by<br />
J µ = δ µ+ δ(x − )ρ1(x⊥) + δ µ− δ(x + )ρ2(x⊥) with ρ1 (ρ2)<br />
be<strong>in</strong>g a color source <str<strong>on</strong>g>of</str<strong>on</strong>g> the right (left) mov<strong>in</strong>g nucleus 1 (2).<br />
After the collisi<strong>on</strong> at t = z = 0, two nuclei receed from<br />
each other still at very high speed, but there appears a highenergy-density<br />
matter <strong>in</strong> between these two. This matter<br />
is predom<strong>in</strong>antly made <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s reflect<strong>in</strong>g the CGC state<br />
before collisi<strong>on</strong> and we expect it to evolve towards QGP.<br />
This transiti<strong>on</strong>al state is now called the ‘glasma’ (= glass<br />
+ plasma) [12]. We describe the very early stage <str<strong>on</strong>g>of</str<strong>on</strong>g> time<br />
evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma by solv<strong>in</strong>g source free Yang-Mills<br />
equati<strong>on</strong>s <strong>in</strong> the forward light c<strong>on</strong>e, with the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong><br />
specified by the CGC fields <str<strong>on</strong>g>of</str<strong>on</strong>g> each nucleus. In short,<br />
the glasma is a weakly-coupled 4 str<strong>on</strong>g Yang-Mills field<br />
that exhibits quite n<strong>on</strong>trivial time evoluti<strong>on</strong> towards QGP<br />
due to n<strong>on</strong>l<strong>in</strong>ear <strong>in</strong>teracti<strong>on</strong>s.<br />
Flux tube structure <strong>in</strong> glasma<br />
Let us discuss a characteristic structure <str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma.<br />
Before the collisi<strong>on</strong>, each CGC has purely transverse 5 E i<br />
and B i that are orthog<strong>on</strong>al to each other E · B = E⊥ · B⊥ =<br />
0. However, just after the collisi<strong>on</strong>, the field strength suddenly<br />
becomes purely l<strong>on</strong>gitud<strong>in</strong>al [12]. Indeed, the zcomp<strong>on</strong>ents<br />
at t = 0 + are given by<br />
E z | τ=0 + = −ig[α i 1, α i 2], B z | τ=0 + = igϵij[α i 1, α j<br />
2 ] , (3)<br />
with α1,2 be<strong>in</strong>g the CGC fields <strong>in</strong> matrix representati<strong>on</strong><br />
αi = α a i T a , while all the transverse comp<strong>on</strong>ents are vanish<strong>in</strong>g.<br />
Notice that the glu<strong>on</strong> field just after the collisi<strong>on</strong> is<br />
completely determ<strong>in</strong>ed by the CGC fields. In other words,<br />
CGC provides the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> HIC’s. This also<br />
implies that the glasma field is <strong>in</strong>itially very str<strong>on</strong>g and the<br />
l<strong>on</strong>gitud<strong>in</strong>al color electromagnetic fields E z and B z are <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the order <str<strong>on</strong>g>of</str<strong>on</strong>g> Q 2 s/g, the same order as those <str<strong>on</strong>g>of</str<strong>on</strong>g> the CGC<br />
fields. Some comments are <strong>in</strong> order:<br />
4 Recall that most <str<strong>on</strong>g>of</str<strong>on</strong>g> the glu<strong>on</strong>s <strong>in</strong> CGC have momenta around Qs,<br />
and Qs <strong>in</strong>creases with <strong>in</strong>creas<strong>in</strong>g energy. Thus, <strong>on</strong>e can treat the CGC <strong>in</strong><br />
weak-coupl<strong>in</strong>g technique αs(Qs) ≪ 1. This is true for the glasma, too.<br />
5 Only the x and y-comp<strong>on</strong>ents are n<strong>on</strong>zero, i.e., E i = (E x , E y , 0).<br />
Figure 4: Dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> color flux tubes.<br />
• In the high-energy limit, there is no z-dependence 6 <strong>in</strong><br />
the l<strong>on</strong>gitud<strong>in</strong>al fields E z and B z , reflect<strong>in</strong>g the fact<br />
that the CGC is c<strong>on</strong>tracted to an <strong>in</strong>f<strong>in</strong>itly th<strong>in</strong> disk.<br />
• The l<strong>on</strong>gitud<strong>in</strong>al fields are correlated <strong>on</strong> the transverse<br />
plane with<strong>in</strong> the length scale 1/Qs, reflect<strong>in</strong>g the random<br />
c<strong>on</strong>figurati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the color source <strong>on</strong> the disk.<br />
• S<strong>in</strong>ce both E z and B z are <strong>in</strong> general n<strong>on</strong>zero, the product<br />
is n<strong>on</strong>vanish<strong>in</strong>g E · B ̸= 0, suggest<strong>in</strong>g the (local)<br />
emergence <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>zero topological charge.<br />
The first two imply that the glasma has a flux tube structure<br />
as shown <strong>in</strong> Fig. 3. Unlike the color flux tube c<strong>on</strong>nect<strong>in</strong>g<br />
a quark and an antiquark, the glasma flux tube can have a<br />
magnetic field <strong>in</strong> it. In fact, even a purely magnetic flux<br />
tube is possible if <strong>on</strong>e takes appropriate color structure <strong>in</strong><br />
eq. (3). Of course, a purely electric flux tube is also possible.<br />
The third comment is related to the chiral magnetic effects<br />
discussed <strong>in</strong> the previous secti<strong>on</strong>. The glasma allows<br />
local fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the topological charge, which could be<br />
measured with the help <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g magnetic fields.<br />
As far as the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the fields at the classical level<br />
is c<strong>on</strong>cerned, both the electric and magnetic flux tubes<br />
show the same behavior because the electric-magnetic duality<br />
holds <strong>in</strong> the forward light-c<strong>on</strong>e where there is no color<br />
charge [13] (see also [14]). The flux tube expands rapidly<br />
towards transverse directi<strong>on</strong>s just like the fields <strong>in</strong> the ord<strong>in</strong>ary<br />
electrodynamics. This is because the glasma is a<br />
perturbative object without the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>f<strong>in</strong>ement. On<br />
the other hand, behaviors <str<strong>on</strong>g>of</str<strong>on</strong>g> fluctuati<strong>on</strong>s around the classical<br />
c<strong>on</strong>figurati<strong>on</strong> are drastically different <strong>in</strong> the electric and<br />
magnetic flux tubes, as we will see below.<br />
Towards QGP<br />
If the <strong>in</strong>itial glasma field has no z dependence as ideally<br />
realized <strong>in</strong> the high-energy limit (see the first comment<br />
above), it never acquires n<strong>on</strong>trivial pz dependences<br />
after all. This means that the glasma cannot reach thermal<br />
equilibrium (even isotropizati<strong>on</strong>) which does not have<br />
any preference <strong>on</strong> spatial directi<strong>on</strong>s. This is a serious problem<br />
<strong>in</strong> the CGC-glasma descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the heavy-i<strong>on</strong> collisi<strong>on</strong>.<br />
This problem has not been fully resolved, but recently<br />
6 Precisely, this should be the η dependence where η is the coord<strong>in</strong>ate<br />
rapidity, but physically it is the same as the z dependence.
many <strong>in</strong>terest<strong>in</strong>g phenomena were found <strong>in</strong> the dynamics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma, which we expect relevant for thermalizati<strong>on</strong>.<br />
In particular, it turned out that the behaviors <str<strong>on</strong>g>of</str<strong>on</strong>g> fluctuati<strong>on</strong>s<br />
<strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> electric or magnetic flux tubes are very<br />
important. Note that there always appear quantum fluctuati<strong>on</strong>s<br />
even though the glasma field is c<strong>on</strong>stant <strong>in</strong> the zdirecti<strong>on</strong><br />
at the classical level.<br />
Magnetic flux tube: Nielsen-Olesen <strong>in</strong>stability<br />
C<strong>on</strong>sider small fluctuati<strong>on</strong>s <strong>in</strong> a s<strong>in</strong>gle magnetic flux<br />
tube [13, 14, 15]. One can simplify the situati<strong>on</strong> by c<strong>on</strong>sider<strong>in</strong>g<br />
fluctuati<strong>on</strong>s <strong>in</strong> a c<strong>on</strong>stant magnetic field <strong>in</strong> an expand<strong>in</strong>g<br />
coord<strong>in</strong>ate system. This problem has been discussed<br />
<strong>in</strong> the Cartesian coord<strong>in</strong>ates l<strong>on</strong>g time ago, and it is<br />
known that the c<strong>on</strong>stant color magnetic field undergoes the<br />
Nielsen-Olesen <strong>in</strong>stability[16]. This is <strong>in</strong>deed the case even<br />
<strong>in</strong> an expand<strong>in</strong>g coord<strong>in</strong>ate system (see Fig. 4). Fluctuati<strong>on</strong>s<br />
a(x⊥, z) with n<strong>on</strong>trivial z-dependence can grow exp<strong>on</strong>entially<br />
due to n<strong>on</strong>l<strong>in</strong>ear <strong>in</strong>teracti<strong>on</strong>s <strong>in</strong> the Yang-Mills<br />
theory:<br />
√<br />
gBz τ<br />
a(x⊥, z) ∝ e , (4)<br />
where τ is the propertime and B z is the str<strong>on</strong>g color magnetic<br />
field <strong>in</strong> a flux tube B z ∼ Q 2 s/g. S<strong>in</strong>ce Qs is large<br />
at high energy, the time scale for the fluctuati<strong>on</strong> to grow<br />
is very rapid 1/ √ gB z ∼ 1/Qs. Obviously, this k<strong>in</strong>d<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> fluctuati<strong>on</strong> must c<strong>on</strong>tribute to drive the system towards<br />
isotropizati<strong>on</strong>. Note that this picture is c<strong>on</strong>sistent with the<br />
result <str<strong>on</strong>g>of</str<strong>on</strong>g> numerical simulati<strong>on</strong>s [17]. The Nielsen-Olesen<br />
<strong>in</strong>stability has been reexam<strong>in</strong>ed <strong>in</strong> a box (<strong>in</strong> the Cartesian<br />
coord<strong>in</strong>ates) with the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong> similar to the<br />
CGC [18], where a sec<strong>on</strong>dary <strong>in</strong>stability was found to occur<br />
as a c<strong>on</strong>sequence <str<strong>on</strong>g>of</str<strong>on</strong>g> the enhanced fluctuati<strong>on</strong>s.<br />
Electric flux tube: Schw<strong>in</strong>ger mechanism<br />
A completely different phenomen<strong>on</strong> occurs <strong>in</strong> an electric<br />
flux tube. Recall that glu<strong>on</strong>s are massless, and that<br />
light quarks have small masses compared to the strength<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the color electric field <strong>in</strong> the glasma √ gE z ∼ Qs.<br />
Then, producti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s or q¯q pairs are possible <strong>in</strong> the<br />
presence <str<strong>on</strong>g>of</str<strong>on</strong>g> a glasma field, which is the QCD versi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the Schw<strong>in</strong>ger mechanism <strong>in</strong> QED (see Fig. 4). In fact,<br />
Schw<strong>in</strong>ger mechanism <strong>in</strong> QCD has been discussed as <strong>on</strong>e<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the plausible mechanisms for thermalizati<strong>on</strong> <strong>in</strong> HIC’s.<br />
But recently, this problem acquires renewed <strong>in</strong>terests <strong>in</strong><br />
view <str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma. As discussed <strong>in</strong> [19, 20], this effects<br />
could c<strong>on</strong>tribute to thermalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma s<strong>in</strong>ce the<br />
created charged particles will be accelerated <strong>in</strong> the color<br />
electric field to obta<strong>in</strong> n<strong>on</strong>trivial pz dependence. Once<br />
pairs are created, the external color electric field will be<br />
screened. It is possible to <strong>in</strong>clude such k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> ‘backreacti<strong>on</strong>’<br />
<strong>in</strong>to the calculati<strong>on</strong> to see the time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
whole system made <str<strong>on</strong>g>of</str<strong>on</strong>g> particles and the background field<br />
[19]. It is also <strong>in</strong>terest<strong>in</strong>g to evaluate the Schw<strong>in</strong>ger mechanism<br />
<strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic field. In this case, the<br />
lowest Landau level <str<strong>on</strong>g>of</str<strong>on</strong>g> quarks become ‘massless’ and we<br />
expect large enhancement <str<strong>on</strong>g>of</str<strong>on</strong>g> quark pair producti<strong>on</strong>s [21].<br />
SUMMARY<br />
In this talk, I discussed the importance and <strong>in</strong>terest<strong>in</strong>g aspects<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>g field dynamics <strong>in</strong> HIC’s. There are two<br />
different str<strong>on</strong>g fields, i.e., str<strong>on</strong>g electromagnetic fields<br />
and str<strong>on</strong>g Yang-Mills fields, and we can enjoy rich phenomena<br />
caused by them, <strong>in</strong>clud<strong>in</strong>g the <strong>in</strong>terplay between<br />
these two. I hope many people get <strong>in</strong>terested <strong>in</strong> this subject,<br />
and participate <strong>in</strong> the reaseach.<br />
The topics discussed <strong>in</strong> my talk are already broad, but <strong>in</strong><br />
fact there are still many problems left unsolved <strong>in</strong> relati<strong>on</strong><br />
to the physics <str<strong>on</strong>g>of</str<strong>on</strong>g> HIC’s. One <str<strong>on</strong>g>of</str<strong>on</strong>g> the important problems<br />
which I did not touch <strong>in</strong> this talk is the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> ord<strong>in</strong>ary<br />
magnetic fields <strong>on</strong> the phase transiti<strong>on</strong> dynamics <strong>in</strong><br />
QCD. It is discussed that the chiral symmetry break<strong>in</strong>g is<br />
affected by the external magnetic field. Also important is<br />
the applicati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> recent development <strong>on</strong> the QED cascade<br />
discussed <strong>in</strong> [22] to QCD. This would be <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the ideal<br />
subjects for the collaborati<strong>on</strong> between different fields.<br />
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[20] A. Iwazaki, <strong>in</strong> these proceed<strong>in</strong>gs.<br />
[21] Y. Hidaka, <strong>in</strong> these proceed<strong>in</strong>gs.<br />
[22] H. Ruhl and N. Elk<strong>in</strong>a, <strong>in</strong> these proceed<strong>in</strong>gs.
Abstract<br />
Yoctosec<strong>on</strong>d phot<strong>on</strong> pulse generati<strong>on</strong> <strong>in</strong> heavy i<strong>on</strong> collisi<strong>on</strong>s<br />
Heavy i<strong>on</strong> collisi<strong>on</strong>s at RHIC and at the LHC can create<br />
the quark-glu<strong>on</strong> plasma, a state <str<strong>on</strong>g>of</str<strong>on</strong>g> matter at very high<br />
temperatures. Am<strong>on</strong>g a plethora <str<strong>on</strong>g>of</str<strong>on</strong>g> particles that are produced<br />
<strong>in</strong> these collisi<strong>on</strong>s, also light is emitted throughout<br />
the evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma.<br />
In this talk, the properties <str<strong>on</strong>g>of</str<strong>on</strong>g> this light are discussed and<br />
related to recent efforts towards shorter and more energetic<br />
phot<strong>on</strong> pulses <strong>in</strong> laser physics. In particular, the time evoluti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> high-energy phot<strong>on</strong>s is studied. These phot<strong>on</strong>s<br />
orig<strong>in</strong>ate from Compt<strong>on</strong> scatter<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s and quarkantiquark<br />
annihilati<strong>on</strong> <strong>in</strong> the plasma. Due to the <strong>in</strong>ternal<br />
dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma, double pulses at the yoctosec<strong>on</strong>d<br />
time scale can be generated under certa<strong>in</strong> c<strong>on</strong>diti<strong>on</strong>s. Such<br />
double pulses may be utilized for novel pump-probe experiments<br />
at nuclear time scales.<br />
INTRODUCTION<br />
The year 2010 marks the fiftieth anniversary <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>venti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser. S<strong>in</strong>ce its <strong>in</strong>venti<strong>on</strong>, not <strong>on</strong>ly did the<br />
<strong>in</strong>tensity steadily <strong>in</strong>crease, but also the pulse durati<strong>on</strong> became<br />
shorter and shorter. In fact, pulse durati<strong>on</strong> and <strong>in</strong>tensity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> lasers (or derived coherent radiati<strong>on</strong> bursts) turn<br />
out to be correlated over a large range <str<strong>on</strong>g>of</str<strong>on</strong>g> energies and time<br />
scales [1]. This observati<strong>on</strong> provides a good motivati<strong>on</strong> to<br />
present at a c<strong>on</strong>ference <strong>on</strong> <strong>Physics</strong> <strong>in</strong> Intense Fields (PIF<br />
2010) a study <str<strong>on</strong>g>of</str<strong>on</strong>g> the shortest possible light flashes that can<br />
be produced <strong>in</strong> experiments, which is heavy i<strong>on</strong> collisi<strong>on</strong>s<br />
that produce the quark-glu<strong>on</strong> plasma (QGP) [2].<br />
A few selected milest<strong>on</strong>es <str<strong>on</strong>g>of</str<strong>on</strong>g> the development <str<strong>on</strong>g>of</str<strong>on</strong>g> short<br />
laser pulses are depicted <strong>in</strong> Fig. 1. Each <str<strong>on</strong>g>of</str<strong>on</strong>g> the time scales<br />
has enabled to access new systems: At the picosec<strong>on</strong>d<br />
to femtosec<strong>on</strong>d timescale, femtochemistry allows for the<br />
time-resolved study <str<strong>on</strong>g>of</str<strong>on</strong>g> chemical reacti<strong>on</strong>s [3]. For pumpprobe<br />
spectroscopy, it is essential to have two short pulses<br />
<strong>in</strong> close successi<strong>on</strong>: The first laser pulse triggers a chemical<br />
reacti<strong>on</strong>, which may <strong>in</strong>volve an excited state and various<br />
short-lived transiti<strong>on</strong>s, while the sec<strong>on</strong>d pulse takes a<br />
snapshot <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>termediate state. By vary<strong>in</strong>g the <strong>in</strong>terval<br />
between the two pulses, the time-evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a chemical<br />
reacti<strong>on</strong> can be studied.<br />
At shorter timescales, attosec<strong>on</strong>d science is the w<strong>in</strong>dow<br />
to captur<strong>in</strong>g electr<strong>on</strong> moti<strong>on</strong> <strong>in</strong> molecules and atoms [4].<br />
High-order harm<strong>on</strong>ics <str<strong>on</strong>g>of</str<strong>on</strong>g> femtosec<strong>on</strong>d laser radiati<strong>on</strong> have<br />
been shown to be sources <str<strong>on</strong>g>of</str<strong>on</strong>g> tra<strong>in</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> attosec<strong>on</strong>d extremeultraviolet<br />
pulses [5] that can be used to produce s<strong>in</strong>gle<br />
attosec<strong>on</strong>d s<str<strong>on</strong>g>of</str<strong>on</strong>g>t X-ray pulses [6]. For example, such s<strong>in</strong>gle<br />
attosec<strong>on</strong>d X-ray bursts [7] have applicati<strong>on</strong>s <strong>in</strong> molec-<br />
∗ ipp@hep.itp.tuwien.ac.at<br />
A. Ipp ∗ , Vienna University <str<strong>on</strong>g>of</str<strong>on</strong>g> Technology, Austria<br />
Pulse durati<strong>on</strong><br />
ns<br />
ps<br />
fs<br />
as<br />
zs<br />
ys<br />
Nd:glass<br />
CW Dye<br />
CPM<br />
Compressi<strong>on</strong><br />
HHG<br />
Ti:sapphire<br />
QGP<br />
1960 1970 1980 1990 2000 2010 2020<br />
Figure 1: History <str<strong>on</strong>g>of</str<strong>on</strong>g> laser pulse durati<strong>on</strong>. Marked are<br />
selected milest<strong>on</strong>es <str<strong>on</strong>g>of</str<strong>on</strong>g> technologies that allowed to decrease<br />
the laser pulse durati<strong>on</strong>, like Neodymium glass laser<br />
(Nd:glass), C<strong>on</strong>t<strong>in</strong>uous Wave Dye laser (CW Dye), Collid<strong>in</strong>g<br />
Pulse-Mode locked dye laser (CPM), Titanium sapphire<br />
laser (Ti:sapphire), or High-Harm<strong>on</strong>ic Generati<strong>on</strong><br />
(HHG). For comparis<strong>on</strong>, also the lifetime <str<strong>on</strong>g>of</str<strong>on</strong>g> the Quark<br />
Glu<strong>on</strong> Plasma (QGP) is <strong>in</strong>dicated as it is produced nowadays<br />
<strong>in</strong> heavy i<strong>on</strong> colliders like RHIC or LHC.<br />
ular imag<strong>in</strong>g [8], quantum c<strong>on</strong>trol [9], or Raman spectroscopy<br />
[10]. By <strong>in</strong>troduc<strong>in</strong>g a c<strong>on</strong>trolled delay between<br />
two such peaks, the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> systems could be<br />
studied us<strong>in</strong>g pump-probe techniques [6]. Such techniques<br />
also allow for the direct time resoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> many-body dynamics,<br />
like the observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the dress<strong>in</strong>g process <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
charged particles [11]. In the zeptosec<strong>on</strong>d regime, nuclear<br />
processes may become accessible [12]. It has been suggested<br />
that zeptosec<strong>on</strong>d pulses could be created via n<strong>on</strong>l<strong>in</strong>ear<br />
Thoms<strong>on</strong> backscatter<strong>in</strong>g [13, 14], or by employ<strong>in</strong>g relativistic<br />
laser-plasma <strong>in</strong>teracti<strong>on</strong>s [15, 16], A possible detecti<strong>on</strong><br />
scheme for the characterizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> short γ-ray pulses<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> MeV to GeV energy phot<strong>on</strong>s down to the zeptosec<strong>on</strong>d<br />
scale has been proposed <strong>in</strong> Ref. [17].<br />
At even shorter timescales, double pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> yoctosec<strong>on</strong>d<br />
durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> GeV phot<strong>on</strong> energy could be created from the<br />
quark-glu<strong>on</strong> plasma <strong>in</strong> n<strong>on</strong>-central heavy i<strong>on</strong> collisi<strong>on</strong>s [2].<br />
It turns out that the emissi<strong>on</strong> envelope can depend str<strong>on</strong>gly<br />
<strong>on</strong> the <strong>in</strong>ternal dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP. Under certa<strong>in</strong> c<strong>on</strong>diti<strong>on</strong>s,<br />
a double peak structure <strong>in</strong> the emissi<strong>on</strong> envelope<br />
may be observed. This could be the first source for pumpprobe<br />
experiments at the yoctosec<strong>on</strong>d timescale. The delay<br />
between the peaks is directly related to the isotropizati<strong>on</strong><br />
time, and the relative height between the peaks can be<br />
shaped by vary<strong>in</strong>g phot<strong>on</strong> energy and emissi<strong>on</strong> angle. Such<br />
pulses could be utilized, for example, to resolve dynamics<br />
<strong>on</strong> the nuclear timescale such as that <str<strong>on</strong>g>of</str<strong>on</strong>g> bary<strong>on</strong> res<strong>on</strong>ances<br />
[18]. C<strong>on</strong>versely, a time-resolved study <str<strong>on</strong>g>of</str<strong>on</strong>g> the emit-
ted phot<strong>on</strong>s could provide a w<strong>in</strong>dow to the <strong>in</strong>ternal QGP<br />
dynamics throughout its expansi<strong>on</strong>.<br />
QGP PHOTON PRODUCTION<br />
The Large Hadr<strong>on</strong> Collider (LHC) at CERN has begun<br />
its heavy-i<strong>on</strong> program <strong>in</strong> November 2010 [19], just a<br />
few weeks before the PIF 2010 c<strong>on</strong>ference. With center<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass<br />
energies <str<strong>on</strong>g>of</str<strong>on</strong>g> 2.76 TeV per nucle<strong>on</strong>, the collisi<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> lead i<strong>on</strong>s produce hotter and denser plasmas than previously<br />
achievable at the Relativistic Heavy I<strong>on</strong> Collider<br />
(RHIC), where gold i<strong>on</strong>s were used. The temperatures<br />
reached <strong>in</strong> these collisi<strong>on</strong>s are so high that the c<strong>on</strong>stituents<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> atomic nuclei, the neutr<strong>on</strong>s and prot<strong>on</strong>s, are split <strong>in</strong>to<br />
their c<strong>on</strong>stituents, the quarks and glu<strong>on</strong>s. The <strong>in</strong>terest <strong>in</strong><br />
the QGP stems not least from the fact that it is believed to<br />
have filled the entire universe dur<strong>in</strong>g the first few microsec<strong>on</strong>ds<br />
after the Big Bang.<br />
In heavy-i<strong>on</strong> collisi<strong>on</strong>s, the QGP is produced up to the<br />
size <str<strong>on</strong>g>of</str<strong>on</strong>g> a nucleus (∼ 15 fm) for a durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a few tens <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
yoctosec<strong>on</strong>ds (1 ys = 10 −24 s). The plasma is produced<br />
<strong>in</strong>itially <strong>in</strong> a very anisotropic state, and reaches a hydrodynamic<br />
evoluti<strong>on</strong> through <strong>in</strong>ternal <strong>in</strong>teracti<strong>on</strong>s <strong>on</strong>ly after<br />
some isotropizati<strong>on</strong> timeτiso (see Fig. 2). Am<strong>on</strong>g the many<br />
particles that are produced, also phot<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> a few GeV energy<br />
are emitted [20, 21].<br />
It was <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the early surprises that the observed particle<br />
spectra turned out to agree well with ideal hydrodynamical<br />
model predicti<strong>on</strong>s [22]. This led to the <strong>in</strong>itial assumpti<strong>on</strong><br />
that isotropizati<strong>on</strong> times may be as low as τiso ≈ 1 ys.<br />
However, it has been po<strong>in</strong>ted out <strong>in</strong> the mean-time that viscous<br />
hydrodynamic models are still c<strong>on</strong>sistent with RHIC<br />
data if isotropizati<strong>on</strong> times as large as τiso ≈ 7 ys are assumed<br />
[23], even if the expansi<strong>on</strong> before isotropizati<strong>on</strong><br />
is assumed to be collisi<strong>on</strong>less (“free stream<strong>in</strong>g”). This<br />
should be compared to the lifetime <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP, which could<br />
amount to 15 ys at RHIC, and which could be as large as<br />
25 ys at LHC.<br />
Figure 2(a-c) shows a schematic view <str<strong>on</strong>g>of</str<strong>on</strong>g> the collisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
two heavy i<strong>on</strong>s. The i<strong>on</strong>s are illustrated as relativistically<br />
c<strong>on</strong>tracted pancakes. In general, two i<strong>on</strong>s will not collide<br />
head-<strong>on</strong>-head, but will be displaced by an impact parameter<br />
b. Direct phot<strong>on</strong>s are emitted from the expand<strong>in</strong>g QGP<br />
throughout its lifetime [24]. The energy spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
emitted phot<strong>on</strong>s extends to the GeV range, and the upper<br />
limit for the temporal durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the GeV phot<strong>on</strong> pulse is<br />
determ<strong>in</strong>ed by the expansi<strong>on</strong> dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP, which<br />
leads to yoctosec<strong>on</strong>d pulses. At the <strong>in</strong>itial stage <str<strong>on</strong>g>of</str<strong>on</strong>g> the collisi<strong>on</strong>,<br />
even before the plasma is created, prompt phot<strong>on</strong>s are<br />
emitted from nucle<strong>on</strong>-nucle<strong>on</strong> collisi<strong>on</strong>s <strong>in</strong> all directi<strong>on</strong>s,<br />
see Figs. 2(a) and (d). For an <strong>in</strong>termediate time after the<br />
collisi<strong>on</strong> shown <strong>in</strong> Fig.2(b), a momentum anisotropy occurs<br />
due to the l<strong>on</strong>gitud<strong>in</strong>al expansi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma: Those<br />
particles that orig<strong>in</strong>ally had momentum comp<strong>on</strong>ents <strong>in</strong> forward<br />
or backward directi<strong>on</strong> al<strong>on</strong>g the beam axis leave the<br />
central regi<strong>on</strong> quickly, so that ma<strong>in</strong>ly particles with transverse<br />
momenta rema<strong>in</strong> <strong>in</strong> the plasma. High-energy pho-<br />
Figure 2: Early stages <str<strong>on</strong>g>of</str<strong>on</strong>g> a high-energy collisi<strong>on</strong>, <strong>in</strong>volv<strong>in</strong>g<br />
pre-equilibrium (first two columns) and equilibrated<br />
QGP phases (last column). Parts (a)-(c) show three snapshots<br />
<strong>in</strong> time <strong>in</strong> positi<strong>on</strong> space. Shown are the two relativistically<br />
c<strong>on</strong>tracted collid<strong>in</strong>g i<strong>on</strong>s that create the quarkglu<strong>on</strong><br />
plasma <strong>in</strong> the overlap regi<strong>on</strong>. Curly arrows denote<br />
phot<strong>on</strong> emissi<strong>on</strong> and semicircles the detectors. Parts<br />
(d)-(f) are corresp<strong>on</strong>d<strong>in</strong>g pictorial representati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
plasma <strong>in</strong> momentum space. In an <strong>in</strong>termediate stage <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the pre-equilibrium phase, the momentum distributi<strong>on</strong> is<br />
anisotropic, result<strong>in</strong>g <strong>in</strong> a change <strong>in</strong> the angular phot<strong>on</strong><br />
emissi<strong>on</strong> pattern that can give rise to double-peaked phot<strong>on</strong><br />
pulses [2].<br />
t<strong>on</strong>s that are created <strong>in</strong> the plasma through Compt<strong>on</strong> scatter<strong>in</strong>g<br />
or quark-antiquark annihilati<strong>on</strong> carry preferentially<br />
the momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> the orig<strong>in</strong>al participants <str<strong>on</strong>g>of</str<strong>on</strong>g> the collisi<strong>on</strong>.<br />
Because <str<strong>on</strong>g>of</str<strong>on</strong>g> the momentum anisotropy <str<strong>on</strong>g>of</str<strong>on</strong>g> the quarks and<br />
glu<strong>on</strong>s with<strong>in</strong> the plasma, also the emitted phot<strong>on</strong>s are preferentially<br />
emitted perpendicular to the beam axis z, as <strong>in</strong>dicated<br />
<strong>in</strong> Fig. 2(e). F<strong>in</strong>ally, <strong>in</strong> Fig. 2(c), the system had<br />
time to isotropize due to collisi<strong>on</strong>s with<strong>in</strong> the plasma. The<br />
phot<strong>on</strong>s will be emitted aga<strong>in</strong> <strong>in</strong> all directi<strong>on</strong>s, as shown<br />
<strong>in</strong> Fig. 2(f), particularly also <strong>in</strong>to the directi<strong>on</strong> <strong>in</strong> which<br />
the phot<strong>on</strong> emissi<strong>on</strong> was suppressed dur<strong>in</strong>g the anisotropic<br />
stage.<br />
A phot<strong>on</strong> detector placed towards the beam axis would<br />
therefore measure a time-dependent phot<strong>on</strong> flux. Ideally, <strong>in</strong><br />
the <strong>in</strong>termediate stage <strong>in</strong> Fig. 2(e), the phot<strong>on</strong> emissi<strong>on</strong> <strong>in</strong><br />
the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the detector is suppressed highly enough, so<br />
that the phot<strong>on</strong>s emitted <strong>in</strong> the stages Fig. 2(d) and Fig. 2(f)<br />
are dist<strong>in</strong>ct enough <strong>in</strong> time to form two separate phot<strong>on</strong><br />
pulses. In order to quantitatively describe the pulse envelope<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the emitted phot<strong>on</strong>s, detailed calculati<strong>on</strong>s are necessary.<br />
Such a calculati<strong>on</strong> has been performed <strong>in</strong> Ref. [2],<br />
which was based <strong>on</strong> a <strong>on</strong>e-dimensi<strong>on</strong>al expansi<strong>on</strong> model<br />
by Bjorken [25]. This model assumes boost-<strong>in</strong>variant evoluti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the quark-glu<strong>on</strong> plasma <strong>in</strong> a central regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
collisi<strong>on</strong>. The detector is placed away from the beam axis<br />
by an angle θ with<strong>in</strong> the reacti<strong>on</strong> plane. Direct phot<strong>on</strong>s <strong>in</strong><br />
the GeV energy range are emitted from the pre-equilibrium<br />
and equilibrated phases <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP. The lead<strong>in</strong>g c<strong>on</strong>tributi<strong>on</strong><br />
to the phot<strong>on</strong> producti<strong>on</strong> rateR orig<strong>in</strong>ates from quarkglu<strong>on</strong><br />
Compt<strong>on</strong> scatter<strong>in</strong>g and quark-antiquark annihilati<strong>on</strong>.<br />
In pr<strong>in</strong>ciple, higher order s<str<strong>on</strong>g>of</str<strong>on</strong>g>t scatter<strong>in</strong>g processes like
emsstrahlung or <strong>in</strong>elastic pair annihilati<strong>on</strong> would have to<br />
be <strong>in</strong>cluded as well, but their c<strong>on</strong>tributi<strong>on</strong>s become dom<strong>in</strong>ant<br />
<strong>on</strong>ly at lower energies, and are less important at the<br />
higher energies c<strong>on</strong>sidered [26].<br />
For anisotropic momentum distributi<strong>on</strong>s, the phot<strong>on</strong> producti<strong>on</strong><br />
rate R has to be calculated numerically [21]. It<br />
depends <strong>on</strong> the temperature T <str<strong>on</strong>g>of</str<strong>on</strong>g> the medium, the phot<strong>on</strong><br />
energy E and momentum k, the f<strong>in</strong>e structure c<strong>on</strong>stantα,<br />
and the corresp<strong>on</strong>d<strong>in</strong>g quantity for the str<strong>on</strong>g force<br />
αs (with ¯h = c = kB = 1). The rate further depends<br />
<strong>on</strong> the anisotropy, which is described by a parameter<br />
ξ = p2 <br />
2<br />
T / 2 pL − 1 that relates the mean l<strong>on</strong>gitud<strong>in</strong>al<br />
and transverse momenta pL and pT [26, 27]. To <strong>in</strong>tegrate<br />
this rate over time, a time evoluti<strong>on</strong> model for the preequilibrium<br />
and equilibrated QGP has been used [27]. This<br />
model specifies the time evoluti<strong>on</strong> for the energy density<br />
E = E(τ), for the hard scale phard = phard(τ) (which corresp<strong>on</strong>ds<br />
to T <strong>in</strong> the isotropic case), and for the anisotropy<br />
parameter ξ = ξ(τ) as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the proper time τ.<br />
Qualitatively, the model follows the evoluti<strong>on</strong> as outl<strong>in</strong>ed<br />
<strong>in</strong> Fig. 2. For early times, a free stream<strong>in</strong>g phase lets the<br />
anisotropy grow. At late times, the system c<strong>on</strong>verges to<br />
an ideal hydrodynamic phase with vanish<strong>in</strong>g anisotropy.<br />
These two phases are l<strong>in</strong>ked by a smooth transiti<strong>on</strong> which<br />
is c<strong>on</strong>trolled by additi<strong>on</strong>al model parameters. Thermalizati<strong>on</strong><br />
and isotropizati<strong>on</strong> happen c<strong>on</strong>currently <strong>in</strong> this model,<br />
τtherm = τiso. The model is thus able to cover both, the<br />
pre-equilibrium phase and the equilibrated QGP phase <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the expand<strong>in</strong>g plasma.<br />
dNdtdΩdas 1 GeV 1 <br />
100 a p2 GeV<br />
80<br />
b0 fm<br />
60 ΘΠ2<br />
40<br />
20<br />
b<br />
p3 GeV<br />
b0 fm<br />
ΘΠ2<br />
20 0 20 20 0 20<br />
100<br />
80<br />
60<br />
40<br />
20<br />
0<br />
100<br />
80<br />
60<br />
40<br />
20<br />
c Τiso p2 GeV<br />
b9.2 fm<br />
ΘΠ4<br />
4<br />
e p2 GeV<br />
b12.2 fm<br />
ΘΠ8<br />
Τiso 4<br />
d<br />
f<br />
5 p3 GeV<br />
b9.2 fm<br />
Τiso ΘΠ4<br />
20<br />
Τ<br />
p3 GeV<br />
iso<br />
b12.2 fm<br />
ΘΠ8<br />
4<br />
5 0 5 10 155<br />
Τys<br />
0 5 10 15<br />
Figure 3: Phot<strong>on</strong> emissi<strong>on</strong> rate as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> detector<br />
time τ. Solid blue l<strong>in</strong>es show a large isotropizati<strong>on</strong> time<br />
τiso = 6.7 ys while dashed red l<strong>in</strong>es corresp<strong>on</strong>d to a short<br />
isotropizati<strong>on</strong> time τiso = 0.3 ys. Parts (a) and (b) display<br />
emissi<strong>on</strong> at midrapidity (θ = π/2) for a central collisi<strong>on</strong><br />
with impact parameter b = 0. Parts (c)-(f) show<br />
double-peaked phot<strong>on</strong> pulses obta<strong>in</strong>ed for b = 9.2 fm or<br />
12.2 fm, and the vertical dotted l<strong>in</strong>e <strong>in</strong>dicates the positi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the largerτiso = 6.7 ys. In parts (c) and (d), the detector<br />
directi<strong>on</strong> isθ = π/4, <strong>in</strong> (e) and (f), it isθ = π/8 [2].<br />
The numerical parameters suitable for calculat<strong>in</strong>g LHC<br />
parameters are given as follows: The <strong>in</strong>itial temperature<br />
is assumed to be T0 = 845 MeV with a formati<strong>on</strong> time<br />
τ0 = 0.3 ys. The critical temperature, where the QGP<br />
ceases to exist, is taken asTC = 160 MeV. The isotropizati<strong>on</strong><br />
time is varied <strong>in</strong> the range τiso = τ0 to τiso = 6.7 ys,<br />
assum<strong>in</strong>g free-stream<strong>in</strong>g at early times. Both possibilities<br />
are not yet ruled out by RHIC data. In order to ensure<br />
fixed f<strong>in</strong>al multiplicity, the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong>s are adjusted as<br />
a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> τiso such that the same entropy is generated<br />
as forτiso = τ0.<br />
For central collisi<strong>on</strong>s with emissi<strong>on</strong> angle orthog<strong>on</strong>al to<br />
the beam axis (θ = π/2), a typical time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
phot<strong>on</strong> emissi<strong>on</strong> rate is depicted <strong>in</strong> Figs. 3(a) and 3(b). At 2<br />
GeV energy, the phot<strong>on</strong> producti<strong>on</strong> from the QGP at midrapidity<br />
is 3 to 4 times as large as the producti<strong>on</strong> from the<br />
<strong>in</strong>itial collisi<strong>on</strong>s. It is roughly 6 times as large as the producti<strong>on</strong><br />
from the hadr<strong>on</strong> gas [24]. At 3 GeV energy, QGP<br />
phot<strong>on</strong> producti<strong>on</strong> is even 50 times larger than the producti<strong>on</strong><br />
from the hadr<strong>on</strong> gas. In the Figs. 3, the orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
abscissa is the time when a phot<strong>on</strong> emitted from the center<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the collisi<strong>on</strong> arrives at the detector. Phot<strong>on</strong>s arriv<strong>in</strong>g<br />
earlier orig<strong>in</strong>ate from a part <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP that is closer to the<br />
detector. The pulse shape is ma<strong>in</strong>ly determ<strong>in</strong>ed by the geometry<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the lead i<strong>on</strong> with radius7.1 fm. Any structure <strong>on</strong><br />
the yoctosec<strong>on</strong>d timescale is blurred simply by the time for<br />
light to traverse the QGP.<br />
This limit can be overcome <strong>in</strong> the follow<strong>in</strong>g ways: By<br />
c<strong>on</strong>sider<strong>in</strong>g n<strong>on</strong>-central collisi<strong>on</strong>s with impact parameter<br />
b, the physical extent <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP is reduced. Also, an optimizati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the detecti<strong>on</strong> angle can m<strong>in</strong>imize the travel<strong>in</strong>g<br />
time through the plasma. In forward directi<strong>on</strong>, the <strong>in</strong>itial<br />
shape <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP is Lorentz-c<strong>on</strong>tracted, and light leaves<br />
this <strong>in</strong>itial regi<strong>on</strong> quickly. This is partially spoiled due to<br />
the QGP expansi<strong>on</strong> <strong>in</strong> the same directi<strong>on</strong>. Thus <strong>in</strong>termediate<br />
emissi<strong>on</strong> angles are most promis<strong>in</strong>g for which the QGP<br />
appears partly Lorentz c<strong>on</strong>tracted but does not expand towards<br />
the detector.<br />
Figures 3(c)-3(f) show the phot<strong>on</strong> emissi<strong>on</strong> <strong>in</strong> the directi<strong>on</strong>s<br />
θ = π/4 and θ = π/8. For large impact parameters<br />
b = 9.2 fm and b = 12.2 fm, a strik<strong>in</strong>g double-peak structure<br />
appears. The m<strong>in</strong>imum between the two peaks corresp<strong>on</strong>ds<br />
roughly to maximum anisotropy with<strong>in</strong> the plasma.<br />
This follows from the fact that the phot<strong>on</strong> emissi<strong>on</strong> rate<br />
is suppressed for larger values <str<strong>on</strong>g>of</str<strong>on</strong>g> ξ and smaller values <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
θ [26]. The distance between the two peaks is approximately<br />
governed by the isotropizati<strong>on</strong> time τiso, <strong>in</strong>dicated<br />
by the dotted l<strong>in</strong>e <strong>in</strong> Figs. 3(c)-3(f). The first peak corresp<strong>on</strong>ds<br />
to phot<strong>on</strong>s emitted from the blue-shifted approach<strong>in</strong>g<br />
part <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP, while the sec<strong>on</strong>d peak corresp<strong>on</strong>ds to<br />
a slightly red-shifted and time-dilated reced<strong>in</strong>g tail <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
plasma. For a short isotropizati<strong>on</strong> time τiso = τ0 (dashed<br />
l<strong>in</strong>es) the separati<strong>on</strong> <strong>in</strong>to two peaks does not occur. Therefore,<br />
this effect depends delicately <strong>on</strong> the QGP dynamics.<br />
There are a couple <str<strong>on</strong>g>of</str<strong>on</strong>g> caveats to this model calculati<strong>on</strong>:<br />
Apart from the phot<strong>on</strong>s orig<strong>in</strong>at<strong>in</strong>g from the QGP, <strong>in</strong> an<br />
actual experiment there is a background <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s from
various sources [28]. These <strong>in</strong>clude phot<strong>on</strong>s produced by a<br />
jet pass<strong>in</strong>g through the QGP [20], and could dom<strong>in</strong>ate the<br />
effect that is expected from the QGP al<strong>on</strong>e. S<strong>in</strong>ce these<br />
phot<strong>on</strong>s are produced <strong>on</strong> a similar yoctosec<strong>on</strong>d timescale,<br />
they would modify the pulse shape <strong>on</strong> this timescale. Phot<strong>on</strong>s<br />
produced from the <strong>in</strong>itial collisi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the two i<strong>on</strong>s<br />
can be <str<strong>on</strong>g>of</str<strong>on</strong>g> comparable size <strong>in</strong> <strong>in</strong>tensity, but they would<br />
<strong>on</strong>ly enhance the first peak <str<strong>on</strong>g>of</str<strong>on</strong>g> the double peaks depicted<br />
<strong>in</strong> Fig. 3(c)-3(f). Other background phot<strong>on</strong>s are produced<br />
<strong>in</strong> the decay <str<strong>on</strong>g>of</str<strong>on</strong>g> pi<strong>on</strong>s at later stages <str<strong>on</strong>g>of</str<strong>on</strong>g> the collisi<strong>on</strong>. S<strong>in</strong>ce<br />
these are produced at much later time scales, they would<br />
not modify a time structure <strong>on</strong> the yoctosec<strong>on</strong>d timescale.<br />
It would be necessary to take these various phot<strong>on</strong> sources<br />
<strong>in</strong>to account <strong>in</strong> order to obta<strong>in</strong> a quantitative predicti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the expected pulse structure.<br />
What k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> properties should the detector have <strong>in</strong> order<br />
to resolve the double pulses? The time-<strong>in</strong>tegrated highenergetic<br />
phot<strong>on</strong>s are already rout<strong>in</strong>ely detected <strong>in</strong> experiments<br />
[29]. In the GeV energy range, the phot<strong>on</strong> producti<strong>on</strong><br />
rate is <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> a few phot<strong>on</strong>s per collisi<strong>on</strong> [28].<br />
Note that a s<strong>in</strong>gle GeV phot<strong>on</strong> pulse <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 ys durati<strong>on</strong> corresp<strong>on</strong>ds<br />
to a pulse energy <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>on</strong>ly about 100 pJ, but to a<br />
power <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 TW. The phot<strong>on</strong> yield could be enhanced by<br />
c<strong>on</strong>sider<strong>in</strong>g lower energy phot<strong>on</strong>s, but this would also <strong>in</strong>crease<br />
the number <str<strong>on</strong>g>of</str<strong>on</strong>g> unwanted background phot<strong>on</strong>s. Alternatively,<br />
the phot<strong>on</strong> yield could be enhanced by <strong>in</strong>creas<strong>in</strong>g<br />
the collisi<strong>on</strong> energy. This may have the additi<strong>on</strong>al benefit<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>creas<strong>in</strong>g the relative importance <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>tributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
thermal phot<strong>on</strong>s compared to other k<strong>in</strong>ds <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s [28].<br />
The emissi<strong>on</strong> envelope is <strong>in</strong>fluenced by the geometry, emissi<strong>on</strong><br />
angle, and <strong>in</strong>ternal dynamics like the isotropizati<strong>on</strong><br />
time <str<strong>on</strong>g>of</str<strong>on</strong>g> the expand<strong>in</strong>g QGP. The double-peak structure described<br />
may emerge <strong>in</strong> n<strong>on</strong>-central collisi<strong>on</strong>s at an emissi<strong>on</strong><br />
angle close to forward directi<strong>on</strong>, assum<strong>in</strong>g that the<br />
isotropizati<strong>on</strong> time is large. In order to detect such short<br />
pulses, new detecti<strong>on</strong> schemes would be required. Exist<strong>in</strong>g<br />
tools and ideas from attosec<strong>on</strong>d metrology, like pumpprobe<br />
experiments or spectroscopy techniques, may turn<br />
out to be appropriate candidates to be scaled to zepto- or<br />
yoctosec<strong>on</strong>d durati<strong>on</strong> [4, 17]. Also, an experimental determ<strong>in</strong>ati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> emissi<strong>on</strong> envelope would serve as<br />
an additi<strong>on</strong>al probe <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>ternal dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP,<br />
for example by measur<strong>in</strong>g its isotropizati<strong>on</strong> time.<br />
CONCLUSIONS<br />
The steady progress <strong>in</strong> laser physics over the last decades<br />
towards higher <strong>in</strong>tensities and shorter pulse lengths provides<br />
str<strong>on</strong>g motivati<strong>on</strong> to study systems that produce extremely<br />
short flashes <str<strong>on</strong>g>of</str<strong>on</strong>g> light. The QGP is an ideal candidate<br />
because it exhibits n<strong>on</strong>-trivial dynamics <strong>on</strong> the yoctosec<strong>on</strong>d<br />
timescale, dur<strong>in</strong>g which GeV phot<strong>on</strong>s are emitted.<br />
Under certa<strong>in</strong> c<strong>on</strong>diti<strong>on</strong>s, a double-peak structure may<br />
be produced. This could eventually lead to novel pumpprobe<br />
experiments at the GeV energy scale. Alternatively,<br />
measur<strong>in</strong>g the temporal shape <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> emissi<strong>on</strong> envelope,<br />
dynamic properties <str<strong>on</strong>g>of</str<strong>on</strong>g> the QGP, like its isotropizati<strong>on</strong><br />
time, could be probed experimentally.<br />
ACKNOWLEDGMENT<br />
I would like to thank my collaborators J. Evers,<br />
K. Z. Hatsagortsyan, and C. H. Keitel for guidance and<br />
fruitful discussi<strong>on</strong>s that led to the work that has been presented<br />
[2, 17]. I would further like to thank the organizers<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the PIF 2010 for their k<strong>in</strong>d hospitality.<br />
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Abstract<br />
Fields, Instant<strong>on</strong>s, and Currents<br />
Kenji Fukushima<br />
Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Keio University<br />
3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa 223-8522, Japan<br />
A review <strong>on</strong> the chiral magnetic effect is given with special<br />
emphasis put <strong>on</strong> the pseudo <strong>on</strong>e-dimensi<strong>on</strong>al property<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the system under str<strong>on</strong>g magnetic fields. In such a (1+1)dimensi<strong>on</strong>al<br />
descripti<strong>on</strong> as a result <str<strong>on</strong>g>of</str<strong>on</strong>g> the dimensi<strong>on</strong>al reducti<strong>on</strong>,<br />
electric fields can be identified as the topological<br />
charge associated with <strong>in</strong>stant<strong>on</strong>-like gauge c<strong>on</strong>figurati<strong>on</strong>s.<br />
The chiral magnetic current is, <strong>in</strong> this picture, noth<strong>in</strong>g but<br />
a current accord<strong>in</strong>g to familiar Ohm’s law. The currentcurrent<br />
susceptibility is found to be a product <str<strong>on</strong>g>of</str<strong>on</strong>g> the bos<strong>on</strong><br />
mass <strong>in</strong> the Schw<strong>in</strong>ger model and the Landau level density.<br />
CHIRAL MAGNETIC EFFECT<br />
It is well-known that special gauge c<strong>on</strong>figurati<strong>on</strong>s with<br />
n<strong>on</strong>-zero w<strong>in</strong>d<strong>in</strong>g number play an important role <strong>in</strong> the understand<strong>in</strong>g<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum structure <strong>in</strong> the str<strong>on</strong>g <strong>in</strong>teracti<strong>on</strong>s.<br />
The sp<strong>on</strong>taneous break<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> chiral symmetry is attributed<br />
to the QCD <strong>in</strong>stant<strong>on</strong>, which is the orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> dynamical<br />
mass generati<strong>on</strong>. The c<strong>on</strong>f<strong>in</strong>ement nature is also<br />
expla<strong>in</strong>ed <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic m<strong>on</strong>opole c<strong>on</strong>densati<strong>on</strong> <strong>in</strong><br />
a special class <str<strong>on</strong>g>of</str<strong>on</strong>g> the gauge choice.<br />
There is no doubt about the existence <str<strong>on</strong>g>of</str<strong>on</strong>g> topological c<strong>on</strong>figurati<strong>on</strong>s<br />
<strong>in</strong> QCD physics, but it is a highly n<strong>on</strong>-trivial<br />
questi<strong>on</strong> how to “see” such topological effects <strong>in</strong> real experiments.<br />
The chiral magnetic effect is <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the promis<strong>in</strong>g<br />
candidates [1, 2, 3]. Let us imag<strong>in</strong>e the follow<strong>in</strong>g situati<strong>on</strong>;<br />
the QCD vacuum accommodates <strong>on</strong>e <strong>in</strong>stant<strong>on</strong> that<br />
has a topological charge QW and a (QED) magnetic field<br />
B that is as str<strong>on</strong>g as the QCD energy scale ΛQCD is applied<br />
<strong>on</strong> the <strong>in</strong>stant<strong>on</strong>.<br />
Then, the axial anomaly relati<strong>on</strong> (for the s<strong>in</strong>gle-flavor<br />
case),<br />
implies that<br />
∂µj µ<br />
5<br />
g2<br />
= −<br />
8π2 ∫<br />
d 3 x trFµν F µν , (1)<br />
∆N5 = N5(t = ∞) − N5(t = −∞) = −2QW . (2)<br />
This means that, if we start with the chirally neutral situati<strong>on</strong><br />
(i.e. N5(t = −∞) = 0), a f<strong>in</strong>ite amount <str<strong>on</strong>g>of</str<strong>on</strong>g> chirality is<br />
generated by the topological charge. In the chiral limit <strong>in</strong><br />
which Dirac fermi<strong>on</strong>s are all massless, the momentum and<br />
the sp<strong>in</strong> are parallel to each other if the chirality is righthanded,<br />
while they are anti-parallel if the chirality is lefthanded.<br />
Therefore, the sp<strong>in</strong> is aligned by the str<strong>on</strong>g B effect,<br />
which makes the momentum also aligned al<strong>on</strong>g the B<br />
directi<strong>on</strong>, lead<strong>in</strong>g to a n<strong>on</strong>-vanish<strong>in</strong>g value <str<strong>on</strong>g>of</str<strong>on</strong>g> the total momentum<br />
if ∆N5 ̸= 0. In other words, s<strong>in</strong>ce Dirac fermi<strong>on</strong>s<br />
are charged, an electric or bary<strong>on</strong>ic current is produced for<br />
B ̸= 0 and ∆N5 ̸= 0. Such an effect can be expressed<br />
simply as [1]<br />
B<br />
J V = −2QW , (3)<br />
|B|<br />
<strong>in</strong> the str<strong>on</strong>g B limit, where J V represents the vector current<br />
⟨ ¯ ψγ µ ψ⟩ <strong>in</strong>tegrated over the volume.<br />
For the general strength <str<strong>on</strong>g>of</str<strong>on</strong>g> B it is more appropriate<br />
to work <strong>in</strong> the grand can<strong>on</strong>ical ensemble us<strong>in</strong>g the chiral<br />
chemical potential µ5 <strong>in</strong>stead <str<strong>on</strong>g>of</str<strong>on</strong>g> N5. One can actually<br />
prove that [2, 4]<br />
jV = eµ5<br />
B (4)<br />
2π2 holds for any B and µ5. For free Dirac particles under<br />
B ̸= 0 and µ5 ̸= 0, it is possible to c<strong>on</strong>firm that Eq. (4)<br />
is reduced to Eq. (3) <strong>in</strong> the str<strong>on</strong>g B limit us<strong>in</strong>g the relati<strong>on</strong><br />
(2).<br />
What is detectable <strong>in</strong> experiments should not be the current<br />
jV itself because the QCD vacuum has both <strong>in</strong>stant<strong>on</strong>s<br />
and anti-<strong>in</strong>stant<strong>on</strong>s and they always fluctuate. In other<br />
words, the parity (P) and the charge-parity (CP) symmetries<br />
are broken <strong>on</strong>ly locally at the <strong>in</strong>stant<strong>on</strong> or anti<strong>in</strong>stant<strong>on</strong>,<br />
but those symmetries are restored <strong>on</strong> average<br />
over all fluctuati<strong>on</strong>s <strong>in</strong> space and time. Thus, the chiral<br />
magnetic current jV is also a local object and its (ensemble<br />
or spatial) average is vanish<strong>in</strong>g. This is the reas<strong>on</strong> why the<br />
chiral magnetic effect is sometimes referred to as the “local<br />
parity violati<strong>on</strong>” <strong>in</strong> the c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g> the relativistic heavy-i<strong>on</strong><br />
collisi<strong>on</strong>s.<br />
In this sense the most relevant quantity to the experimental<br />
data [5] is the fluctuati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the chiral magnetic current,<br />
i.e. the electric-current susceptibility χj [6]. One-loop<br />
computati<strong>on</strong> results <strong>in</strong>,<br />
χj = e2 |eB|<br />
, (5)<br />
2π2 which does not depend <strong>on</strong> µ5 and comes from the Landau<br />
zero-mode al<strong>on</strong>e, <strong>in</strong>terest<strong>in</strong>gly. The derivati<strong>on</strong> from<br />
explicit calculati<strong>on</strong>s is lengthy but there is an argument to<br />
take a short-cut to the above expressi<strong>on</strong> [6]. For this purpose<br />
let us c<strong>on</strong>sider the electric current generati<strong>on</strong> rate,<br />
d(eJV )<br />
dt = V e2 |eB|E<br />
2π2 , (6)<br />
which orig<strong>in</strong>ates from the corresp<strong>on</strong>dence between the chirality<br />
generati<strong>on</strong> and the particle producti<strong>on</strong> when fields are<br />
str<strong>on</strong>g enough [7]. The same quantity can be expressed <strong>in</strong><br />
the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> the l<strong>in</strong>ear resp<strong>on</strong>se theory as<br />
∫<br />
d(eJV )<br />
= −<br />
dt<br />
d 3 x d 4 x ′ ⟨ d(ejV )(x)<br />
jV (x<br />
dt<br />
′ ⟩<br />
) eA(x ′ ∫<br />
),<br />
= d 3 x d 4 x ′ e 2 ⟨jV (x)jV (x ′ )⟩ E, (7)
where A(x) denotes a vector potential comp<strong>on</strong>ent parallel<br />
to B and thus J V . From the first l<strong>in</strong>e to the sec<strong>on</strong>d l<strong>in</strong>e<br />
above, we used E = ˙ A. By equat<strong>in</strong>g Eqs. (6) and (7), we<br />
can f<strong>in</strong>d chij given by Eq. (5) immediately.<br />
What is addressed <strong>in</strong> this article is that we can quickly<br />
derive these expressi<strong>on</strong>s related to the chiral magnetic effect<br />
<strong>on</strong>ce we take the str<strong>on</strong>g B limit and the associated<br />
dimensi<strong>on</strong>al reducti<strong>on</strong>.<br />
DIMENSIONAL REDUCTION<br />
Under a str<strong>on</strong>g magnetic field, <strong>in</strong> general, the transverse<br />
moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles is equivalent to the <strong>on</strong>e <strong>in</strong> the<br />
harm<strong>on</strong>ic oscillator. The energy level is then discrete due<br />
to the Landau quantizati<strong>on</strong>. Sp<strong>in</strong>-1/2 particles have the<br />
Landau zero-mode which would dom<strong>in</strong>ate <strong>in</strong> the dynamics<br />
at energies below the scale ∼ √ |eB|. Such a str<strong>on</strong>g B<br />
enables us to use the so-called lowest Landau-level (LLL)<br />
approximati<strong>on</strong> and to drop the transverse moti<strong>on</strong> at all. In<br />
this limit, thus, we can reduce the (3+1)-dimensi<strong>on</strong>al theory<br />
<strong>in</strong>to a form <str<strong>on</strong>g>of</str<strong>on</strong>g> the (1+1)-dimensi<strong>on</strong>al <strong>on</strong>e multiplied by<br />
the Landau level density.<br />
In M<strong>in</strong>kowskian space-time we use the metric g 00 =<br />
−g 11 = 1, g 01 = g 10 = 0, and the 2 × 2 γ-matrices<br />
which satisfy {γ µ , γ ν } = 2g µν . Chirality is characterized<br />
by γ 5 = γ 0 γ 1 = diag(1, −1) <strong>in</strong> the chiral representati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the γ-matrices. Therefore the upper (lower) element<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> two-comp<strong>on</strong>ent sp<strong>in</strong>or ψ = (ψR, ψL) t represents the<br />
right-handed (left-handed) particle. In (1+1) dimensi<strong>on</strong>s<br />
the particle–anti-particle difference is correlated with the<br />
chirality. That is, <strong>in</strong> momentum space, the right-handed<br />
comp<strong>on</strong>ent corresp<strong>on</strong>ds to a right-mov<strong>in</strong>g (p > 0) particle<br />
and a left-mov<strong>in</strong>g (p < 0) anti-particle. One can understand<br />
the left-handed comp<strong>on</strong>ent <strong>in</strong> the same way, i.e. a<br />
left-mov<strong>in</strong>g (p < 0) particle and a right-mov<strong>in</strong>g (p > 0)<br />
anti-particle.<br />
In (1+1) dimensi<strong>on</strong>s the follow<strong>in</strong>g relati<strong>on</strong> am<strong>on</strong>g the<br />
γ-matrices plays an <strong>in</strong>terest<strong>in</strong>g role for the topological currents;<br />
γ µ γ 5 = −ϵ µν γν, (8)<br />
where ϵ 01 = −ϵ 10 = −ϵ01 = ϵ10 = 1, which relates the<br />
vector and the axial-vector currents. As usual, we can write<br />
the vector and the axial-vector currents as<br />
j µ<br />
V = ¯ ψγ µ ψ, j µ<br />
5 = ¯ ψγ µ γ 5 ψ. (9)<br />
Us<strong>in</strong>g the relati<strong>on</strong> (8), we have a relati<strong>on</strong>, j µ<br />
5 = −ϵµν jν,<br />
that is explicitly written as [8]<br />
j 1 V = j 0 5, j 1 5 = j 0 . (10)<br />
TOPOLOGICAL CURRENTS<br />
The relati<strong>on</strong> between the vector and axial-vector currents<br />
is very useful because, as we will see <strong>in</strong> this secti<strong>on</strong>, it captures<br />
the essential feature <str<strong>on</strong>g>of</str<strong>on</strong>g> the topologically <strong>in</strong>duced currents<br />
<strong>in</strong> (3+1) dimensi<strong>on</strong>s.<br />
Let us c<strong>on</strong>sider the anomaly relati<strong>on</strong> <strong>in</strong> (1+1) dimensi<strong>on</strong>s.<br />
It is well-known that the axial anomaly leads to<br />
∂µj µ<br />
5<br />
= e<br />
2π ϵµν Fµν = e<br />
π E = −2qW , (11)<br />
where the electric field is E = F 10 <strong>in</strong> our c<strong>on</strong>venti<strong>on</strong>. Note<br />
that there is no magnetic field but <strong>on</strong>ly the electric field E <strong>in</strong><br />
(1+1) dimensi<strong>on</strong>s. We here def<strong>in</strong>ed the (1+1)-dimensi<strong>on</strong>al<br />
topological charge density as qW = −(e/2π)E <strong>in</strong> accord<br />
to the c<strong>on</strong>venti<strong>on</strong>. By <strong>in</strong>tegrat<strong>in</strong>g Eq. (11) over space-time<br />
and assum<strong>in</strong>g that the current falls sufficiently fast at spatial<br />
<strong>in</strong>f<strong>in</strong>ity, we can recover Eq. (2) easily. We can also prove<br />
that the topological charge, QW = ∫ d 2 x qW (x), takes an<br />
<strong>in</strong>teger number so that the boundary c<strong>on</strong>diti<strong>on</strong> <strong>in</strong> the xdirecti<strong>on</strong><br />
can be ma<strong>in</strong>ta<strong>in</strong>ed.<br />
We also note that we can express Eq. (11) <strong>in</strong> the form <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
∂µj µ e<br />
5 (x) = −<br />
π (∂0A 1 − ∂ 1 A 0 ) = −2∂µK µ (x) (12)<br />
with the (1+1)-dimensi<strong>on</strong>al Chern-Sim<strong>on</strong>s current density<br />
def<strong>in</strong>ed by K µ = −(e/2π)ϵ µνAν. From this identificati<strong>on</strong>,<br />
the Chern-Sim<strong>on</strong>s number <strong>in</strong> this system is deduced as<br />
∫<br />
ν(t) =<br />
dx K 0 (t, x) = e<br />
2π<br />
∫<br />
dx A 1 (t, x). (13)<br />
Comb<strong>in</strong><strong>in</strong>g these expressi<strong>on</strong>s with the relati<strong>on</strong> j1 V = j0 5<br />
(where N5 is the volume <strong>in</strong>tegral <str<strong>on</strong>g>of</str<strong>on</strong>g> j0 5), we can immediately<br />
write the vector current <strong>in</strong>tegrated over space;<br />
J 1 ∫<br />
V (t) = N5(t) = −2 dt dx qW (t, x), (14)<br />
assum<strong>in</strong>g that N5 was zero at the <strong>in</strong>itial time (i.e. N5(t =<br />
−∞) = 0). This simple relati<strong>on</strong> leads to the current at late<br />
time as given by<br />
J 1 V = −2QW . (15)<br />
This is a result expected when the sp<strong>in</strong> is fully polarized <strong>in</strong><br />
the (3+1)-dimensi<strong>on</strong>al chiral magnetic effect under a str<strong>on</strong>g<br />
magnetic field (see Eq. (3)). Note that, <strong>in</strong> (1+1) dimensi<strong>on</strong>s<br />
the sp<strong>in</strong> is always fully polarized because there is <strong>on</strong>ly <strong>on</strong>e<br />
spatial directi<strong>on</strong> and thus the mov<strong>in</strong>g directi<strong>on</strong> (either p ><br />
0 or p < 0) and the chirality <str<strong>on</strong>g>of</str<strong>on</strong>g> particles have <strong>on</strong>e-to-<strong>on</strong>e<br />
corresp<strong>on</strong>dence. Here Eq. (14) is noth<strong>in</strong>g but Ohm’s law<br />
because the (1+1)-dimensi<strong>on</strong>al topological charge density<br />
is proporti<strong>on</strong>al to the electric field as <strong>in</strong> Eq. (11).<br />
If the spatial comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the Chern-Sim<strong>on</strong>s current<br />
falls sufficiently fast, the topological charge is written as<br />
QW = ν(t = ∞) − ν(t = −∞). Therefore, (the spatial<br />
average <str<strong>on</strong>g>of</str<strong>on</strong>g>) A1 is the Chern-Sim<strong>on</strong>s number and the boundary<br />
c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> A1 <strong>in</strong> the t-directi<strong>on</strong> gives the topological<br />
w<strong>in</strong>d<strong>in</strong>g number. Supposed that ν(t = −∞) = N5(t =<br />
−∞) = 0, the topologically <strong>in</strong>duced current is<br />
J 1 V (t) = − e<br />
π<br />
∫<br />
dx A 1 (t, x). (16)<br />
If we identify −eA 0 as the chemical potential µ (regard<strong>in</strong>g<br />
the sign, remember the covariant derivative p 0 − eA 0 and
the dispersi<strong>on</strong> relati<strong>on</strong> p0 = Ep − µ for particles). Equati<strong>on</strong><br />
(8) implies that eA1γ1 = eA1γ0γ 5 and thus −eA1 can<br />
be identified as the axial (or chiral) chemical potential µ5.<br />
Therefore, we can c<strong>on</strong>clude;<br />
J 1 V = 1<br />
π<br />
∫<br />
dx µ5, (17)<br />
which correctly recovers the (3+1)-dimensi<strong>on</strong>al chiral<br />
magnetic current (4) <strong>on</strong>ce we multiply this by the Landau<br />
level density, eB/(2π). That is,<br />
jV = µ5<br />
π<br />
−→ jV = |eB|<br />
2π<br />
(<strong>in</strong> (1+1) dimensi<strong>on</strong>s)<br />
· µ5<br />
π<br />
(<strong>in</strong> (3+1) dimensi<strong>on</strong>s), (18)<br />
which co<strong>in</strong>cides with Eq. (4).<br />
Here, it is clear that the l<strong>on</strong>gitud<strong>in</strong>al gauge field A 1 ,<br />
which is the Chern-Sim<strong>on</strong>s number <strong>in</strong> (1+1) dimensi<strong>on</strong>s,<br />
plays the role <str<strong>on</strong>g>of</str<strong>on</strong>g> the chiral chemical potential µ5 <strong>in</strong> (3+1)<br />
dimensi<strong>on</strong>s. We note, however, that there is an important<br />
difference; usually µ5 is <strong>in</strong>troduced by hand as a c<strong>on</strong>stant,<br />
but <strong>in</strong> (1+1) dimensi<strong>on</strong>s A 1 must have t-dependence to allow<br />
for n<strong>on</strong>zero QW . We can th<strong>in</strong>k <str<strong>on</strong>g>of</str<strong>on</strong>g> a c<strong>on</strong>crete “<strong>in</strong>stant<strong>on</strong>”<br />
c<strong>on</strong>figurati<strong>on</strong> <strong>in</strong> (1+1) dimensi<strong>on</strong>s simply as<br />
A 1 (t, x) = 2πQW t<br />
= −Et, (19)<br />
eL T<br />
where we limit ourselves to the spatially homogeneous case<br />
and denote the spatial and temporal extents as L and T ,<br />
respectively, and then we have<br />
j 1 V (t) = J 1 V (t) eE<br />
= t. (20)<br />
L π<br />
From this, aga<strong>in</strong>, if multiplied by the Landau-level degeneracy<br />
we can correctly recover the current generati<strong>on</strong> rate<br />
given by Eq. (6), i.e.<br />
d(ejV )<br />
dt = e2E (<strong>in</strong> (1+1) dimensi<strong>on</strong>s)<br />
π<br />
−→ d(ejV ) |eB|<br />
=<br />
dt 2π · e2E (<strong>in</strong> (3+1) dimensi<strong>on</strong>s), (21)<br />
π<br />
which co<strong>in</strong>cides with Eq. (6).<br />
In the same way we can get a f<strong>in</strong>ite axial-vector current<br />
at f<strong>in</strong>ite quark chemical potential µ. To see the anomalous<br />
nature the important fact is that the relati<strong>on</strong> between the<br />
density and the chemical potential is given by the quantum<br />
anomaly <strong>in</strong> (1+1) dimensi<strong>on</strong>s, i.e.<br />
n = − eA0<br />
, (22)<br />
π<br />
which results from the anomaly. One can derive this expressi<strong>on</strong><br />
directly from n = ⟨ψ † (x)ψ(x)⟩ by <strong>in</strong>sert<strong>in</strong>g<br />
the gauge field as lim y 0 →x 0 ψ † (y) exp[−ie ∫ dtA 0 ]ψ(x).<br />
From this we can immediately reach,<br />
J 1 ∫<br />
5 = dx n = 1<br />
∫<br />
dx µ, (23)<br />
π<br />
which represents the chiral separati<strong>on</strong> effect. This is aga<strong>in</strong><br />
the anomaly relati<strong>on</strong> exactly same as that <strong>in</strong> (3+1) dimensi<strong>on</strong>s<br />
<strong>on</strong>ce multiplied by the Landau level density eB/2π.<br />
SCHWINGER MODEL<br />
So far the arguments and the result<strong>in</strong>g expressi<strong>on</strong>s are<br />
quite general. From now <strong>on</strong> we shall go <strong>in</strong>to the dynamical<br />
properties calculat<strong>in</strong>g microscopic quantities <strong>in</strong> a solvable<br />
(1+1)-dimensi<strong>on</strong>al model, i.e. the massless Schw<strong>in</strong>ger<br />
model. The easiest way to accomplish a calculati<strong>on</strong> <strong>in</strong> the<br />
Schw<strong>in</strong>ger model is to use mapp<strong>in</strong>g <strong>on</strong>to a free bos<strong>on</strong>ic<br />
theory. In our case, however, the bos<strong>on</strong>izati<strong>on</strong> rule is a<br />
bit more complicated than usual because we deal with not<br />
<strong>on</strong>ly fermi<strong>on</strong>ic fields (such as the chiral c<strong>on</strong>densate) but<br />
also gauge fields (such as the electric field). So, the Lagrangian<br />
density <str<strong>on</strong>g>of</str<strong>on</strong>g> the corresp<strong>on</strong>d<strong>in</strong>g theory should be<br />
L = 1<br />
2 (∂µ θ)(∂µθ) − mγ(∂ µ θ)(∂µϕ) − 1<br />
2 (∂µ ϕ)∂ 2 (∂µϕ)<br />
(24)<br />
with the bos<strong>on</strong> mass,<br />
m 2 γ = e2<br />
. (25)<br />
π<br />
If the ϕ-field is <strong>in</strong>tegrated out, we get a theory <strong>on</strong>ly <strong>in</strong> terms<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the θ-field that is free and has a mass by mγ. Such a<br />
scalar theory is usually used with the bos<strong>on</strong>izati<strong>on</strong> rule;<br />
j µ<br />
V = ¯ ψγ µ ψ = 1<br />
√ π ϵ µν ∂νθ, (26)<br />
j µ<br />
5 = ¯ ψγ µ γ 5 ψ = − 1<br />
√ π ∂ µ θ, (27)<br />
¯ψψ = −c mγ : cos(2 √ πθ) : (28)<br />
with the normal order<strong>in</strong>g : :. Now we remark that ϕ <strong>in</strong><br />
the Lagrangian density (24) comes from the gauge field,<br />
A µ = −ϵ µν∂νϕ (where ϕ <strong>in</strong>cludes an <strong>in</strong>stant<strong>on</strong>-like c<strong>on</strong>figurati<strong>on</strong><br />
∼ 1<br />
2Et2 which does not satisfy the periodic<br />
boundary c<strong>on</strong>diti<strong>on</strong> <strong>in</strong> the t-directi<strong>on</strong>). Then the electric<br />
field takes a form E = ∂2ϕ. Once we <strong>in</strong>tegrate the θ-field<br />
out from the theory, after the Gaussian <strong>in</strong>tegrati<strong>on</strong> <strong>in</strong> the<br />
functi<strong>on</strong>al formalism, Eq. (27) is replaced by<br />
j µ<br />
5<br />
= − 1<br />
√ π ∂ µ θ → − mγ<br />
√π ∂ µ ϕ = − e<br />
π ∂µ ϕ. (29)<br />
The anomaly relati<strong>on</strong> is then derived as<br />
∂µj µ<br />
5<br />
= − e<br />
π ∂2 ϕ = − e<br />
π E = −2qW , (30)<br />
which is fully c<strong>on</strong>sistent with the anomaly relati<strong>on</strong> (11).<br />
In the same manner we can express the vector current <strong>in</strong><br />
terms <str<strong>on</strong>g>of</str<strong>on</strong>g> ϕ, and then we f<strong>in</strong>d,<br />
j µ<br />
V<br />
e<br />
=<br />
π ϵµν µν ∂ν<br />
∂νϕ = 2ϵ<br />
∂2 qW . (31)<br />
It is easy to c<strong>on</strong>firm that this result is fully c<strong>on</strong>sistent with<br />
the previous relati<strong>on</strong>. That is, after the spatial <strong>in</strong>tegrati<strong>on</strong><br />
for the µ = 1 comp<strong>on</strong>ent (or ϕ and qW ), the spatial derivative<br />
∂1 drops and the right-hand side simplifies as −2/∂0,<br />
that is just a t-<strong>in</strong>tegrati<strong>on</strong>. Therefore the right-hand side f<strong>in</strong>ally<br />
becomes −2QW together with the spatial <strong>in</strong>tegrati<strong>on</strong>,<br />
and hence we obta<strong>in</strong> J 1 V = −2QW .
The above equati<strong>on</strong> gives a microscopic structure <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
current <strong>in</strong> more general cases with spatial modulati<strong>on</strong>. In<br />
momentum space we can re-express this as follows;<br />
j 1 V (ω, k) = −2iω<br />
ω 2 − k 2 qW (ω, k). (32)<br />
This is an <strong>in</strong>terest<strong>in</strong>g relati<strong>on</strong>. If ω → 0 is taken first, we<br />
see that j1 V (0, k) is vanish<strong>in</strong>g. To get the chiral magnetic<br />
current or a f<strong>in</strong>ite chiral magnetic c<strong>on</strong>ductivity, it is necessary<br />
to take the zero-momentum limit <strong>in</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> k → 0<br />
first and then ω → 0. This observati<strong>on</strong> is <strong>in</strong> fact c<strong>on</strong>sistent<br />
with the result <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>on</strong>e-loop calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the chiral<br />
magnetic c<strong>on</strong>ductivity [9].<br />
We po<strong>in</strong>t out that the structure <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (32) naturally appears<br />
from the transverse projecti<strong>on</strong>. That is, after the <strong>on</strong>eloop<br />
<strong>in</strong>tegrati<strong>on</strong> with the gauge potential source, <strong>in</strong> momentum<br />
space <strong>on</strong>e can f<strong>in</strong>d,<br />
j µ<br />
(<br />
V (ω, k) = −<br />
g µν − qµ q ν<br />
q 2<br />
) e<br />
π Aν(ω, k) (33)<br />
with q = (ω, k) and q 2 = ω − k 2 , from which <strong>on</strong>e can<br />
easily f<strong>in</strong>d that<br />
j 1 V (ω, k) = − ω2<br />
ω2 − k2 e<br />
π A1 (ω, k). (34)<br />
Because qW = (e/π)∂ 0 A 1 , <strong>on</strong>e can substitute A 1 =<br />
i(2π/e)qW /ω for A 1 above, from which we can immediately<br />
c<strong>on</strong>firm that the above expressi<strong>on</strong> is equivalent to<br />
Eq. (32).<br />
From the equivalence to the bos<strong>on</strong>ized theory it is very<br />
easy to read the electric current-current fluctuati<strong>on</strong> too. To<br />
this end we <strong>in</strong>tegrate the ϕ-field first, and then what we<br />
have is a free massive scalar theory <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> θ al<strong>on</strong>e.<br />
Then we trivially get,<br />
χj(x − y) = e 2 ⟨j 1 (x)j 1 (y)⟩ = m 2 γ ∂ x 0 ∂ y<br />
0 ⟨θ(x)θ(y)⟩,<br />
(35)<br />
or <strong>in</strong> momentum space we can express this as<br />
m 2 γ ω 2<br />
χj(ω, k) =<br />
ω2 − k2 − m2 . (36)<br />
γ + iϵ<br />
At a first glance this expressi<strong>on</strong> looks different from<br />
Eq. (5). This is because the above expressi<strong>on</strong> (36) is a result<br />
after resummati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble-type diagrams, while<br />
Eq. (5) is the <strong>on</strong>e-loop result. Roughly speak<strong>in</strong>g m 2 γ appears<br />
<strong>in</strong> the denom<strong>in</strong>ator <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (36) as a result <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>f<strong>in</strong>ite<br />
<strong>in</strong>serti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the polarizati<strong>on</strong> diagram. This <strong>in</strong>dicates that<br />
we can extract the <strong>on</strong>e-loop result from the lead<strong>in</strong>g-order<br />
Taylor expansi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (36) <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> m 2 γ. That actually<br />
leads to<br />
χ <strong>on</strong>e-loop<br />
j (ω, k) = m2γ ω2 ω2 − k2 → m 2 γ = e2<br />
π<br />
(k → 0). (37)<br />
Therefore,<br />
χ <strong>on</strong>e-loop<br />
j<br />
= e2<br />
π<br />
−→ χ <strong>on</strong>e-loop<br />
j<br />
(<strong>in</strong> (1+1) dimensi<strong>on</strong>s)<br />
= |eB|<br />
2π<br />
· e2<br />
π<br />
(<strong>in</strong> (3+1) dimensi<strong>on</strong>s), (38)<br />
which aga<strong>in</strong> co<strong>in</strong>cides with the previous result (5).<br />
SUMMARY<br />
The chiral magnetic effect is <str<strong>on</strong>g>of</str<strong>on</strong>g> theoretical and experimental<br />
<strong>in</strong>terest <strong>in</strong> the c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g> the relativistic heavy-i<strong>on</strong><br />
collisi<strong>on</strong>s where the str<strong>on</strong>g fields and the topological excitati<strong>on</strong>s<br />
(<strong>in</strong>stant<strong>on</strong>s and sphaler<strong>on</strong>s) exist, which leads to<br />
the (electric) current. Such an effect could be observed as<br />
charge asymmetry <strong>in</strong> experiments.<br />
There are a number <str<strong>on</strong>g>of</str<strong>on</strong>g> works that aim to clarify the<br />
microscopic properties <str<strong>on</strong>g>of</str<strong>on</strong>g> QCD matter related to the chiral<br />
magnetic effect. The explicit computati<strong>on</strong> is usually<br />
lengthy and cumbersome due to the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> the external<br />
magnetic field. We here discussed the dimensi<strong>on</strong>al reducti<strong>on</strong><br />
and dem<strong>on</strong>strated that many <str<strong>on</strong>g>of</str<strong>on</strong>g> known results can<br />
be reproduced without tedious calculati<strong>on</strong>s. As a model<br />
study we picked up the Schw<strong>in</strong>ger model, which is useful<br />
but drops the (3+1)-dimensi<strong>on</strong>al gauge dynamics. It is<br />
highly demanded to c<strong>on</strong>struct a full effective descripti<strong>on</strong><br />
valid for genu<strong>in</strong>e (3+1) dimensi<strong>on</strong>al gauge dynamics <strong>in</strong> the<br />
str<strong>on</strong>g magnetic field limit.<br />
The author thanks D.E. Kharzeev and H.J. Warr<strong>in</strong>ga for<br />
fruitful discussi<strong>on</strong>s and also he is grateful to G.V. Dunne<br />
for giv<strong>in</strong>g useful comments dur<strong>in</strong>g the workshop.<br />
REFERENCES<br />
[1] D. E. Kharzeev, L. D. McLerran and H. J. Warr<strong>in</strong>ga, Nucl.<br />
Phys. A 803, 227 (2008).<br />
[2] K. Fukushima, D. E. Kharzeev and H. J. Warr<strong>in</strong>ga, Phys. Rev.<br />
D 78, 074033 (2008).<br />
[3] D. E. Kharzeev, Annals Phys. 325, 205 (2010);<br />
arXiv:1010.0943 [hep-ph].<br />
[4] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D 72,<br />
045011 (2005).<br />
[5] B. I. Abelev et al. [STAR Collaborati<strong>on</strong>], Phys. Rev. Lett.<br />
103, 251601 (2009). N. N. Ajitanand, R. A. Lacey, A. Taranenko<br />
and J. M. Alexander, arXiv:1009.5624 [nucl-ex].<br />
[6] K. Fukushima, D. E. Kharzeev and H. J. Warr<strong>in</strong>ga, Nucl.<br />
Phys. A 836, 311 (2010).<br />
[7] K. Fukushima, D. E. Kharzeev and H. J. Warr<strong>in</strong>ga, Phys. Rev.<br />
Lett. 104, 212001 (2010).<br />
[8] G. Basar, G. V. Dunne, D. E. Kharzeev, Phys. Rev. Lett. 104,<br />
232301 (2010).<br />
[9] D. E. Kharzeev and H. J. Warr<strong>in</strong>ga, Phys. Rev. D 80, 034028<br />
(2009).
CRITICAL BEHAVIOR OF CHARMONIUM: QCD SECOND ORDER<br />
STARK EFFECT ∗<br />
Abstract<br />
Kenji Morita, GSI, Helmholzzentrum für Schweri<strong>on</strong>enforschung, Darmstadt, Germany †<br />
Su Houng Lee ‡ , Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong> and Applied <strong>Physics</strong>, Y<strong>on</strong>sei University, Seoul, Korea<br />
We study a mass shift <str<strong>on</strong>g>of</str<strong>on</strong>g> charm<strong>on</strong>ia <strong>in</strong> hot QCD medium<br />
near and below the critical temperature. We <strong>in</strong>troduce<br />
a formula for the mass shift by the QCD sec<strong>on</strong>d order<br />
Stark effect based <strong>on</strong> the operator producti<strong>on</strong> expansi<strong>on</strong>.<br />
Then we discuss the behavior <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field square<br />
<strong>on</strong> the basis <str<strong>on</strong>g>of</str<strong>on</strong>g> results from lattice QCD. We dem<strong>on</strong>strate<br />
the mass shift <str<strong>on</strong>g>of</str<strong>on</strong>g> J/ψ serves as a good <strong>in</strong>dicator <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
c<strong>on</strong>f<strong>in</strong>ement-dec<strong>on</strong>f<strong>in</strong>ement transiti<strong>on</strong>. We propose a res<strong>on</strong>ance<br />
gas model for the c<strong>on</strong>densate to describe the medium<br />
with n<strong>on</strong>zero bary<strong>on</strong>ic chemical potential. We discuss the<br />
mass shift at hadr<strong>on</strong>izati<strong>on</strong> temperature and chemical potential<br />
and possible implicati<strong>on</strong> for heavy i<strong>on</strong> experiments.<br />
INTRODUCTION<br />
Understand<strong>in</strong>g the c<strong>on</strong>f<strong>in</strong>ement phenomen<strong>on</strong> <strong>in</strong> QCD is<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> fundamental subject <strong>in</strong> modern physics. Despite the difficulty<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> solv<strong>in</strong>g QCD ow<strong>in</strong>g to the str<strong>on</strong>gly coupled nature,<br />
recent development <strong>in</strong> high-performance comput<strong>in</strong>g<br />
enables us to calculate bulk property <str<strong>on</strong>g>of</str<strong>on</strong>g> the hot QCD matter<br />
at T ∼ 200 MeV ∼ 10 12 K by M<strong>on</strong>te-Carlo simulati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> QCD <strong>on</strong> the lattice. The result at physical quark masses<br />
reveals that the matter at high temperature c<strong>on</strong>sists <str<strong>on</strong>g>of</str<strong>on</strong>g> dec<strong>on</strong>f<strong>in</strong>ed<br />
quarks and glu<strong>on</strong>s, called quark-glu<strong>on</strong> plasma,<br />
and that it undergoes a transiti<strong>on</strong> <strong>in</strong>to a hadr<strong>on</strong>ic gas at<br />
T ∼ 170 MeV. Currently relativistic heavy i<strong>on</strong> collisi<strong>on</strong>s<br />
provide unique opportunity to produce the matter <strong>on</strong> earth.<br />
However, the matter is far from ideal situati<strong>on</strong> because the<br />
created matter has <strong>on</strong>ly short lifetime ∼ 10 −22 sec and we<br />
can detect <strong>on</strong>ly f<strong>in</strong>ally produced particles such as pi<strong>on</strong>s and<br />
nucle<strong>on</strong>s. Therefore it is important to f<strong>in</strong>d an observable<br />
which carries <strong>in</strong>formati<strong>on</strong> <strong>on</strong> the transiti<strong>on</strong> from QGP to<br />
hadr<strong>on</strong> gas that would happen <strong>in</strong> the collisi<strong>on</strong> process. In<br />
this work, we focus <strong>on</strong> charm<strong>on</strong>ium and how it reacts with<br />
the change <str<strong>on</strong>g>of</str<strong>on</strong>g> the matter property. Indeed such an idea was<br />
proposed many years ago [1, 2] based <strong>on</strong> facts that the charm<strong>on</strong>ium<br />
spectrum can be well expla<strong>in</strong>ed by c<strong>on</strong>f<strong>in</strong>ement<br />
force [3] and that the force could be modified by decrease<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>in</strong>g tensi<strong>on</strong> and Debye screen<strong>in</strong>g <strong>in</strong> medium. We<br />
have been elaborat<strong>in</strong>g a method <strong>on</strong> the basis <str<strong>on</strong>g>of</str<strong>on</strong>g> the operator<br />
product expansi<strong>on</strong> (OPE) and <strong>in</strong>-medium change <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the glu<strong>on</strong> c<strong>on</strong>densates which gives lead<strong>in</strong>g c<strong>on</strong>tributi<strong>on</strong> to<br />
the OPE. The aim <str<strong>on</strong>g>of</str<strong>on</strong>g> this talk is to give an explanati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the role <str<strong>on</strong>g>of</str<strong>on</strong>g> the temperature dependent glu<strong>on</strong> c<strong>on</strong>den-<br />
∗ Work supported by FIAS and Korea M<strong>in</strong>istry <str<strong>on</strong>g>of</str<strong>on</strong>g> Educati<strong>on</strong><br />
† k.morita@gsi.de<br />
‡ suhoug@phya.y<strong>on</strong>sei.ac.kr<br />
sates as an effective order parameter <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>f<strong>in</strong>ementdec<strong>on</strong>f<strong>in</strong>ement<br />
transiti<strong>on</strong> <strong>in</strong> QCD and to discuss its c<strong>on</strong>sequence<br />
<strong>on</strong> the <strong>in</strong>-medium modificati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charm<strong>on</strong>ium.<br />
QCD SECOND ORDER STARK EFFECT<br />
Interacti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a heavy quark<strong>on</strong>ium with glu<strong>on</strong>s based <strong>on</strong><br />
the OPE was formulated by Pesk<strong>in</strong> [4, 5]. The primary<br />
po<strong>in</strong>t is to <strong>in</strong>troduce the separati<strong>on</strong> scale k, which is set<br />
to the b<strong>in</strong>d<strong>in</strong>g energy k ∼ ϵ <str<strong>on</strong>g>of</str<strong>on</strong>g> the quark<strong>on</strong>ium. S<strong>in</strong>ce<br />
the size <str<strong>on</strong>g>of</str<strong>on</strong>g> the Coulombic bound state with heavy quark<br />
mass m is a0 = 4/(Ncαsm) and the b<strong>in</strong>d<strong>in</strong>g energy is<br />
ϵ = 1/(ma 2 0), the separati<strong>on</strong> scale for sufficiently heavy<br />
quark can be large enough for perturbative treatment. Then<br />
the matrix element is calculated through the OPE <strong>in</strong> which<br />
the short distance process is implemented <strong>in</strong>to the Wils<strong>on</strong><br />
coefficients and the s<str<strong>on</strong>g>of</str<strong>on</strong>g>t <strong>on</strong>e is accounted for gauge <strong>in</strong>variant<br />
local operators. While the formulati<strong>on</strong> can be applied to<br />
scatter<strong>in</strong>g cross secti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the quark<strong>on</strong>ium-hadr<strong>on</strong> <strong>in</strong>teracti<strong>on</strong>s<br />
[5, 6], it reduces to the formula for a mass shift <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
quark<strong>on</strong>ium at rest by change <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field squared<br />
[7]. The formula with a normalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave functi<strong>on</strong><br />
∫ d 3 k<br />
(2π) 3 |ψ(k)| 2 = 1 reads<br />
∆m ¯ QQ = − 1<br />
18<br />
= − 7π2<br />
18<br />
∫∞<br />
dk 2<br />
<br />
<br />
<br />
∂ψ(k) <br />
<br />
∂k <br />
0<br />
a 2 0<br />
ϵ<br />
2<br />
k<br />
k2 ⟨<br />
αs<br />
/m + ϵ π ∆E2⟩<br />
med<br />
(1)<br />
⟨<br />
αs<br />
π ∆E2⟩ , (2)<br />
med<br />
where the sec<strong>on</strong>d l<strong>in</strong>e holds for the Coulombic wave functi<strong>on</strong>.<br />
As we will see below, the electric c<strong>on</strong>densate <strong>in</strong>creases<br />
as temperature rises up. This implies a downward<br />
mass shift <str<strong>on</strong>g>of</str<strong>on</strong>g> 1S quark<strong>on</strong>ium irrespective to the detail <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
wave functi<strong>on</strong>. In the formula with Coulomb wave functi<strong>on</strong>,<br />
<strong>on</strong>e sees that a dipole nature <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>teracti<strong>on</strong> explicitly<br />
appears as a 2 0 scal<strong>in</strong>g. This implies the mass shift does<br />
not depend <strong>on</strong> the detail <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave functi<strong>on</strong> but <strong>on</strong> the size<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the quark<strong>on</strong>ium. We fix the parameters as mc = 1704<br />
MeV and a0 = 0.271 fm by fitt<strong>in</strong>g with J/ψ mass, 3097<br />
MeV and the size <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave functi<strong>on</strong> <strong>in</strong> the Cornell potential<br />
model, ⟨r 2 ⟩ 1/2 = 0.47 fm, imply<strong>in</strong>g αs = 0.57.<br />
In the next secti<strong>on</strong>, we discuss the behavior <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric<br />
c<strong>on</strong>densate <strong>in</strong> the pure glu<strong>on</strong>ic system.<br />
PURE GLUONIC CASE<br />
We start with the pure glu<strong>on</strong>ic system as a vivid example<br />
for the relati<strong>on</strong> between the electric c<strong>on</strong>densate E 2 and
c<strong>on</strong>f<strong>in</strong>ement-dec<strong>on</strong>f<strong>in</strong>ement transiti<strong>on</strong>. It is c<strong>on</strong>venient to<br />
<strong>in</strong>troduce the follow<strong>in</strong>g quantities [8];<br />
⟨<br />
β(g)<br />
M0(T ) =<br />
2g Ga µνG aµν<br />
⟩<br />
, (3)<br />
T<br />
(<br />
uαuβ − 1<br />
4 gαβ<br />
) ⟨<br />
M2(T ) = −ST G a αµG aµ<br />
⟩<br />
β . (4)<br />
T<br />
These denote the decompositi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the glu<strong>on</strong>ic operator<br />
<strong>in</strong>to the trace anomaly part and traceless and symmetric<br />
<strong>on</strong>e, i.e., the scalar glu<strong>on</strong> c<strong>on</strong>densate and the twist-2 operator<br />
which stand for the lead<strong>in</strong>g c<strong>on</strong>tributi<strong>on</strong> to the OPE.<br />
Note that the twist-2 operator must be taken <strong>in</strong>to account<br />
for c<strong>on</strong>sistency <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> medium <strong>in</strong> which Lorentz<br />
<strong>in</strong>variance is absent. The M0 and M2 <strong>in</strong> the pure gauge<br />
theory are related to the energy density ε(T ) and pressure<br />
p(T ) which were obta<strong>in</strong>ed <strong>in</strong> lattice calculati<strong>on</strong>s as<br />
M0 = ε − 3p and M2 = ϵ + p. Then tak<strong>in</strong>g the <strong>on</strong>e-loop<br />
expressi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the beta functi<strong>on</strong> leads to the electric c<strong>on</strong>densate<br />
⟨<br />
αs<br />
π ∆E2⟩ =<br />
T<br />
2<br />
11 − 2<br />
3 Nf<br />
M0(T ) + 3<br />
4<br />
α eff<br />
s<br />
π M2(T ). (5)<br />
The equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> state is adopted from Ref. [9]. As for α eff<br />
s ,<br />
we use the effective coupl<strong>in</strong>g c<strong>on</strong>stant αqq(T ) which was<br />
measured by heavy quark free energy [10]. Putt<strong>in</strong>g Nf =<br />
0 <strong>in</strong>to Eq. (5), we plot the temperature dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electric c<strong>on</strong>densate near Tc = 264 MeV <strong>in</strong> Fig. 1. One<br />
sees a rapid rise <strong>in</strong> the vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g> the transiti<strong>on</strong> temperature<br />
which reflects the first order phase transiti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pure SU(3)<br />
theory. Indeed this behavior could be related with that <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the space-time Wils<strong>on</strong> loop which shows change from the<br />
area law to the perimeter law across Tc [11] through OPE<br />
for the Wils<strong>on</strong> loop calculated by Shifman [12].<br />
Putt<strong>in</strong>g Eq. (5) <strong>in</strong>to Eq. (2), we obta<strong>in</strong>ed the mass shift<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> J/ψ shown <strong>in</strong> Fig. 2. The downward mass shift occurs<br />
abruptly <strong>in</strong> the vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g> Tc, signal<strong>in</strong>g the phase transiti<strong>on</strong>.<br />
It reaches 40 MeV at Tc and 100 MeV at 1.05Tc.<br />
We note, however, that applicability <str<strong>on</strong>g>of</str<strong>on</strong>g> this method for the<br />
charm<strong>on</strong>ium is questi<strong>on</strong>able bey<strong>on</strong>d this temperature [8].<br />
RESONANCE GAS MODEL<br />
Now we turn to the realistic case <strong>in</strong>clud<strong>in</strong>g light quarks.<br />
It is known that QCD with physical quark mass exhibits<br />
a crossover transiti<strong>on</strong> such that the change <str<strong>on</strong>g>of</str<strong>on</strong>g> the thermodynamic<br />
quantity becomes smoother <strong>in</strong> the vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g> Tc.<br />
From an experimental po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> view, the system produced<br />
<strong>in</strong> heavy i<strong>on</strong> collisi<strong>on</strong>s will be QGP, which undergoes a<br />
transiti<strong>on</strong> <strong>in</strong>to a hadr<strong>on</strong>ic gas. The temperature and chemical<br />
potential at the hadr<strong>on</strong>izati<strong>on</strong> can be extracted from statistical<br />
model analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> particle ratios [13]. We c<strong>on</strong>sider<br />
the electric c<strong>on</strong>densate at these po<strong>in</strong>ts which are just below<br />
Tc.<br />
There are two difficulties to extract the c<strong>on</strong>densate from<br />
the lattice data as d<strong>on</strong>e <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the pure gauge theory.<br />
One is the so-called sign problem. Namely, lattice QCD<br />
(α s /π)E 2 [GeV 4 ]<br />
0.004<br />
0.002<br />
0<br />
-0.002<br />
-0.004<br />
(αs /π)E<br />
0.8 0.9 1<br />
T/Tc 1.1 1.2<br />
2<br />
Figure 1: Electric c<strong>on</strong>densates <str<strong>on</strong>g>of</str<strong>on</strong>g> pure glu<strong>on</strong>ic case near<br />
Tc. The value at low temperature limit is obta<strong>in</strong>ed from<br />
that <str<strong>on</strong>g>of</str<strong>on</strong>g> the glu<strong>on</strong> c<strong>on</strong>densate <strong>in</strong> vacuum, ⟨(αs/π)G 2 ⟩ =<br />
(0.35GeV) 4 .<br />
∆m [MeV]<br />
0<br />
-50<br />
-100<br />
-150<br />
-200<br />
-250<br />
J/ψ<br />
0.9 0.95 1 1.05<br />
T/Tc 1.1 1.15 1.2<br />
Figure 2: Mass shift <str<strong>on</strong>g>of</str<strong>on</strong>g> J/ψ from the sec<strong>on</strong>d order Stark<br />
effect with the electric c<strong>on</strong>densate shown <strong>in</strong> Fig. 1.<br />
cannot perform a simulati<strong>on</strong> at f<strong>in</strong>ite chemical potential.<br />
The other is that <strong>on</strong>e has to separate glu<strong>on</strong>ic c<strong>on</strong>tributi<strong>on</strong> to<br />
the thermodynamic quantities from those <strong>in</strong>clud<strong>in</strong>g quarks.<br />
Therefore, we use a res<strong>on</strong>ance gas model based <strong>on</strong> the l<strong>in</strong>ear<br />
density approximati<strong>on</strong> which has been used to estimate<br />
the glu<strong>on</strong> c<strong>on</strong>densate <strong>in</strong> the nuclear matter. We def<strong>in</strong>e M0<br />
and M2 for a res<strong>on</strong>ance gas, <strong>in</strong>troduced <strong>in</strong> previous secti<strong>on</strong>,<br />
as<br />
M had<br />
0 (T, µ) = ∑<br />
ρi(T, µ)m<br />
i=hadr<strong>on</strong>s<br />
0 i (6)<br />
M had<br />
2 (T, µ) = ∑<br />
ρi(T, µ)miA<br />
i=hadr<strong>on</strong>s<br />
i G. (7)<br />
One sees this expressi<strong>on</strong> is l<strong>in</strong>ear <strong>in</strong> the number density <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
i−th hadr<strong>on</strong>s, ρi. If we put ρi as normal nuclear density<br />
ρ0 = 0.17 fm −3 and pick up <strong>on</strong>ly nucle<strong>on</strong> c<strong>on</strong>tributi<strong>on</strong><br />
<strong>in</strong> the summati<strong>on</strong>, this formula reduces to the glu<strong>on</strong> c<strong>on</strong>densates<br />
<strong>in</strong> the nuclear matter [14]. In Eq. (6), the chiral<br />
limit is taken for the mass <str<strong>on</strong>g>of</str<strong>on</strong>g> hadr<strong>on</strong>s m 0 i to isolate<br />
the glu<strong>on</strong>ic c<strong>on</strong>tributi<strong>on</strong> to the trace anomaly. The sec<strong>on</strong>d<br />
moment <str<strong>on</strong>g>of</str<strong>on</strong>g> the glu<strong>on</strong> distributi<strong>on</strong> functi<strong>on</strong> A i G plays<br />
a similar role that picks up the glu<strong>on</strong>ic part <str<strong>on</strong>g>of</str<strong>on</strong>g> the twist-2<br />
term. Here we take the all res<strong>on</strong>ances given <strong>in</strong> the Particle<br />
Data Group [15] <strong>in</strong>to account. Masses <strong>in</strong> the chiral<br />
limit, however, are not known for most hadr<strong>on</strong>s. We use
different masses m 0 i ̸= mi <strong>on</strong>ly for the Nambu-Goldst<strong>on</strong>e<br />
bos<strong>on</strong>s and ground state octet and decouplet bary<strong>on</strong>s. We<br />
put m0 π = m0 K = 0 and m0N = 750 MeV [16]; these are<br />
the most important <strong>in</strong>puts as the c<strong>on</strong>tributi<strong>on</strong>s to the thermodynamic<br />
quantities are dom<strong>in</strong>ated by these hadr<strong>on</strong>s. For<br />
the vector and axial vector mes<strong>on</strong>s, we assume m0 ρ = mρ<br />
and m0 a1 = ma1. We also assume m0 ∆ = m∆. Furthermore,<br />
tak<strong>in</strong>g the flavor SU(3) limit, we also put m0 f0 = m0σ, m0 ϕ = m0ω = m0 K∗ = m0ρ. m0 Λ = m0Ξ = m0Σ = m0N ,<br />
m0 Σ∗ = m0Ξ ∗ = m0Ω = m0∆ . In general AiG can differ<br />
for hadr<strong>on</strong>s. We, however, put Ai G (8m2c) = 0.9 for all the<br />
can be shown to deviate little from<br />
hadr<strong>on</strong>s because Aπ G<br />
this value at such a high energy scale.<br />
At vanish<strong>in</strong>g chemical potential, lattice calculati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> state with physical quark mass are available<br />
[17, 18, 19]. The results have been compared with<br />
the res<strong>on</strong>ance gas. However, there is still uncerta<strong>in</strong>ty <strong>in</strong> the<br />
lattice results due to the discretizati<strong>on</strong> [20]. To take this<br />
uncerta<strong>in</strong>ty <strong>in</strong>to account, we <strong>in</strong>troduce an excluded volume<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> hadr<strong>on</strong>s v0 as a fitt<strong>in</strong>g parameter <strong>in</strong> the model by follow<strong>in</strong>g<br />
a prescripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Ref. [21]. While vanish<strong>in</strong>g excluded<br />
volume, v0 = 0, fits the lattice equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> state by<br />
the Wuppertal-Budapest collaborati<strong>on</strong> well [19], v0 = 1.19<br />
fm3 reproduces the scalar glu<strong>on</strong> c<strong>on</strong>densates by HotQCD<br />
collaborati<strong>on</strong> [18]. Figure 3 shows the comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
lattice data and the res<strong>on</strong>ance gas model. From this result,<br />
we may regard our v0 = 0 and v0 = 1.19 fm3 results as<br />
maximum and m<strong>in</strong>imum M0, respectively. After fix<strong>in</strong>g parameters,<br />
we can extend the model to <strong>in</strong>clude f<strong>in</strong>ite chemical<br />
potential.<br />
Table 1: Temperatures and chemical potentials at chemical<br />
freeze-out <strong>in</strong> heavy i<strong>on</strong> collisi<strong>on</strong>s at various energies. Data<br />
are taken from Ref. [13].<br />
√ sNN [GeV] T [MeV] µB[MeV]<br />
8.76 156 403<br />
12.3 154 298<br />
17.3 160 240<br />
130 165.5 38<br />
200 160.5 20<br />
Here we c<strong>on</strong>sider several sets <str<strong>on</strong>g>of</str<strong>on</strong>g> temperature and chemical<br />
potential which are estimated by the statistical model<br />
[13] and summarized <strong>in</strong> Table 1. The electric c<strong>on</strong>densates<br />
are obta<strong>in</strong>ed by putt<strong>in</strong>g M0(T, µB) and M2(T, µB)<br />
[Eqs. (6) and (7)] <strong>in</strong>to Eq. (5) with Nf = 3. We depict<br />
some examples <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric c<strong>on</strong>densate <strong>in</strong> Fig. 4 for illustrat<strong>in</strong>g<br />
the effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the chemical potential. One sees at<br />
µB = 403 MeV, the change <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric c<strong>on</strong>densate is<br />
much larger than the <strong>on</strong>e at µB = 20 MeV. Hence <strong>on</strong>e expects<br />
a larger mass shift at lower collisi<strong>on</strong> energies.<br />
Figure 5 shows the mass shift obta<strong>in</strong>ed from the electric<br />
c<strong>on</strong>densate <str<strong>on</strong>g>of</str<strong>on</strong>g> the res<strong>on</strong>ance gas and the sec<strong>on</strong>d order<br />
Stark effect. The band <strong>in</strong>dicates the possible range <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
mass shift <strong>in</strong>corporat<strong>in</strong>g the uncerta<strong>in</strong>ty <str<strong>on</strong>g>of</str<strong>on</strong>g> the lattice data<br />
M 0 [GeV 4 ]<br />
(ε-3p)/T 4<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0.004<br />
0.003<br />
0.002<br />
0.001<br />
0<br />
HotQCD N τ =8, p4<br />
HotQCD N τ =8, asqtad<br />
WB, C<strong>on</strong>t<strong>in</strong>uum limit<br />
Model, v 0 =1.19 fm 3<br />
0.57 fm 3<br />
0 fm 3<br />
HotQCD N τ =8, p4<br />
HotQCD N τ =8, asqtad<br />
WB full e-3p (not M 0 )<br />
Model, v 0 =1.19 fm 3<br />
0.57 fm 3<br />
0 fm 3<br />
140 145 150 155 160 165 170 175 180<br />
T [MeV]<br />
Figure 3: Upper: <strong>in</strong>teracti<strong>on</strong> measure (ε − 3p)/T 4 . Lower<br />
: its glu<strong>on</strong>ic part M0. The po<strong>in</strong>ts are taken from Refs. [18]<br />
for the “HotQCD” data and [19] for the “WB” (Wuppertal-<br />
Budapest) data while each l<strong>in</strong>e shows the result corresp<strong>on</strong>d<strong>in</strong>g<br />
to various v0.<br />
through vary<strong>in</strong>g v0. S<strong>in</strong>ce the hadr<strong>on</strong>izati<strong>on</strong> temperature is<br />
not m<strong>on</strong>ot<strong>on</strong>ic aga<strong>in</strong>st collid<strong>in</strong>g energy, the mass shift does<br />
not exhibit so <strong>in</strong> spite <str<strong>on</strong>g>of</str<strong>on</strong>g> the decreas<strong>in</strong>g chemical potential.<br />
Nevertheless, <strong>on</strong>e sees the largest mass shift at the lowest<br />
collid<strong>in</strong>g energy and the magnitude is 10–20 MeV at the<br />
higher <strong>on</strong>es.<br />
IMPLICATION FOR EXPERIMENTS<br />
F<strong>in</strong>ally we discuss possible implicati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the results<br />
for heavy i<strong>on</strong> experiments. Unfortunately, the complexity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the collisi<strong>on</strong> process prevents us from directly detect<strong>in</strong>g<br />
the mass shift through the shift <str<strong>on</strong>g>of</str<strong>on</strong>g> the peak <strong>in</strong> the dilept<strong>on</strong><br />
spectrum, even if the detector resoluti<strong>on</strong> is f<strong>in</strong>e enough to<br />
cover the magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the shift. In this case, a dynamical<br />
model is <strong>in</strong>dispensable to describe the space-time evoluti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the created matter and <strong>on</strong>e needs to estimate the number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> charm<strong>on</strong>ium which decay <strong>in</strong>side the medium [2, 22].<br />
Here we c<strong>on</strong>sider the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> the statistical<br />
hadr<strong>on</strong>izati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charm<strong>on</strong>ium [23, 24] and <strong>in</strong>fluence <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
mass shift <strong>on</strong> the charm<strong>on</strong>ium producti<strong>on</strong>. In fact, a measurement<br />
<strong>in</strong> the Pb+Pb collisi<strong>on</strong>s at CERN-SPS seems to<br />
support this scenario, because ψ ′ ratio to J/ψ can be well<br />
reproduced by the statistical producti<strong>on</strong> while that <strong>in</strong> the<br />
elementary collisi<strong>on</strong> such as p + p cannot be d<strong>on</strong>e so [25].<br />
We do not have to take <strong>in</strong>to account charm c<strong>on</strong>servati<strong>on</strong> if<br />
we restrict ourselves to the charm<strong>on</strong>ium-charm<strong>on</strong>ium ratio.<br />
To compare the experimental data, we need to <strong>in</strong>clude<br />
J/ψ from decay <str<strong>on</strong>g>of</str<strong>on</strong>g> higher res<strong>on</strong>ances. S<strong>in</strong>ce the<br />
sec<strong>on</strong>d order Stark effect applies to such res<strong>on</strong>ances, pro-
(α s /π)∆E 2 [GeV 4 ]<br />
0.003<br />
0.002<br />
0.001<br />
v 0 =0 fm 3 , µ B =403 MeV<br />
v 0 =1.19 fm 3 , µ B =403 MeV<br />
v 0 =0 fm 3 , µ B =20 MeV<br />
v 0 =1.19 fm 3 , µ B =20 MeV<br />
0<br />
130 135 140 145 150 155 160 165 170<br />
T [MeV]<br />
Figure 4: Electric c<strong>on</strong>densates as functi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> temperature.<br />
Each l<strong>in</strong>e stands for different chemical potential and the<br />
excluded volume.<br />
0<br />
∆m [MeV]<br />
-10<br />
-20<br />
-30<br />
-40<br />
J/ψ<br />
v 0 =1.19fm 3<br />
v 0 =0fm 3<br />
Stark<br />
10 1/2 100<br />
sNN [GeV]<br />
Figure 5: Mass shift <str<strong>on</strong>g>of</str<strong>on</strong>g> J/ψ at hadr<strong>on</strong>izati<strong>on</strong> temperature<br />
and chemical potentials for various collid<strong>in</strong>g energies.<br />
ducti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> those res<strong>on</strong>ances is also <strong>in</strong>fluenced by the mass<br />
shift. Based <strong>on</strong> the dipole nature (2), we assume the mass<br />
shift scales with the size <str<strong>on</strong>g>of</str<strong>on</strong>g> the res<strong>on</strong>ance. Then we have<br />
∆mχc ≃ −24 ∼ −49 MeV and ∆mψ ′ ≃ −40 ∼ −82<br />
MeV, respectively. In fact, this crude estimate for χc is<br />
close to more precise <strong>on</strong>e obta<strong>in</strong>ed from QCD sum rules<br />
[26]. Assum<strong>in</strong>g the same branch<strong>in</strong>g ratio <strong>in</strong> medium as <strong>in</strong><br />
vacuum, we calculate the number ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> ψ ′ to J/ψ <strong>in</strong>clud<strong>in</strong>g<br />
decay c<strong>on</strong>tributi<strong>on</strong> to J/ψ from mass-shifted χc<br />
and ψ ′ . The result is shown <strong>in</strong> Fig. 6 together with the experimental<br />
data. We plot the result as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> ψ ′ mass<br />
shift which is not known well. One sees an enhancement<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the ratio for large ψ ′ mass shift. While larger mass shift<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> ψ ′ than 100 MeV does not seem c<strong>on</strong>sistent with the experimental<br />
data, our rough estimati<strong>on</strong> is still <strong>in</strong>side the experimental<br />
band. If such an enhancement is c<strong>on</strong>firmed, it<br />
will prove the mass shift <str<strong>on</strong>g>of</str<strong>on</strong>g> the charm<strong>on</strong>ia as a precursor <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the c<strong>on</strong>f<strong>in</strong>ement-dec<strong>on</strong>f<strong>in</strong>ement transiti<strong>on</strong>. Details <strong>in</strong>clud<strong>in</strong>g<br />
analyses with QCD sum rules have been presented <strong>in</strong><br />
Ref. [26].<br />
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Statistical producti<strong>on</strong> w/ mass shift<br />
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-200 -150 -100 -50 0<br />
ψ′ mass shift [MeV]<br />
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ON THE UNRUH EFFECT ∗<br />
Ralf Schützhold † , Fakultät für Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany<br />
Abstract<br />
After a brief <strong>in</strong>troducti<strong>on</strong> <strong>in</strong>to the Unruh effect and its<br />
generalizati<strong>on</strong> to n<strong>on</strong>-uniform (here circular) accelerati<strong>on</strong>,<br />
we discuss prospects for measur<strong>in</strong>g signatures <str<strong>on</strong>g>of</str<strong>on</strong>g> this effect<br />
<strong>in</strong> str<strong>on</strong>g lasers.<br />
INTRODUCTION<br />
The Unruh effect [1] describes the strik<strong>in</strong>g discovery<br />
that an accelerated observer/detector experiences the<br />
M<strong>in</strong>kowski vacuum as a thermal bath – imply<strong>in</strong>g that<br />
the particle c<strong>on</strong>cept depends <strong>on</strong> the <strong>in</strong>ertial state <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
observer/detector. After important preparatory works by<br />
Full<strong>in</strong>g [2] and Davies [3], Unruh [1] realized this phenomen<strong>on</strong><br />
while try<strong>in</strong>g to understand how black holes can<br />
evaporate by emitt<strong>in</strong>g Hawk<strong>in</strong>g radiati<strong>on</strong> [4]. Around the<br />
same time, the mathematical foundati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this strik<strong>in</strong>g fact<br />
was established by Bisognano & Wichmann [5] by show<strong>in</strong>g<br />
the relati<strong>on</strong> between the R<strong>in</strong>dler Hamilt<strong>on</strong>ian and thermality<br />
– but apparently without immediately realiz<strong>in</strong>g the<br />
broad physical significance.<br />
However, so far this predicti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum field theory<br />
has eluded a direct experimental verificati<strong>on</strong>, see also [6].<br />
There are some observati<strong>on</strong>s regard<strong>in</strong>g the imperfect polarizability<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s <strong>in</strong> storage r<strong>in</strong>gs which are related to<br />
the Sokolov-Ternov effect [7] and can be <strong>in</strong>terpreted as an<br />
<strong>in</strong>direct verificati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh effect generalized to the<br />
case <str<strong>on</strong>g>of</str<strong>on</strong>g> circular accelerati<strong>on</strong>, see, e.g., [8, 9].<br />
In the follow<strong>in</strong>g, we briefly discuss a recent proposal<br />
[10, 11] for directly observ<strong>in</strong>g signatures <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh<br />
effect <strong>in</strong> the form <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> pairs created by electr<strong>on</strong>s<br />
which are accelerated <strong>in</strong> an ultra-str<strong>on</strong>g laser field, see also<br />
[12, 13]. We start with a short <strong>in</strong>troducti<strong>on</strong> <strong>in</strong>to the Unruh<br />
effect for uniform accelerati<strong>on</strong> and discuss the generalizati<strong>on</strong><br />
to circular accelerati<strong>on</strong>, which is relevant for electr<strong>on</strong>s<br />
<strong>in</strong> storage r<strong>in</strong>gs and the Sokolov-Ternov effect.<br />
DETECTOR PLUS FIELD<br />
Let us c<strong>on</strong>sider the follow<strong>in</strong>g acti<strong>on</strong> for the field ϕ and<br />
the detector with excitati<strong>on</strong> energy E, coupled with the<br />
coupl<strong>in</strong>g strength g (we use = c = 1)<br />
A = Adetector + Afield<br />
∫ [<br />
E<br />
= dτ<br />
2 σz<br />
]<br />
+ gσxϕ (x[τ])<br />
+ 1<br />
∫<br />
2<br />
d 4 x [(∂µϕ)(∂ µ ϕ)] , (1)<br />
∗ Work supported by DFG under grant SCHU 1557/1.<br />
† ralf.schuetzhold@uni-due.de<br />
where σz and σx are the Pauli matrices and τ is the proper<br />
time al<strong>on</strong>g the detector trajectory. In the <strong>in</strong>teracti<strong>on</strong> picture,<br />
the transiti<strong>on</strong> Hamilt<strong>on</strong>ian reads (with 2σ± = σx ± iσy)<br />
]<br />
ˆϕ (x[τ]) , (2)<br />
ˆH<strong>in</strong>t(τ) = g(τ) [ e iEτ σ+ + e −iEτ σ−<br />
where g(τ) is smooth switch<strong>in</strong>g (<strong>on</strong> and <str<strong>on</strong>g>of</str<strong>on</strong>g>f) functi<strong>on</strong>. Initially<br />
(where the <strong>in</strong>teracti<strong>on</strong> is switched <str<strong>on</strong>g>of</str<strong>on</strong>g>f) both, detector<br />
and field, are <strong>in</strong> their ground state<br />
|Ψ<strong>in</strong>⟩ = |Ψ(τ ↓ −∞)⟩<br />
= |Ψdetector⟩ ⊗ |Ψfield⟩ = |↓⟩ ⊗ |0⟩ . (3)<br />
Assum<strong>in</strong>g small coupl<strong>in</strong>g g, we may derive the f<strong>in</strong>al state<br />
via perturbati<strong>on</strong> theory<br />
|Ψout⟩ = |Ψ(τ ↑ +∞)⟩<br />
∫<br />
= |Ψ<strong>in</strong>⟩ − i<br />
dτ ˆ H<strong>in</strong>t(τ) |Ψ<strong>in</strong>⟩ + O(g 2 ) .(4)<br />
This yields the excitati<strong>on</strong> probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the detector<br />
P↑ = ⟨Ψout| ↑⟩ ⟨↑ |Ψout⟩<br />
=<br />
∫ ∫<br />
dτ dτ ′ g(τ) g(τ ′ ) e iE(τ−τ ′ )<br />
×<br />
× ⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ . (5)<br />
For simplicity, we c<strong>on</strong>sider the Wightmann functi<strong>on</strong> for a<br />
massless scalar field <strong>in</strong> 3+1 dimensi<strong>on</strong>s<br />
⟨0| ˆ ϕ (x) ˆ ϕ (x ′ ) |0⟩ = − 1<br />
(2π) 2<br />
1<br />
(t − t ′ ) 2 − (r − r ′ , (6)<br />
) 2<br />
where the pole structure (at the light-c<strong>on</strong>e) is understood <strong>in</strong><br />
such a way that the Fourier transform <str<strong>on</strong>g>of</str<strong>on</strong>g> the Wightmann<br />
functi<strong>on</strong> <strong>on</strong>ly c<strong>on</strong>ta<strong>in</strong>s n<strong>on</strong>-negative energies. Roughly<br />
speak<strong>in</strong>g, (t − t ′ ) 2 is replaced by (t − t ′ − iε) 2 with ε ↓ 0.<br />
UNIFORM ACCELERATION<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> (eternal) uniform accelerati<strong>on</strong> a, the detector<br />
trajectory <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the proper time τ reads<br />
t[τ] = 1<br />
a<br />
s<strong>in</strong>h(aτ) , x[τ] = 1<br />
a<br />
cosh(aτ) , y = z = 0 . (7)<br />
Evaluat<strong>in</strong>g the two-po<strong>in</strong>t functi<strong>on</strong> al<strong>on</strong>g this trajectory<br />
⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ = − 1<br />
2(2π) 2<br />
a2 cosh(a[τ − τ ′ , (8)<br />
]) − 1<br />
we f<strong>in</strong>d a stati<strong>on</strong>ary expressi<strong>on</strong> which is periodic al<strong>on</strong>g the<br />
imag<strong>in</strong>ary τ− = τ − τ ′ -axis and possesses double poles at<br />
this axis<br />
a[τ − τ ′ ] ∈ 2πi N . (9)
In the stati<strong>on</strong>ary case, it is useful to change the <strong>in</strong>tegrati<strong>on</strong><br />
variables <strong>in</strong> Eq. (5) from τ and τ ′ to τ− = τ − τ ′ and<br />
τ+ = (τ + τ ′ )/2. For positive E, we may deform the τ−<strong>in</strong>tegrati<strong>on</strong><br />
<strong>in</strong>to the upper complex half plane ℑ(τ−) > 0<br />
and close the <strong>in</strong>tegrati<strong>on</strong> c<strong>on</strong>tour at <strong>in</strong>f<strong>in</strong>ity. As a result, the<br />
τ−-<strong>in</strong>tegral is just the sum over all poles with ℑ(τ−) > 0.<br />
Due to periodicity, all residuals are equal and so their sum<br />
yields a geometric series<br />
∞∑<br />
{<br />
P↑ ∝ exp − 2πnE<br />
}<br />
1<br />
=<br />
. (10)<br />
a exp{2πE/a} − 1<br />
n=1<br />
Thus the excitati<strong>on</strong> probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the detector is given by a<br />
thermal spectrum with the Unruh temperature (kB = 1)<br />
TUnruh = a<br />
. (11)<br />
2π<br />
CIRCULAR ACCELERATION<br />
For comparis<strong>on</strong>, let us c<strong>on</strong>sider the case <str<strong>on</strong>g>of</str<strong>on</strong>g> circular moti<strong>on</strong><br />
where the magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the accelerati<strong>on</strong> rema<strong>in</strong>s c<strong>on</strong>stant<br />
but its directi<strong>on</strong> changes all the time. In this case, the<br />
detector trajectory is given by<br />
x[τ] = R s<strong>in</strong>(γωτ) , y[τ] = R cos(γωτ) , z = 0 ,<br />
t[τ] = γτ =<br />
τ<br />
√ . (12)<br />
1 − R2ω2 The two-po<strong>in</strong>t functi<strong>on</strong> al<strong>on</strong>g this trajectory<br />
− 1<br />
(2π) 2<br />
⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ =<br />
1<br />
γ 2 (τ − τ ′ ) 2 − 4R 2 s<strong>in</strong> 2 (γω[τ − τ ′ ]/2)<br />
(13)<br />
is aga<strong>in</strong> stati<strong>on</strong>ary but no l<strong>on</strong>ger periodic. Now the poles<br />
lie at γ(τ − τ ′ ) = ±2R s<strong>in</strong>(γω[τ − τ ′ ]/2). Introduc<strong>in</strong>g<br />
the abbreviati<strong>on</strong> φ = γω[τ − τ ′ ]/2, we have φ = β s<strong>in</strong> φ<br />
where β = ±Rω.<br />
Ultra-relativistic Case<br />
The above transcendental equati<strong>on</strong> for the locati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the poles simplifies if the detector velocity approaches the<br />
speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light. Us<strong>in</strong>g a Taylor expansi<strong>on</strong> (assum<strong>in</strong>g β > 0)<br />
(<br />
φ = β s<strong>in</strong> φ = β φ − φ3<br />
6 + O(φ5 )<br />
) , (14)<br />
we f<strong>in</strong>d apart from the trivial pole at φ = 0<br />
φ 2 β − 1 3<br />
= 6 ≈ −<br />
β γ2 ❀ τ − τ ′ √<br />
12<br />
= ±<br />
γ2 i .<br />
ω<br />
(15)<br />
This yields the excitati<strong>on</strong> probability, see also [9]<br />
{ √<br />
12 E<br />
P↑ ∝ exp −<br />
γ2 }<br />
. (16)<br />
ω<br />
Even though the spectrum is not exactly thermal, we may<br />
identify an effective temperature (for large E) via<br />
Teff ≈ γ2 ω<br />
√ 12 ≈ a<br />
√ 12 , (17)<br />
where we have used that a ≈ γ 2 ω for β ↑ 1.<br />
SIGNATURES<br />
Instead <str<strong>on</strong>g>of</str<strong>on</strong>g> the excitati<strong>on</strong> probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the detector <strong>in</strong><br />
Eq. (5), we may also study the excitati<strong>on</strong> probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
quantum field itself via the particle number operator ˆ Nk<br />
Pk = ⟨Ψout| ˆ Nk |Ψout⟩<br />
∫<br />
<br />
∝ <br />
dτ g(τ) e iEτ+ikt[τ]+ik·r[τ]<br />
<br />
<br />
<br />
<br />
2<br />
. (18)<br />
Compar<strong>in</strong>g the structure <str<strong>on</strong>g>of</str<strong>on</strong>g> the two expressi<strong>on</strong>s for P↑ and<br />
Pk, we f<strong>in</strong>d that for each excitati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the detector, exactly<br />
<strong>on</strong>e particle has been created [14]. Now, if the detector<br />
decays aga<strong>in</strong> after some time, another particle is created.<br />
So <strong>in</strong> total, the state <str<strong>on</strong>g>of</str<strong>on</strong>g> the detector is the same aga<strong>in</strong> – but<br />
a pair <str<strong>on</strong>g>of</str<strong>on</strong>g> particles has been created.<br />
Tak<strong>in</strong>g the limit <str<strong>on</strong>g>of</str<strong>on</strong>g> the time between excitati<strong>on</strong> and decay<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the detector go<strong>in</strong>g to zero, we effectively have a scatter<strong>in</strong>g<br />
event. In this limit, we may replace the detector by an<br />
electr<strong>on</strong>, which can scatter phot<strong>on</strong>s via Thoms<strong>on</strong> (Compt<strong>on</strong>)<br />
scatter<strong>in</strong>g. Thus an accelerated electr<strong>on</strong> would create<br />
pairs <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s – which can be <strong>in</strong>terpreted as a signature<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh effect [10, 11], see also [12, 13, 15]. As we<br />
have seen before, this effect is not restricted to uniform accelerati<strong>on</strong>,<br />
but also occurs <strong>in</strong> the more general case.<br />
REFERENCES<br />
[1] W. G. Unruh, Phys. Rev. D 14, 870 (1976).<br />
[2] S. A. Full<strong>in</strong>g, Phys. Rev. D 7, 2850 (1973).<br />
[3] P. C. W. Davies, J. Phys. A 8, 609 (1975).<br />
[4] S. W. Hawk<strong>in</strong>g, Nature 248, 30 (1974); Comm. Math. Phys.<br />
43, 199 (1975).<br />
[5] J.J. Bisognano, E.H. Wichmann, J. Math. Phys. 17, 303<br />
(1976).<br />
[6] H. C. Rosu, Grav. Cosmol. 7, 1 (2001); Int. J. Mod. Phys. D<br />
3, 545 (1994); <strong>Physics</strong> World, October 1999, 21-22.<br />
[7] A.A. Sokolov, I.M. Ternov, Sov. Phys. Dokl. 8, 1203 (1964).<br />
[8] J. S. Bell and J. M. Le<strong>in</strong>aas, Nucl. Phys. B 284, 488 (1987);<br />
W. G. Unruh, Phys. Rept. 307, 163 (1998).<br />
[9] E. T. Akhmedov, D. S<strong>in</strong>glet<strong>on</strong>, Pisma Zh. Eksp. Teor. Fiz.<br />
86, 702-706 (2007); Int. J. Mod. Phys. A22, 4797-4823<br />
(2007).<br />
[10] R. Schützhold, G. Schaller, D. Habs, Phys. Rev. Lett. 97,<br />
121302 (2006).<br />
[11] R. Schützhold, G. Schaller, D. Habs, Phys. Rev. Lett. 100,<br />
091301 (2008).<br />
[12] P. Chen and T. Tajima, Phys. Rev. Lett. 83, 256 (1999).<br />
[13] R. Schützhold, C. Maia, Eur. Phys. J. D55, 375 (2009).<br />
[14] W. G. Unruh and R. M. Wald, Phys. Rev. D 29, 1047 (1984).<br />
[15] Ya.B. Zeldovich, L.V. Rozhanskii, A.A. Starob<strong>in</strong>skii, Pisma<br />
Zh. Eksp. Teor. Fiz. 43, 407 (1986).
Abstract<br />
Can we detect ”Unruh radiati<strong>on</strong>” <strong>in</strong> the high <strong>in</strong>tensity lasers? ∗<br />
Satoshi Iso † , Yasuhiro Yamamoto ‡ and Sen Zhang § , <strong>KEK</strong>, Tsukuba, Japan<br />
An accelerated particle sees the M<strong>in</strong>kowski vacuum as<br />
thermally excited, which is called the Unruh effect. Due<br />
to an <strong>in</strong>teracti<strong>on</strong> with the thermal bath, the particle moves<br />
stochastically like the Brownian moti<strong>on</strong> <strong>in</strong> a heat bath. It<br />
has been discussed that the accelerated charged particle<br />
may emit extra radiati<strong>on</strong> (the Unruh radiati<strong>on</strong> [2]) besides<br />
the Larmor radiati<strong>on</strong>, and experiments are under plann<strong>in</strong>g<br />
to detect such radiati<strong>on</strong> by us<strong>in</strong>g ultrahigh <strong>in</strong>tensity lasers<br />
[3, 4]. There are, however, counterarguments that the radiati<strong>on</strong><br />
is canceled by an <strong>in</strong>terference effect between the<br />
vacuum fluctuati<strong>on</strong> and the radiati<strong>on</strong> from the fluctuat<strong>in</strong>g<br />
moti<strong>on</strong>. In this and another reports [5], we review our recent<br />
analysis <strong>on</strong> the issue <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong>. In this<br />
report, we particularly c<strong>on</strong>sider the thermalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an<br />
accelerated particle <strong>in</strong> the scalar QED, and derive the relaxati<strong>on</strong><br />
time <str<strong>on</strong>g>of</str<strong>on</strong>g> the thermalizati<strong>on</strong>. The <strong>in</strong>terference effect<br />
is discussed separately <strong>in</strong> [5].<br />
UNRUH EFFECT AND UNRUH<br />
RADIATION<br />
Quantum field theories <strong>in</strong> the space-time with horiz<strong>on</strong>s<br />
exhibit <strong>in</strong>terest<strong>in</strong>g thermodynamic behavior. The most<br />
prom<strong>in</strong>ent phenomen<strong>on</strong> is the Hawk<strong>in</strong>g radiati<strong>on</strong> and the<br />
fundamental laws <str<strong>on</strong>g>of</str<strong>on</strong>g> thermodynamics hold <strong>in</strong> the black<br />
hole background. A similar phenomen<strong>on</strong> occurs for a<br />
uniformly accelerated observer <strong>in</strong> the ord<strong>in</strong>ary M<strong>in</strong>kowski<br />
vacuum [6]. This is called the Unruh effect. If a particle<br />
is uniformly accelerated <strong>in</strong> the M<strong>in</strong>kowski space with an<br />
accelerati<strong>on</strong> a, there is a causal horiz<strong>on</strong> (the R<strong>in</strong>dler horiz<strong>on</strong>)<br />
and no <strong>in</strong>formati<strong>on</strong> can be transmitted from the other<br />
side <str<strong>on</strong>g>of</str<strong>on</strong>g> the horiz<strong>on</strong>. Because <str<strong>on</strong>g>of</str<strong>on</strong>g> the existence <str<strong>on</strong>g>of</str<strong>on</strong>g> the R<strong>in</strong>dler<br />
horiz<strong>on</strong>, the accelerated observer sees the M<strong>in</strong>kowski vac-<br />
uum as thermally excited with the Unruh temperature<br />
TU = a<br />
2πckB<br />
= 4 × 10 −23<br />
(<br />
a<br />
1 cm/s 2<br />
)<br />
[K]. (1)<br />
S<strong>in</strong>ce the Unruh temperature is very small for ord<strong>in</strong>ary<br />
accelerati<strong>on</strong>, it was very difficult to detect the Unruh effect.<br />
But with the ultra-high <strong>in</strong>tensity lasers, the Unruh effect<br />
can be experimentally accessible. In the electro-magnetic<br />
field <str<strong>on</strong>g>of</str<strong>on</strong>g> a laser with <strong>in</strong>tensity I, the Unruh temperature is<br />
given by<br />
TU = 8 × 10 −11<br />
√<br />
I<br />
2 [K]. (2)<br />
1 W/cm<br />
∗ The report is based <strong>on</strong> a talk by S.Zhang and [1].<br />
† satoshi.iso@kek.jp<br />
‡ yamayasu@post.kek.jp<br />
§ zhangsen@post.kek.jp<br />
The Extreme Light Infrastructure project [4] is plann<strong>in</strong>g to<br />
c<strong>on</strong>struct Peta Watt lasers with an <strong>in</strong>tensity as high as 5 ×<br />
10 26 [W/cm 2 ]. The expected Unruh temperature becomes<br />
more than 10 3 K. So it is time to ask ourselves how we can<br />
experimentally observe such high Unruh temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> an<br />
accelerated electr<strong>on</strong> <strong>in</strong> the laser field.<br />
Chen and Tajima proposed that <strong>on</strong>e may be able to detect<br />
the Unruh effect by observ<strong>in</strong>g quantum radiati<strong>on</strong> [2] from<br />
the electr<strong>on</strong>. It is called the Unruh radiati<strong>on</strong>. S<strong>in</strong>ce a uniformly<br />
accelerated electr<strong>on</strong> feels the vacuum (with quantum<br />
virtual pair creati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> particles and anti-particles) as<br />
thermally excited with the Unruh temperature, the moti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> fluctuates and is expected to become thermalized(Fig.<br />
1). This fluctuat<strong>in</strong>g moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong><br />
Figure 1: Stochastic trajectories <str<strong>on</strong>g>of</str<strong>on</strong>g> a uniformly accelerated<br />
electr<strong>on</strong> affected by quantum field fluctuati<strong>on</strong>s.<br />
changes the accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> and may produce<br />
additi<strong>on</strong>al radiati<strong>on</strong> (the Unruh radiati<strong>on</strong>) to the ord<strong>in</strong>ary<br />
Larmor radiati<strong>on</strong>. The rough estimati<strong>on</strong> [2] suggested that<br />
the strength <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong> is much smaller than the<br />
classical <strong>on</strong>e by 10 −5 , but the angular dependence becomes<br />
quite different. Especially <strong>in</strong> the directi<strong>on</strong> al<strong>on</strong>g the accelerati<strong>on</strong><br />
there is a bl<strong>in</strong>d spot for the Larmor radiati<strong>on</strong> while<br />
the Unruh radiati<strong>on</strong> is expected to be radiated more spherically.<br />
Hence they proposed to detect the Unruh radiati<strong>on</strong><br />
by sett<strong>in</strong>g a phot<strong>on</strong> detector <strong>in</strong> this directi<strong>on</strong>.<br />
The above argument seems <strong>in</strong>tuitively correct, but there<br />
are two problems that should be clarified. The first problem<br />
is the thermalizati<strong>on</strong> time <str<strong>on</strong>g>of</str<strong>on</strong>g> the fluctuati<strong>on</strong>. The electromagnetic<br />
field <str<strong>on</strong>g>of</str<strong>on</strong>g> laser are not c<strong>on</strong>stant but oscillat<strong>in</strong>g. One<br />
may approximate the electr<strong>on</strong>’s moti<strong>on</strong> around the turn<strong>in</strong>g<br />
po<strong>in</strong>ts by a uniform accelerati<strong>on</strong>. This approximati<strong>on</strong> is<br />
valid <strong>on</strong>ly when the period <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser is large enough compared<br />
to the relaxati<strong>on</strong> time (or thermalizati<strong>on</strong> time) <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
particle’s fluctuati<strong>on</strong>. Us<strong>in</strong>g a stochastic approach, we ob-
ta<strong>in</strong>ed the relaxati<strong>on</strong> time <str<strong>on</strong>g>of</str<strong>on</strong>g> the fluctuati<strong>on</strong> and showed that<br />
the relaxati<strong>on</strong> time is l<strong>on</strong>ger than the period <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser. In<br />
such a case, we must fully analyze the transient dynamics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the fluctuati<strong>on</strong> to calculate the radiati<strong>on</strong> <strong>in</strong> the laser field.<br />
The sec<strong>on</strong>d problem is the <strong>in</strong>terference effect. S<strong>in</strong>ce the<br />
Unruh radiati<strong>on</strong> orig<strong>in</strong>ates <strong>in</strong> the <strong>in</strong>teracti<strong>on</strong> with the particle<br />
with the quantum fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum, we cannot<br />
neglect the <strong>in</strong>terference <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong> and the vacuum<br />
quantum fluctuati<strong>on</strong>s. In a simpler model, it has been<br />
known that the Unruh radiati<strong>on</strong> is completely canceled by<br />
the <strong>in</strong>terference effect. The cancellati<strong>on</strong> was shown for<br />
the Unruh detector <strong>in</strong> both 1+1 and 3+1 dimensi<strong>on</strong>s[7, 8].<br />
There was no calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference effect <strong>in</strong> the<br />
case <str<strong>on</strong>g>of</str<strong>on</strong>g> the uniformly accelerated charged particle s<strong>in</strong>ce the<br />
calculati<strong>on</strong> needs some technicalities. In the paper [1] we<br />
calculated the <strong>in</strong>terference effect for the charged particle<br />
<strong>in</strong> the scalar QED and found that some <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong><br />
is canceled by the <strong>in</strong>terference effect, but the cancellati<strong>on</strong><br />
occurs <strong>on</strong>ly partially. So we still have a possibility<br />
to detect additi<strong>on</strong>al radiati<strong>on</strong> from the uniformly accelerated<br />
charged particle, but the complete understand<strong>in</strong>g<br />
needs more detailed analysis.<br />
In the rest <str<strong>on</strong>g>of</str<strong>on</strong>g> this report we first briefly review the<br />
stochastic model <str<strong>on</strong>g>of</str<strong>on</strong>g> a uniformly accelerated charged particle<br />
and then show how the thermalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the fluctuati<strong>on</strong><br />
occurs by solv<strong>in</strong>g the stochastic equati<strong>on</strong>. F<strong>in</strong>ally we<br />
briefly sketch the calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong>, particularly<br />
put emphasis <strong>on</strong> the <strong>in</strong>terference effect. More details <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference effect and the Unruh radiati<strong>on</strong><br />
are reviewed <strong>in</strong> another report <str<strong>on</strong>g>of</str<strong>on</strong>g> the same authors <strong>in</strong><br />
the proceed<strong>in</strong>gs [5].<br />
THERMALIZATION<br />
We c<strong>on</strong>sider the scalar QED. The model is analyzed<br />
<strong>in</strong> [9] and here we briefly review the sett<strong>in</strong>gs and the derivati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the stochastic Abraham-Lorentz-Dirac (ALD) equati<strong>on</strong>.<br />
The system composes <str<strong>on</strong>g>of</str<strong>on</strong>g> a relativistic particle z µ (τ)<br />
and the scalar field ϕ(x). The acti<strong>on</strong> is given by<br />
∫<br />
S[z, ϕ, h] = − m<br />
∫<br />
+<br />
dτ √ ˙z µ ∫<br />
˙zµ +<br />
d 4 x 1 2<br />
(∂µϕ)<br />
2<br />
d 4 x j(x; z)ϕ(x). (3)<br />
The scalar current j(x; z) is def<strong>in</strong>ed as<br />
∫<br />
j(x; z) = e dτ √ ˙z µ ˙zµ δ 4 (x − z(τ)), (4)<br />
We choose the parametrizati<strong>on</strong> τ to satisfy ˙z 2 = 1.<br />
The Stochastic Equati<strong>on</strong><br />
The equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle is given by<br />
m¨z µ = F µ ∫<br />
−<br />
d 4 x<br />
δj(x; z)<br />
ϕ(x) (5)<br />
δzµ(τ)<br />
where we have added the external force F µ so as to accelerate<br />
the particle uniformly; F µ = ma( ˙z 1 , ˙z 0 , 0, 0). Then<br />
a classical soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle (<strong>in</strong> the absence <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
coupl<strong>in</strong>g to ϕ) is given by<br />
z µ<br />
0<br />
1 1<br />
= ( s<strong>in</strong>h aτ, cosh aτ, 0, 0). (6)<br />
a a<br />
The equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> field ∂ 2 ϕ = j is<br />
solved by us<strong>in</strong>g the retarded Green functi<strong>on</strong> GR as<br />
ϕ(x) = ϕh(x) + ϕI(x),<br />
∫<br />
ϕI(x) =<br />
d 4 x ′ GR(x, x ′ )j(x ′ ; z) (7)<br />
where ϕh is the homogeneous soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
moti<strong>on</strong> and represents the vacuum fluctuati<strong>on</strong>. It is resp<strong>on</strong>sible<br />
for the particle’s fluctuat<strong>in</strong>g moti<strong>on</strong> under a uniform<br />
accelerati<strong>on</strong>. Insert<strong>in</strong>g the soluti<strong>on</strong> (7) <strong>in</strong>to (5), we have the<br />
follow<strong>in</strong>g stochastic equati<strong>on</strong> for the particle<br />
m¨z µ (τ) =F µ (z(τ)) − e⃗ω µ<br />
(<br />
∫<br />
× ϕh(z(τ)) + e<br />
dτ ′ GR(z(τ), z(τ ′ ))<br />
(8)<br />
)<br />
,<br />
where ⃗ωµ = ˙z ν ˙z [ν∂ µ]−¨zµ, which comes from the deviati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the current<br />
∫<br />
(9)<br />
d 4 δj(x; z)<br />
x<br />
δz µ (τ) f(x) = e⃗ωµf(x)| x=z(τ). (10)<br />
The homogeneous part ϕh(z(τ)) <str<strong>on</strong>g>of</str<strong>on</strong>g> the scalar field describes<br />
the Gaussian fluctuati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum, hence, the<br />
first term <strong>in</strong> the parenthesis <str<strong>on</strong>g>of</str<strong>on</strong>g> (8) can be <strong>in</strong>terpreted as random<br />
noise to the particle’s moti<strong>on</strong><br />
⟨ϕh(x)ϕh(x ′ )⟩ = − 1<br />
4π2 1<br />
(t − t ′ − iϵ) 2 . (11)<br />
− r2 It is essentially quantum mechanical, but if it is evaluated<br />
<strong>on</strong> a world l<strong>in</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> a uniformly accelerated particle<br />
x = z(τ), x ′ = z(τ ′ ), it behaves as the ord<strong>in</strong>ary f<strong>in</strong>ite<br />
temperature noise. The sec<strong>on</strong>d term <strong>in</strong> the parenthesis <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
(8) is a functi<strong>on</strong>al <str<strong>on</strong>g>of</str<strong>on</strong>g> the total history <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle’s moti<strong>on</strong><br />
z(τ ′ ) for τ ′ ≤ τ, but it can be reduced to the so<br />
called radiati<strong>on</strong> damp<strong>in</strong>g term <str<strong>on</strong>g>of</str<strong>on</strong>g> a charged particle coupled<br />
with radiati<strong>on</strong> field. It is generally n<strong>on</strong>local, but s<strong>in</strong>ce the<br />
Green functi<strong>on</strong> damps rapidly as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the distance<br />
r, the term is approximated by local derivative terms. After<br />
the mass renormalizati<strong>on</strong>, we get the follow<strong>in</strong>g generalized<br />
Langev<strong>in</strong> equati<strong>on</strong> for the charged particle,<br />
m ˙v µ − F µ − e2<br />
12π (vµ ˙v 2 + ¨v µ ) = −e⃗ω µ ϕh(z) (12)<br />
where v µ = ˙z µ . This equati<strong>on</strong> is an analog <str<strong>on</strong>g>of</str<strong>on</strong>g> the ALD<br />
equati<strong>on</strong> for a charged particle <strong>in</strong>teract<strong>in</strong>g with the electromagnetic<br />
field. The dissipati<strong>on</strong> term is <strong>in</strong>duced by the<br />
effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the backreacti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle’s radiati<strong>on</strong> to the<br />
particle’s moti<strong>on</strong>. Note that, if the noise term is absent, the<br />
classical soluti<strong>on</strong> (6) with a c<strong>on</strong>stant accelerati<strong>on</strong> is still a<br />
soluti<strong>on</strong> to the equati<strong>on</strong> (12).
Equipartiti<strong>on</strong> Theorem<br />
The stochastic equati<strong>on</strong> (12) is n<strong>on</strong>l<strong>in</strong>ear and difficult to<br />
solve. Here we c<strong>on</strong>sider small fluctuati<strong>on</strong>s around the classical<br />
trajectory <strong>in</strong>duced by the vacuum fluctuati<strong>on</strong> ϕh. Especially<br />
we c<strong>on</strong>sider fluctuati<strong>on</strong>s <strong>in</strong> the transverse directi<strong>on</strong>s.<br />
First we expand the particle’s moti<strong>on</strong> around the<br />
classical trajectory z µ<br />
0 as<br />
z µ (τ) = z µ<br />
0 (τ) + δzµ (τ). (13)<br />
The particle is accelerated al<strong>on</strong>g the x directi<strong>on</strong>. In the follow<strong>in</strong>g<br />
we c<strong>on</strong>sider small fluctuati<strong>on</strong> <strong>in</strong> transverse directi<strong>on</strong>s.<br />
By expand<strong>in</strong>g the stochastic equati<strong>on</strong> (12), we can<br />
obta<strong>in</strong> a l<strong>in</strong>earized stochastic equati<strong>on</strong> for the transverse<br />
velocity fluctuati<strong>on</strong> δv i ≡ δ ˙z i as,<br />
mδ ˙v i = e∂iϕh + e2<br />
12π (δ¨vi − a 2 δv i ). (14)<br />
Perform<strong>in</strong>g the Fourier transformati<strong>on</strong> with respect to the<br />
trajectory’s parameter τ<br />
δv i ∫<br />
dω<br />
(τ) =<br />
2π δ˜vi (ω)e −iωτ , (15)<br />
∫<br />
dω<br />
∂iϕh(τ) =<br />
2π ∂iφ(ω)e −iωτ , (16)<br />
the stochastic equati<strong>on</strong> can be solved as<br />
where<br />
δ˜v i (ω) = eh(ω)∂iφ(ω), (17)<br />
h(ω) =<br />
1<br />
−imω + e2 (ω 2 +a 2 )<br />
12π<br />
. (18)<br />
The vacuum 2-po<strong>in</strong>t functi<strong>on</strong> al<strong>on</strong>g the classical trajectory<br />
can be evaluated from (11) as<br />
⟨∂iϕh(x)∂jϕh(x ′ )⟩| x=z(τ),x ′ =z(τ ′ )<br />
= 1<br />
2π 2<br />
= a4<br />
32π 2<br />
δij<br />
((t − t ′ − iϵ) 2 − r 2 ) 2<br />
δij<br />
s<strong>in</strong>h 4 ( a(τ−τ ′ . (19)<br />
−iϵ)<br />
2 )<br />
It has orig<strong>in</strong>ated from the quantum fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum,<br />
but it can be <strong>in</strong>terpreted as f<strong>in</strong>ite temperature noise<br />
if it is evaluated <strong>on</strong> the accelerated particle’s trajectory [6].<br />
The Fourier transformati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the symmetrized two po<strong>in</strong>t<br />
functi<strong>on</strong> is evaluated as<br />
where<br />
⟨∂iϕ(x)∂jϕ(x ′ )⟩S = ⟨{∂iϕ(x), ∂jϕ(x ′ )}⟩/2<br />
= 2πδ(ω + ω ′ )δijIS(ω), (20)<br />
IS(ω) = 1<br />
12π coth<br />
(<br />
πω<br />
)<br />
(ω<br />
a<br />
3 + ωa 2 ), (21)<br />
which is an even functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> ω. The correlator IS(ω)<br />
should be regularized at the UV, which is large ω or short<br />
proper time difference, where quantum field theoretic effects<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> become important. Full QED treatment is<br />
necessary there.<br />
For small ω, it is expanded as<br />
IS(ω) = a<br />
12π 2 (a2 + O(ω 2 )). (22)<br />
The expansi<strong>on</strong> corresp<strong>on</strong>ds to the derivative expansi<strong>on</strong><br />
⟨∂iϕh(x)∂jϕh(x ′ )⟩S = a3<br />
12π 2 δijδ(τ − τ ′ ) + · · · . (23)<br />
With this expansi<strong>on</strong>, the expectati<strong>on</strong> value <str<strong>on</strong>g>of</str<strong>on</strong>g> the square <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the transverse velocity fluctuati<strong>on</strong> can be evaluated as<br />
⟨δv i (τ)δv j (τ ′ )⟩S<br />
= e 2<br />
∫ ′ dωdω<br />
(2π) 2 ⟨∂iφ(ω)∂jφ(ω ′ )⟩S h(ω)h(ω ′ )e −i(ωτ+ω′ τ ′ )<br />
∼ e 2 ∫<br />
dω<br />
δij<br />
24π3 a3e−iω(τ−τ ′ )<br />
) . (24)<br />
(ω2 + a2 ) 2<br />
(mω) 2 + ( e 2<br />
12π<br />
Here we c<strong>on</strong>sider the accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> to be at<br />
the order 0.1 eV, which is much smaller than the electr<strong>on</strong><br />
mass 0.5 MeV. With the assumpti<strong>on</strong> m ≫ a, <strong>on</strong>e can evaluate<br />
the <strong>in</strong>tegral and get the follow<strong>in</strong>g result,<br />
m<br />
2 ⟨δvi (τ)δv j (τ)⟩ = 1 a<br />
2<br />
2πc δij<br />
( 1 + O(a 2 /m 2 ) ) . (25)<br />
Here we have recovered c and . This gives the equipartiti<strong>on</strong><br />
relati<strong>on</strong> for the transverse momentum fluctuati<strong>on</strong>s <strong>in</strong><br />
the Unruh temperature TU = a/2πc.<br />
Relaxati<strong>on</strong> Time<br />
The thermalizati<strong>on</strong> process <str<strong>on</strong>g>of</str<strong>on</strong>g> the stochastic equati<strong>on</strong><br />
(14) can be also discussed. For simplicity, we approximate<br />
the stochastic equati<strong>on</strong> by dropp<strong>in</strong>g the sec<strong>on</strong>d derivative<br />
term. Then it is solved as<br />
δv i (τ) =e −Ω−τ δv i (0)<br />
+ e<br />
m<br />
∫ τ<br />
where Ω− is given by<br />
0<br />
dτ ′ ∂iϕ(z(τ ′ ))e −Ω−(τ−τ ′ ) , (26)<br />
Ω− = a2 e 2<br />
12πm<br />
(27)<br />
The relaxati<strong>on</strong> time is τR = 1/Ω−. The velocity square<br />
can be also calculated as<br />
⟨δv i (τ)δv j (τ)⟩ =e −2Ω−τ δv i (0)δv j (0)<br />
+ aδij<br />
2πm (1 − e−2Ω−τ ). (28)<br />
For τ → ∞, it approaches the thermalized average (25).<br />
The relaxati<strong>on</strong> time <strong>in</strong> the proper time can be estimated,<br />
for the parameter a = 0.1 eV and m = 0.5 MeV,<br />
τR = 12πm<br />
a 2 e 2 ∼ 10−5 sec. (29)
Let’s compare this relaxati<strong>on</strong> time with the laser frequency.<br />
The planned wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser at ELI is around<br />
10 3 nm and the oscillati<strong>on</strong> period <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field is very<br />
short; 3 × 10 −15 sec<strong>on</strong>ds. So the relaxati<strong>on</strong> time is much<br />
l<strong>on</strong>ger and the charged particle cannot become thermalized<br />
dur<strong>in</strong>g each oscillati<strong>on</strong>. Hence the assumpti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the uniform<br />
accelerati<strong>on</strong> <strong>in</strong> the laser field is not good. Even <strong>in</strong><br />
such a situati<strong>on</strong>, if the electr<strong>on</strong> is accelerated <strong>in</strong> the laser<br />
field for a l<strong>on</strong>g time, an electr<strong>on</strong> may feel an averaged temperature.<br />
The positi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle <strong>in</strong> the transverse directi<strong>on</strong>s<br />
also fluctuates like the ord<strong>in</strong>ary Brownian moti<strong>on</strong> <strong>in</strong> a<br />
heat bath. The mean square <str<strong>on</strong>g>of</str<strong>on</strong>g> the transverse coord<strong>in</strong>ate<br />
R 2 (τ) = ∑<br />
i=y,z ⟨(zi (τ) − z i (0)) 2 ⟩ is calculated as<br />
R 2 (τ) = 2D<br />
(<br />
τ − 3 − 4e−Ω−τ + e −2Ω−τ<br />
2Ω−<br />
)<br />
. (30)<br />
with the diffusi<strong>on</strong> c<strong>on</strong>stant D = 2TU/(Ω−m) =<br />
12/ae 2 ∼ 8 × 10 4 m 2 /s. In the Ballistic regi<strong>on</strong> where<br />
τ < τR, the mean square becomes R 2 (τ) = 2TUτ 2 /m<br />
while <strong>in</strong> the diffusive regi<strong>on</strong> (τ > τR), it is proporti<strong>on</strong>al to<br />
the proper time as R 2 (τ) = 2Dτ. As the ord<strong>in</strong>ary Brownian<br />
moti<strong>on</strong>, the mean square <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle’s transverse<br />
positi<strong>on</strong> grows l<strong>in</strong>early with time. If it becomes possible to<br />
accelerate the particle for a sufficiently l<strong>on</strong>g period, it may<br />
be possible to detect such a Brownian moti<strong>on</strong> <strong>in</strong> future laser<br />
experiments.<br />
RADIATION AND INTERFERENCE<br />
Now we are ready to calculate the radiati<strong>on</strong> emanated<br />
from the uniformly accelerated charged particle. An important<br />
po<strong>in</strong>t is the <strong>in</strong>terference effect between the quantum<br />
fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum ϕh and the radiati<strong>on</strong> <strong>in</strong>duced by<br />
the fluctuat<strong>in</strong>g moti<strong>on</strong> <strong>in</strong> the transverse directi<strong>on</strong>s ϕI. First<br />
let’s c<strong>on</strong>sider the two po<strong>in</strong>t functi<strong>on</strong><br />
⟨ϕ(x)ϕ(x ′ )⟩ − ⟨ϕh(x)ϕh(x ′ )⟩ (31)<br />
= ⟨ϕI(x)ϕh(x ′ )⟩ + ⟨ϕh(x)ϕI(x ′ )⟩ + ⟨ϕI(x)ϕI(x ′ )⟩.<br />
The Unruh radiati<strong>on</strong> estimated <strong>in</strong> [2] corresp<strong>on</strong>ds to calculat<strong>in</strong>g<br />
the 2-po<strong>in</strong>t correlati<strong>on</strong> functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>homogeneous<br />
terms ⟨ϕIϕI⟩. (The same term also c<strong>on</strong>ta<strong>in</strong>s the<br />
Larmor radiati<strong>on</strong>.) However, this is not the end <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
story. As it has been discussed <strong>in</strong> [7], the <strong>in</strong>terference terms<br />
⟨ϕIϕh⟩ + ⟨ϕhϕI⟩ may possibly cancel the Unruh radiati<strong>on</strong><br />
<strong>in</strong> ⟨ϕIϕI⟩ after the thermalizati<strong>on</strong> occurs. This is shown<br />
for an <strong>in</strong>ternal detector, but it is not obvious whether the<br />
same cancellati<strong>on</strong> occurs for the case <str<strong>on</strong>g>of</str<strong>on</strong>g> a charged particle<br />
we are c<strong>on</strong>sider<strong>in</strong>g.<br />
The energy-momentum tensor <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> field can<br />
be obta<strong>in</strong>ed from the 2-po<strong>in</strong>t functi<strong>on</strong><br />
⟨Tµν⟩ = ⟨: ∂µϕ∂νϕ − 1<br />
2 gµν∂ α ϕ∂αϕ :⟩S. (32)<br />
It is written as a sum <str<strong>on</strong>g>of</str<strong>on</strong>g> the classical part and the fluctuat<strong>in</strong>g<br />
part Tµν = Tcl,µν + Tfluc,µν. The classical part corresp<strong>on</strong>ds<br />
to the Larmor radiati<strong>on</strong> while the fluctuat<strong>in</strong>g part<br />
c<strong>on</strong>ta<strong>in</strong>s both <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong> and the <strong>in</strong>terference<br />
terms.<br />
In [1] we calculated the 2-po<strong>in</strong>t functi<strong>on</strong> <strong>in</strong>clud<strong>in</strong>g the<br />
<strong>in</strong>terference term, and obta<strong>in</strong>ed the energy-momentum tensor.<br />
The result we have obta<strong>in</strong>ed is summarized <strong>in</strong> [5] <strong>in</strong><br />
this proceed<strong>in</strong>gs. Some terms are partially canceled but not<br />
all. Hence, it seems that the uniformly accelerated charged<br />
particle emits additi<strong>on</strong>al radiati<strong>on</strong> besides the Larmor radiati<strong>on</strong>.<br />
The rema<strong>in</strong><strong>in</strong>g terms after the partial cancellati<strong>on</strong><br />
are proporti<strong>on</strong>al to a 3 and suppressed compared to the Larmor<br />
radiati<strong>on</strong>. It has a different angular distributi<strong>on</strong>, but the<br />
additi<strong>on</strong>al radiati<strong>on</strong> also vanishes <strong>in</strong> the forward directi<strong>on</strong>.<br />
SUMMARY<br />
We have systematically studied the thermalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
uniformly accelerated charged particle <strong>in</strong> the scalar QED<br />
us<strong>in</strong>g the stochastic method, and calculated the radiati<strong>on</strong><br />
by the particle. Two ma<strong>in</strong> messages <str<strong>on</strong>g>of</str<strong>on</strong>g> are<br />
1. ”L<strong>on</strong>g relaxati<strong>on</strong> time compared to the laser period”<br />
2. ”Importance <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference”<br />
The fluctuati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle doesn’t become thermalized<br />
dur<strong>in</strong>g the period <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser, and we need to study transient<br />
dynamics to obta<strong>in</strong> the radiati<strong>on</strong> from an electr<strong>on</strong> accelerated<br />
<strong>in</strong> the oscillat<strong>in</strong>g laser field. The issue <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference<br />
terms are more subtle. S<strong>in</strong>ce the particle fluctuati<strong>on</strong><br />
orig<strong>in</strong>ates <strong>in</strong> the quantum vacuum fluctuati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
radiati<strong>on</strong> field, it can be by no means neglected. Our result<br />
shows that the <strong>in</strong>terference term partially cancels the Unruh<br />
radiati<strong>on</strong>, but some <str<strong>on</strong>g>of</str<strong>on</strong>g> them survives. The rema<strong>in</strong><strong>in</strong>g<br />
Unruh radiati<strong>on</strong> is smaller compared to the Larmor radiati<strong>on</strong><br />
by a factor a (accelerati<strong>on</strong>) and has a different angular<br />
distributi<strong>on</strong>. In this sense, it is qualitatively c<strong>on</strong>sistent with<br />
the proposal [2]. But as we briefly review <strong>in</strong> [5], the additi<strong>on</strong>al<br />
radiati<strong>on</strong> also vanishes <strong>in</strong> the forward directi<strong>on</strong>. and<br />
it seems difficult to detect such additi<strong>on</strong>al radiati<strong>on</strong> experimentally<br />
so far as the transverse fluctuati<strong>on</strong> is c<strong>on</strong>cerned.<br />
Please beware that the l<strong>on</strong>gitud<strong>in</strong>al fluctuati<strong>on</strong>s which we<br />
have not calculated yet (because <str<strong>on</strong>g>of</str<strong>on</strong>g> technical difficulties related<br />
to a choice <str<strong>on</strong>g>of</str<strong>on</strong>g> gauge) may change the situati<strong>on</strong>.<br />
REFERENCES<br />
[1] S. Iso, Y. Yamamoto and S. Zhang, arXiv:1011.4191 [hep-th].<br />
[2] P. Chen and T. Tajima, Phys. Rev. Lett. 83 (1999) 256.<br />
[3] P.G. Thirolf, et.al. Eur. Phys. J. D 55, 379-389 (2009).<br />
[4] http://www.extreme-light-<strong>in</strong>frastructure.eu/<br />
[5] S. Iso, Y. Yamamoto and S. Zhang, <strong>in</strong> the same proceed<strong>in</strong>gs,<br />
“Unruh radiati<strong>on</strong> and Interference effect”<br />
[6] W. G. Unruh, Phys. Rev. D 14, 870 (1976).<br />
[7] D. J. Ra<strong>in</strong>e, D. W. Sciama and P. G. Grove, Proc. R. Soc.<br />
L<strong>on</strong>d. A (1991) 435, 205-215<br />
[8] A. Raval, B. L. Hu, J. Angl<strong>in</strong>, Phys. Rev. D 53 (1996) 7003.<br />
[9] P. R. Johns<strong>on</strong> and B. L. Hu, arXiv:quant-ph/0012137. Phys.<br />
Rev. D 65 (2002) 065015 [arXiv:quant-ph/0101001].
Abstract<br />
Quantum fields and static <strong>in</strong>teracti<strong>on</strong>s <strong>in</strong> accelerated frames<br />
Frieder Lenz<br />
Institute for Theoretical <strong>Physics</strong> III, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Erlangen-Nürnberg<br />
91058 Erlangen, Germany<br />
Properties <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum fields <strong>in</strong> R<strong>in</strong>dler space or equivalently<br />
<strong>in</strong> accelerated frames are explored. C<strong>on</strong>sequences <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the <strong>in</strong>ertial forces for the k<strong>in</strong>ematics and the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
scalar particles and phot<strong>on</strong>s are discussed and results <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>vestigati<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>teracti<strong>on</strong> energies generated by scalar and<br />
vector particle exchange <strong>in</strong> R<strong>in</strong>dler space are presented.<br />
INTRODUCTION<br />
Quantum fields observed <strong>in</strong> accelerated and <strong>in</strong> <strong>in</strong>ertial<br />
frames are related to each other by a coord<strong>in</strong>ate transformati<strong>on</strong>.<br />
Nevertheless their properties differ significantly<br />
which is due to the existence <str<strong>on</strong>g>of</str<strong>on</strong>g> a horiz<strong>on</strong> <strong>in</strong> accelerated<br />
frames (R<strong>in</strong>dler space). In the follow<strong>in</strong>g I will present the<br />
results <str<strong>on</strong>g>of</str<strong>on</strong>g> studies [1, 2] <str<strong>on</strong>g>of</str<strong>on</strong>g> both k<strong>in</strong>ematical and dynamical<br />
properties <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum fields. The k<strong>in</strong>ematics <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong><strong>in</strong>teract<strong>in</strong>g<br />
quantum fields <strong>in</strong> R<strong>in</strong>dler space [3, 4, 5] and<br />
their relati<strong>on</strong> to fields <strong>in</strong> M<strong>in</strong>kowski space together with the<br />
<strong>in</strong>terpretati<strong>on</strong> <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> f<strong>in</strong>ite temperature quantum fields<br />
[6] have been the subject <str<strong>on</strong>g>of</str<strong>on</strong>g> many <strong>in</strong>vestigati<strong>on</strong>s [7]. Here<br />
the focus will be <strong>on</strong> the peculiar properties <str<strong>on</strong>g>of</str<strong>on</strong>g> the spectrum<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the particles <strong>in</strong> R<strong>in</strong>dler space and their signatures <strong>in</strong> the<br />
Unruh radiati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s <strong>in</strong> comparis<strong>on</strong> to blackbody radiati<strong>on</strong>.<br />
The impact <str<strong>on</strong>g>of</str<strong>on</strong>g> the unusual k<strong>in</strong>ematics <strong>in</strong> accelerated<br />
frames <strong>on</strong> the dynamics will be dem<strong>on</strong>strated <strong>in</strong> the<br />
discussi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> static <strong>in</strong>teracti<strong>on</strong>s.<br />
KINEMATICS<br />
A uniformly accelerated observer (accelerati<strong>on</strong> a) at fixed<br />
coord<strong>in</strong>ates transverse to the accelerati<strong>on</strong> x⊥ moves al<strong>on</strong>g<br />
a hyperbola [8]<br />
x 2 − t 2 = 1<br />
a 2 , x⊥ = 0 . (1)<br />
Quantum fields as seen by accelerated observers are most<br />
c<strong>on</strong>veniently described <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the coord<strong>in</strong>ates obta<strong>in</strong>ed<br />
by transformati<strong>on</strong> to the rest frame t, x, x⊥ → τ, ξ, x⊥<br />
t(τ, ξ) = 1<br />
a eaξ s<strong>in</strong>h aτ , x(τ, ξ) = 1<br />
a eaξ cosh aτ . (2)<br />
By c<strong>on</strong>structi<strong>on</strong>, ξ = 0 corresp<strong>on</strong>ds to the hyperbolic moti<strong>on</strong><br />
(1). More generally, a particle at rest <strong>in</strong> the observers<br />
system at ξ = ξ0 =c<strong>on</strong>st. corresp<strong>on</strong>ds to the uniformly<br />
accelerated moti<strong>on</strong> <strong>in</strong> M<strong>in</strong>kowski space with accelerati<strong>on</strong><br />
a exp{−aξ0}. Trajectories <str<strong>on</strong>g>of</str<strong>on</strong>g> uniformly accelerated particles<br />
for different values <str<strong>on</strong>g>of</str<strong>on</strong>g> ξ0 are shown <strong>in</strong> Fig. 1 together<br />
with the l<strong>in</strong>es τ =c<strong>on</strong>st. .<br />
II<br />
t<br />
10<br />
5<br />
+<br />
III x I<br />
10 5 5 10<br />
5<br />
10<br />
IV<br />
→<br />
ξ = −τ = −∞<br />
→<br />
←<br />
ξ = c<strong>on</strong>st.<br />
← τ = c<strong>on</strong>st.<br />
ξ = τ = −∞<br />
Figure 1: K<strong>in</strong>ematics <str<strong>on</strong>g>of</str<strong>on</strong>g> uniform accelerati<strong>on</strong><br />
The coord<strong>in</strong>ate transformati<strong>on</strong> (2) is not <strong>on</strong>e-to-<strong>on</strong>e. The<br />
coord<strong>in</strong>ates −∞ < τ, ξ < ∞ cover <strong>on</strong>ly <strong>on</strong>e quarter <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
M<strong>in</strong>kowski space, the right ”R<strong>in</strong>dler wedge”R+ (regi<strong>on</strong> I)<br />
R± = x µ |t| ≤ ±x . (3)<br />
Up<strong>on</strong> reversi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the sign <str<strong>on</strong>g>of</str<strong>on</strong>g> x <strong>in</strong> Eq. (2) the left R<strong>in</strong>dler<br />
wedge R− (regi<strong>on</strong> III) is covered by a corresp<strong>on</strong>d<strong>in</strong>g<br />
parametrizati<strong>on</strong>. As illustrated <strong>in</strong> the Figure, the boundaries<br />
x = |t| corresp<strong>on</strong>d<strong>in</strong>g to ξ = ±τ = −∞ form an<br />
event horiz<strong>on</strong>. The light c<strong>on</strong>es shown <strong>in</strong> the Figure <strong>in</strong>dicate<br />
that <strong>in</strong> regi<strong>on</strong> I signals can be transmitted to regi<strong>on</strong> II<br />
but not received from it. Signals received from regi<strong>on</strong> IV<br />
appear to have orig<strong>in</strong>ated from the horiz<strong>on</strong> ξ = τ = −∞.<br />
The space-time def<strong>in</strong>ed by the coord<strong>in</strong>ate transformati<strong>on</strong><br />
(2) is called R<strong>in</strong>dler space and its metric is given by<br />
ds 2 = gµν(ξ)dx µ dx ν = e 2aξ (dτ 2 − dξ 2 ) − dx 2 ⊥ . (4)<br />
The R<strong>in</strong>dler metric derives its importance from the fact that<br />
essentially any static metric which possesses a horiz<strong>on</strong> can<br />
be approximated near the horiz<strong>on</strong> by the R<strong>in</strong>dler metric.<br />
This is the case for <strong>in</strong>stance for the Schwarzschild metric.<br />
Accelerati<strong>on</strong> and Schwarzschild radius, or black hole mass,<br />
are related by<br />
a = 1 1<br />
= . (5)<br />
2R 4GM<br />
QUANTUM FIELDS IN RINDLER SPACES<br />
Quantizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> scalar fields<br />
Quantizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> a n<strong>on</strong>-<strong>in</strong>teract<strong>in</strong>g scalar field <strong>in</strong> R<strong>in</strong>dler<br />
space with the acti<strong>on</strong>
S = 1<br />
<br />
2<br />
dτ dξ d d−1 <br />
x⊥ (∂τ φ) 2 − (∂ξφ) 2<br />
−(m 2 φ 2 + (∂⊥φ) 2 ) e 2aξ , (6)<br />
is standard. The expansi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the fields <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
normal modes (proporti<strong>on</strong>al to the Mc D<strong>on</strong>ald functi<strong>on</strong>s)<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the associated wave equati<strong>on</strong> is given by<br />
<br />
dω<br />
φ(τ, ξ, x⊥) ∼ √ d<br />
2ω d−1 k⊥(a(ω, k⊥)e −iωτ+ik⊥x⊥<br />
<br />
m⊥ e aξ<br />
, m 2 ⊥ = (m 2 + k 2 ⊥)/a 2 , (7)<br />
+h.c.) Ki ω<br />
a<br />
and the normalizati<strong>on</strong> is chosen such that the commutator<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> creati<strong>on</strong> and annihilati<strong>on</strong> operators a (†) (ω, k⊥) is standard.<br />
The repulsive exp<strong>on</strong>ential barrier <strong>in</strong> uniformly accelerated<br />
frames is <str<strong>on</strong>g>of</str<strong>on</strong>g> similar orig<strong>in</strong> as the centrifugal barrier<br />
<strong>in</strong> rotat<strong>in</strong>g frames. It prevents unlimited propagati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
wave <strong>in</strong> positive ξ directi<strong>on</strong>. This repulsi<strong>on</strong> accounts for the<br />
fact that a particle mov<strong>in</strong>g with arbitrary c<strong>on</strong>stant speed <strong>in</strong><br />
M<strong>in</strong>kowski space is seen by the accelerated observer to approach<br />
ξ = −∞ and the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light for large times τ. In<br />
the accelerated frame, the transverse velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle<br />
vanishes exp<strong>on</strong>entially for large times (∼ exp{−2aτ})<br />
as a result <str<strong>on</strong>g>of</str<strong>on</strong>g> the forever <strong>in</strong>creas<strong>in</strong>g time dilati<strong>on</strong> <strong>in</strong>duced<br />
by the accelerati<strong>on</strong>. S<strong>in</strong>ce m 2 ⊥<br />
appears <strong>in</strong> the acti<strong>on</strong> (6) as<br />
a coupl<strong>in</strong>g c<strong>on</strong>stant <str<strong>on</strong>g>of</str<strong>on</strong>g> the exp<strong>on</strong>ential “potential” , the energy<br />
eigenvalues <str<strong>on</strong>g>of</str<strong>on</strong>g> the normal modes do not depend <strong>on</strong> the<br />
transverse momentum and the mass, though the eigenfuncti<strong>on</strong>s<br />
do. This is rem<strong>in</strong>iscent <str<strong>on</strong>g>of</str<strong>on</strong>g> the degeneracy <str<strong>on</strong>g>of</str<strong>on</strong>g> the Landau<br />
levels <str<strong>on</strong>g>of</str<strong>on</strong>g> a particle mov<strong>in</strong>g <strong>in</strong> a c<strong>on</strong>stant magnetic field.<br />
The high degeneracy <str<strong>on</strong>g>of</str<strong>on</strong>g> the eigenstates <str<strong>on</strong>g>of</str<strong>on</strong>g> the Hamilt<strong>on</strong>ian,<br />
<strong>in</strong>clud<strong>in</strong>g the ground state, is due to the <strong>in</strong>ertial force and<br />
has far reach<strong>in</strong>g c<strong>on</strong>sequences. It <strong>in</strong>dicates the presence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
a symmetry <str<strong>on</strong>g>of</str<strong>on</strong>g> the R<strong>in</strong>dler space Hamilt<strong>on</strong>ian. In [1] the<br />
<strong>in</strong>variance <str<strong>on</strong>g>of</str<strong>on</strong>g> the Hamilt<strong>on</strong>ian under scale transformati<strong>on</strong>s<br />
valid even <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> a mass term has been identified<br />
as the source <str<strong>on</strong>g>of</str<strong>on</strong>g> the degeneracy.<br />
The Unruh heat bath<br />
Start<strong>in</strong>g po<strong>in</strong>t for establish<strong>in</strong>g the relati<strong>on</strong> between observables<br />
<strong>in</strong> <strong>in</strong>ertial and accelerated frames is the identity <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
(scalar) fields (φ) and ( ˜ φ) <strong>in</strong> the two frames<br />
φ(τ, ξ, x⊥) = ˜ <br />
<br />
φ(t, x) . (8)<br />
t,x=t,x(τ,ξ)<br />
Projecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this equati<strong>on</strong> <strong>on</strong>to the R<strong>in</strong>dler space normal<br />
modes (6) yields the follow<strong>in</strong>g relati<strong>on</strong> (Bogoliubov transformati<strong>on</strong>)<br />
between the creati<strong>on</strong> and annihilati<strong>on</strong> operators<br />
<strong>in</strong> the two frames<br />
a(Ω, k⊥) = <br />
a s<strong>in</strong>h π Ω<br />
a<br />
1<br />
∞<br />
−∞<br />
dk Ω i √ e a<br />
4πωk<br />
βk<br />
<br />
· e πΩ<br />
πΩ<br />
2a −<br />
ã(k, k⊥) + e 2a ã † (k, −k⊥)<br />
<br />
. (9)<br />
Observati<strong>on</strong>s <strong>in</strong> the accelerated frame are performed <strong>in</strong> the<br />
M<strong>in</strong>kowski vacuum |0M 〉 rather than <strong>in</strong> the R<strong>in</strong>dler space<br />
vacuum. A fundamental quantity is the number <str<strong>on</strong>g>of</str<strong>on</strong>g> particles<br />
measured <strong>in</strong> the accelerated frame which, with the help <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
(9), is found to be<br />
〈0M |a † (Ω, k⊥)a(Ω ′ , k ′ ⊥)|0M 〉<br />
1<br />
=<br />
Ω<br />
e2π a − 1 δ(Ω − Ω′ )δ(k⊥ − k ′ ⊥) . (10)<br />
In the accelerated frame, a thermal distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> (R<strong>in</strong>dler)<br />
particles is observed with the temperature determ<strong>in</strong>ed by<br />
the accelerati<strong>on</strong><br />
T = a<br />
. (11)<br />
2π<br />
For a black hole (cf. Eq. (5)) this temperature agrees with<br />
the black hole temperature<br />
TBH =<br />
1<br />
8πGM .<br />
Although the above derivati<strong>on</strong> makes use <str<strong>on</strong>g>of</str<strong>on</strong>g> the properties<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-<strong>in</strong>teract<strong>in</strong>g fields, relati<strong>on</strong>s between observables <strong>in</strong><br />
accelerated frames and at f<strong>in</strong>ite temperature can be derived<br />
for <strong>in</strong>teract<strong>in</strong>g fields (cf. [9] for a derivati<strong>on</strong> with<strong>in</strong> the path<br />
<strong>in</strong>tegral approach). Here we c<strong>on</strong>sider the two po<strong>in</strong>t functi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a self-<strong>in</strong>teract<strong>in</strong>g scalar field and use its <strong>in</strong>variance<br />
under Lorentz transformati<strong>on</strong>s. The relati<strong>on</strong> (8) between<br />
the fields <strong>in</strong> R<strong>in</strong>dler and M<strong>in</strong>kowski space implies that for<br />
arbitrary po<strong>in</strong>ts <strong>in</strong> the right R<strong>in</strong>dler wedge (cf. Fig. 1) the<br />
values <str<strong>on</strong>g>of</str<strong>on</strong>g> the 2-po<strong>in</strong>t functi<strong>on</strong>s <strong>in</strong> M<strong>in</strong>kowski space and <strong>in</strong><br />
R<strong>in</strong>dler space are equal. The two po<strong>in</strong>t functi<strong>on</strong>s depend<br />
<strong>on</strong>ly <strong>on</strong> (x − x ′ ) 2 . We express the distance <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
R<strong>in</strong>dler space coord<strong>in</strong>ates<br />
(x − x ′ ) 2 = 2ea(ξ+ξ′ )<br />
a2 ′<br />
cosh a(τ − τ ) − cosh η , (12)<br />
with<br />
cosh η = 1 +<br />
and obta<strong>in</strong><br />
<br />
aξ aξ e − e ′2 2 + a x⊥ − x ′ 2 ⊥<br />
2e a(ξ+ξ′ )<br />
, (13)<br />
D (x − x ′ ) 2 = D(τ − τ ′ , ξ, ξ ′ , x⊥ − x ′ ⊥)<br />
<br />
= i〈0M<br />
T φ(τ, ξ, x⊥)φ(τ ′ , ξ ′ , x ′ ⊥) 0M 〉<br />
a(ξ+ξ 2e<br />
= D<br />
′ )<br />
a2 ′<br />
cosh a(τ − τ ) − cosh η <br />
. (14)<br />
After a Wick rotati<strong>on</strong> to imag<strong>in</strong>ary R<strong>in</strong>dler time,<br />
τ → τE = −iτ ,<br />
the two po<strong>in</strong>t functi<strong>on</strong> is periodic with period<br />
β = 2π<br />
a ,<br />
and therefore is a f<strong>in</strong>ite temperature 2-po<strong>in</strong>t functi<strong>on</strong> with<br />
T = 1/β, <strong>in</strong> agreement with the result (11) for n<strong>on</strong><strong>in</strong>teract<strong>in</strong>g<br />
fields.
The relati<strong>on</strong> between accelerati<strong>on</strong> and temperature is subtle.<br />
The R<strong>in</strong>dler space propagator def<strong>in</strong>ed with respect<br />
to the M<strong>in</strong>kowski ground state co<strong>in</strong>cides with the R<strong>in</strong>dler<br />
space f<strong>in</strong>ite temperature propagator with the value <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
temperature determ<strong>in</strong>ed by the accelerati<strong>on</strong> a (cf. Eq. (11)).<br />
This identity makes also manifest that a change <strong>in</strong> the accelerati<strong>on</strong><br />
a does not corresp<strong>on</strong>d to a change <strong>in</strong> temperature<br />
<strong>on</strong>ly. The accelerati<strong>on</strong> appears as temperature <strong>in</strong> the<br />
Boltzmann factor and as a parameter <str<strong>on</strong>g>of</str<strong>on</strong>g> the Hamilt<strong>on</strong>ian. It<br />
c<strong>on</strong>trols the range <str<strong>on</strong>g>of</str<strong>on</strong>g> the exp<strong>on</strong>ential barrier (cf. Eq. (6)).<br />
Accelerated versus f<strong>in</strong>ite temperature phot<strong>on</strong>s<br />
Can<strong>on</strong>ical quantizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electromagnetic field <strong>in</strong><br />
R<strong>in</strong>dler space is straightforward though technically <strong>in</strong>volved<br />
[1]. If carried out <strong>in</strong> Weyl gauge, with the l<strong>on</strong>gitud<strong>in</strong>al<br />
fields determ<strong>in</strong>ed by the Gauß law, <strong>on</strong>ly physical, i.e.,<br />
transverse degrees <str<strong>on</strong>g>of</str<strong>on</strong>g> freedom appear. The relati<strong>on</strong> (10) can<br />
be shown to apply separately for the two polarizati<strong>on</strong>s. Although<br />
Eq. (10) is <str<strong>on</strong>g>of</str<strong>on</strong>g> the same structure as the corresp<strong>on</strong>d<strong>in</strong>g<br />
expressi<strong>on</strong> for the number <str<strong>on</strong>g>of</str<strong>on</strong>g> M<strong>in</strong>kowski space phot<strong>on</strong>s<br />
at f<strong>in</strong>ite temperature, the different dispersi<strong>on</strong> law <str<strong>on</strong>g>of</str<strong>on</strong>g> R<strong>in</strong>dler<br />
phot<strong>on</strong>s<br />
∂ ω<br />
∂ k⊥<br />
= 0 gives rise to significant changes <strong>in</strong> Un-<br />
ruh as compared to blackbody radiati<strong>on</strong>. I illustrate this by<br />
a discussi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the expectati<strong>on</strong> value <str<strong>on</strong>g>of</str<strong>on</strong>g> the energy density<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong> (g(ξ) = det[gµν(ξ)]) ,<br />
1<br />
|g(ξ)| 〈0M | : HE(ξ, x⊥) + HB(ξ, x⊥) : |0M 〉<br />
= 1<br />
<br />
e−4aξ<br />
π2 ωdω ω2 + a2 e ω<br />
11π2<br />
=<br />
Ta − 1 15 T 4 ξ . (15)<br />
As <strong>in</strong> f<strong>in</strong>ite temperature field theory, divergences <strong>in</strong> H are<br />
avoided by normal order<strong>in</strong>g with respect to the (R<strong>in</strong>dler)<br />
ground state. The <strong>in</strong>tegrand <str<strong>on</strong>g>of</str<strong>on</strong>g> the frequency <strong>in</strong>tegral <strong>in</strong><br />
Eq. (15) is shown <strong>in</strong> Fig. 2 and compared with the corresp<strong>on</strong>d<strong>in</strong>g<br />
energy density <str<strong>on</strong>g>of</str<strong>on</strong>g> the blackbody radiati<strong>on</strong>. This<br />
Figure dem<strong>on</strong>strates the qualitative difference <strong>in</strong> the spectrum<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s at f<strong>in</strong>ite temperature <strong>in</strong> M<strong>in</strong>kowski and<br />
R<strong>in</strong>dler space. The degeneracy with respect to the transverse<br />
momenta gives rise to a n<strong>on</strong>-vanish<strong>in</strong>g energy density<br />
at vanish<strong>in</strong>g frequency.<br />
The quantity Tξ <strong>in</strong> (15) denotes the Tolman temperature,<br />
Tξ = a<br />
2π e−aξ . (16)<br />
With the local accelerati<strong>on</strong> (cf. the K<strong>in</strong>ematics secti<strong>on</strong>) also<br />
the local temperature Tξ varies with the coord<strong>in</strong>ate ξ. It sat-<br />
√<br />
isfies Tolman’s law [10], Tξ g00 = c<strong>on</strong>st., which is valid<br />
<strong>in</strong> any static space-time. The Tolman temperature is up to<br />
a factor <str<strong>on</strong>g>of</str<strong>on</strong>g> 2π given by the <strong>in</strong>verse <str<strong>on</strong>g>of</str<strong>on</strong>g> the proper distance to<br />
the horiz<strong>on</strong><br />
dH(ξ) = 1<br />
a eaξ , (17)<br />
i.e. the Tolman temperature diverges when approach<strong>in</strong>g the<br />
horiz<strong>on</strong>. The spatial variati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> temperature is necessary<br />
for thermal equilibrium. It compensates the red or blue<br />
shifts <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s mov<strong>in</strong>g away from or towards the horiz<strong>on</strong><br />
respectively.<br />
0.15<br />
0.10<br />
0.05<br />
Unruh<br />
Blackbody<br />
0.5 1.0 1.5 2.0<br />
Figure 2: Energy density <str<strong>on</strong>g>of</str<strong>on</strong>g> blackbody radiati<strong>on</strong> at T =<br />
a/2π and <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong> at accelerati<strong>on</strong> a as a functi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the frequency ω <strong>in</strong> units <str<strong>on</strong>g>of</str<strong>on</strong>g> a.<br />
STATIC INTERACTIONS<br />
ω/a<br />
The propagator <str<strong>on</strong>g>of</str<strong>on</strong>g> a massless n<strong>on</strong>-<strong>in</strong>teract<strong>in</strong>g scalar field <strong>in</strong><br />
R<strong>in</strong>dler space is given by (cf. Eq. (14))<br />
D(x, x ′ ) =<br />
1<br />
4iπ 2 (x − x ′ ) 2 − iδ = D(τ, ξ, ξ′ , x⊥ − x ′ ⊥)<br />
= a2 e −a(ξ+ξ′ )<br />
8iπ 2<br />
Integrati<strong>on</strong> over the R<strong>in</strong>dler time τ,<br />
˜D(ξ, ξ ′ , x⊥ − x ′ ⊥) =<br />
=<br />
∞<br />
−∞<br />
1<br />
. (18)<br />
cosh aτ − cosh η − iδ<br />
a<br />
4πe a(ξ+ξ′ )<br />
dτD(τ, ξ, ξ ′ , x⊥ − x ′ ⊥)<br />
1<br />
<br />
s<strong>in</strong>h η<br />
1 + iη<br />
π<br />
<br />
, (19)<br />
yields the static <strong>in</strong>teracti<strong>on</strong> between two scalar sources<br />
Vsc = −κ 2 e a(ξ+ξ′ )<br />
D˜ ′<br />
ξ, ξ , x⊥ − x ′ =<br />
<br />
⊥<br />
− aκ2<br />
<br />
1 +<br />
4π s<strong>in</strong>h η<br />
iη<br />
<br />
,<br />
π<br />
(20)<br />
with the coupl<strong>in</strong>g c<strong>on</strong>stant κ. The exp<strong>on</strong>ential factors arise<br />
as a c<strong>on</strong>sequence <str<strong>on</strong>g>of</str<strong>on</strong>g> the volume factor |g(ξ)| and <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
transformati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> proper time <str<strong>on</strong>g>of</str<strong>on</strong>g> the accelerated sources to<br />
R<strong>in</strong>dler time.<br />
The static <strong>in</strong>teracti<strong>on</strong> energy is a complex quantity. The<br />
static propagator satisfies the Poiss<strong>on</strong> equati<strong>on</strong> for a po<strong>in</strong>tlike<br />
source <strong>in</strong> R<strong>in</strong>dler space. S<strong>in</strong>ce the source is real, the<br />
imag<strong>in</strong>ary part <str<strong>on</strong>g>of</str<strong>on</strong>g> the propagator satisfies the corresp<strong>on</strong>d<strong>in</strong>g<br />
(homogeneous) Laplace equati<strong>on</strong> and can be written<br />
therefore as a l<strong>in</strong>ear superpositi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> zero modes [2]<br />
Im ˜ D(ξ, ξ ′ , x⊥ − x ′ a η<br />
⊥) =<br />
4π2ea(ξ+ξ′ =<br />
) s<strong>in</strong>h η 1<br />
4π3a <br />
· d 2 ′<br />
ik⊥(x⊥−x<br />
k⊥e ⊥ ) <br />
k⊥<br />
K0<br />
a eaξ<br />
<br />
k⊥<br />
K0<br />
a eaξ′ . (21)<br />
Thus the imag<strong>in</strong>ary part is due to <strong>on</strong>-shell propagati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
zero-energy massless particles. Unlike <strong>in</strong> M<strong>in</strong>kowski space
Abstract<br />
Sh<strong>in</strong><strong>in</strong>g Light through Walls: en Route towards a new<br />
Particle <strong>Physics</strong> Fr<strong>on</strong>tier<br />
Axel L<strong>in</strong>dner for the ALPS Collaborati<strong>on</strong>, DESY, Hamburg, Germany<br />
“Light-sh<strong>in</strong><strong>in</strong>g-through-a-wall” experiments search for<br />
Weakly Interact<strong>in</strong>g Slim Particles (WISPs). L<strong>on</strong>g stand<strong>in</strong>g<br />
quest <strong>in</strong> particle physics and cosmology may f<strong>in</strong>d their<br />
soluti<strong>on</strong> <strong>in</strong> the discovery <str<strong>on</strong>g>of</str<strong>on</strong>g> this new species <str<strong>on</strong>g>of</str<strong>on</strong>g> particles.<br />
In the recent years experiments have achieved unprecedented<br />
sensitivities. The experience ga<strong>in</strong>ed provides a firm<br />
foundati<strong>on</strong> for future enterprises <strong>in</strong>to unexplored parameter<br />
spaces.<br />
INTRODUCTION TO AXIONS AND WISPS<br />
S<strong>in</strong>ce about four decades the so called Standard Model<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> elementary particle physics faces a triumphant success.<br />
There is not a s<strong>in</strong>gle earth-bound experiment which really<br />
questi<strong>on</strong>s the model. In c<strong>on</strong>trast, precisi<strong>on</strong> tests have<br />
probed its c<strong>on</strong>stituents and forces to the per mill level or<br />
better. However, evidence is mount<strong>in</strong>g that the known<br />
c<strong>on</strong>stituents <str<strong>on</strong>g>of</str<strong>on</strong>g> matter and forces do not fully describe the<br />
world around us. Such arguments arise from astrophysical<br />
and cosmological observati<strong>on</strong>s as well as from theoretical<br />
c<strong>on</strong>siderati<strong>on</strong>s. There are str<strong>on</strong>g c<strong>on</strong>victi<strong>on</strong>s am<strong>on</strong>g<br />
scientists that new experiments at the high energy fr<strong>on</strong>tier<br />
at LHC will provide <strong>in</strong>sight <strong>in</strong>to physics bey<strong>on</strong>d the Standard<br />
Model. Although theoretically well motivated, focus<strong>in</strong>g<br />
the search for new physics <strong>on</strong>to highest available energies<br />
neglects evidences po<strong>in</strong>t<strong>in</strong>g at the opposite energy<br />
scale. Extensi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the Standard Model may manifest<br />
themselves also at meV energy scales, n<strong>in</strong>e orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude<br />
below the mass <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>.<br />
Generally, new very light and very weakly <strong>in</strong>teract<strong>in</strong>g<br />
particles denoted as WISPs (Weakly Interact<strong>in</strong>g Slim Particles)<br />
occur naturally <strong>in</strong> str<strong>in</strong>g-theory-motivated extensi<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Standard Model. There could be bos<strong>on</strong>s and<br />
fermi<strong>on</strong>s, charged and uncharged particles. WISPs may <strong>in</strong>teract<br />
with ord<strong>in</strong>ary matter via the exchange <str<strong>on</strong>g>of</str<strong>on</strong>g> very heavy<br />
particles related to very high energy scales and thus give<br />
<strong>in</strong>sight <strong>in</strong>to physics at highest energy scales. The reader is<br />
referred to [1, 2, 3] and references there<strong>in</strong> for a more detailed<br />
view.<br />
One prime example for a WISP is the QCD axi<strong>on</strong> [4, 5,<br />
6] <strong>in</strong>vented to expla<strong>in</strong> the vanish<strong>in</strong>g electric dipole moment<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the neutr<strong>on</strong>, which is a signature <str<strong>on</strong>g>of</str<strong>on</strong>g> CP c<strong>on</strong>servati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the str<strong>on</strong>g <strong>in</strong>teracti<strong>on</strong>. From astrophysical observati<strong>on</strong>s its<br />
mass should be below about 1 eV. For the QCD axi<strong>on</strong> such<br />
a low mass implies very weak <strong>in</strong>teracti<strong>on</strong>s with the other<br />
c<strong>on</strong>stituents <str<strong>on</strong>g>of</str<strong>on</strong>g> the Standard Model [7]. It is strik<strong>in</strong>g that a<br />
QCD axi<strong>on</strong> with a mass around 1 µeV is a perfect candidate<br />
for cold dark matter <strong>in</strong> the Universe [8, 9]. A discovery<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the axi<strong>on</strong> could solve l<strong>on</strong>g last<strong>in</strong>g questi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> particle<br />
physics and cosmology simultaneously. It is worthwhile<br />
to note that also dark energy might be attributed to new<br />
physics at the meV scale [10].<br />
Besides theoretical c<strong>on</strong>siderati<strong>on</strong>s the existence <str<strong>on</strong>g>of</str<strong>on</strong>g> very<br />
light axi<strong>on</strong>-like particles is suggested by different astrophysical<br />
observati<strong>on</strong>s. For example, the cool<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> white<br />
dwarfs can be modeled significantly better if an additi<strong>on</strong>al<br />
energy loss due to axi<strong>on</strong>-like particles is taken <strong>in</strong>to account<br />
[11]. The surpris<strong>in</strong>gly high transparency <str<strong>on</strong>g>of</str<strong>on</strong>g> the Universe<br />
to TeV phot<strong>on</strong>s from AGNs at cosmological distances<br />
may be expla<strong>in</strong>ed by back and forth oscillati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s<br />
<strong>in</strong>to axi<strong>on</strong>-like particles [12]. The heat<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the solar<br />
cor<strong>on</strong>a is not understood, but may be attributed to an energy<br />
flow mediated by axi<strong>on</strong>-like particles [13].<br />
SEARCHING FOR WISPS<br />
WISPs and especially ALPs are searched for <strong>in</strong> astrophysics<br />
phenomena and laboratory experiments. At present<br />
for most <str<strong>on</strong>g>of</str<strong>on</strong>g> the WISPs the most str<strong>in</strong>gent limits <strong>on</strong> their existence<br />
orig<strong>in</strong>ate from astrophysics c<strong>on</strong>siderati<strong>on</strong>s. The existence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> WISPs would for example open up new energy<br />
loss channels for hot envir<strong>on</strong>ments <strong>in</strong> stars and thus shorten<br />
lifetimes or cool<strong>in</strong>g cycles [14]. Limits <strong>on</strong> axi<strong>on</strong>s are also<br />
derived from cosmology [15]. Direct searches for axi<strong>on</strong>like<br />
particles produced <strong>in</strong> the sun [16] or as c<strong>on</strong>stituents <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
galactic dark matter [17] have greatly progressed <strong>in</strong> recent<br />
years and reached impressive sensitivities.<br />
However, <strong>in</strong>terpretati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> astrophysics data are always<br />
hampered by the unc<strong>on</strong>trolled producti<strong>on</strong> mechanism <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
WISPs. Effective theories have been presented, where the<br />
producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> some WISP species is suppressed <strong>in</strong> hot envir<strong>on</strong>ments<br />
[18, 19]. If such scenarios are true, astrophysics<br />
experiments might fail to detect WISPs while laboratory<br />
experiments could open up this new physics w<strong>in</strong>dow. Literally,<br />
astrophysics deals with “astr<strong>on</strong>omical” or “cosmological”<br />
distances <strong>on</strong> the <strong>on</strong>e hand and microscopic distances<br />
<strong>in</strong> hot dense plasmas <strong>on</strong> the other hand. Intermediate distances<br />
are <strong>on</strong>ly probed <strong>in</strong> the laboratory.<br />
The are numerous experimental efforts to probe for<br />
WISPs <strong>in</strong> the laboratory. Typically they are searched for<br />
by look<strong>in</strong>g for new effects <strong>in</strong> gravitati<strong>on</strong>al or QED envir<strong>on</strong>ments.<br />
The latter <strong>on</strong>es comprise atomic physics like Lamb<br />
shift, positr<strong>on</strong>ium decay, Casimir forces or phot<strong>on</strong>-phot<strong>on</strong><br />
<strong>in</strong>teracti<strong>on</strong>. Only phot<strong>on</strong>-phot<strong>on</strong> processes are addressed<br />
<strong>in</strong> the follow<strong>in</strong>g.<br />
Phot<strong>on</strong>-WISPs <strong>in</strong>teracti<strong>on</strong>s<br />
WISPs may <strong>in</strong>teract with phot<strong>on</strong>s <strong>in</strong> different manners as<br />
shown <strong>in</strong> Fig. 1(see [1]). These <strong>in</strong>teracti<strong>on</strong> may give rise to
=<br />
γ WISP<br />
; ;<br />
ALP HP(mγ ′ > 0)<br />
...<br />
MCP<br />
HP(mγ ′ = 0)<br />
Figure 1: A collecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> some Feynman diagrams resp<strong>on</strong>sible for the mix<strong>in</strong>g term between phot<strong>on</strong>s and different hypothetical<br />
“weakly <strong>in</strong>teract<strong>in</strong>g slim particles” (WISPs). Phot<strong>on</strong> oscillati<strong>on</strong>s <strong>in</strong>to axi<strong>on</strong>-like particles (ALPs) and massless<br />
hidden phot<strong>on</strong>s (HPs) via m<strong>in</strong>i-charged particles (MCP) require the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> a background electromagnetic field, denoted<br />
by crossed circles.<br />
subtle polarizati<strong>on</strong> phenomena [20, 21], but also to a very<br />
spectacular “light-sh<strong>in</strong><strong>in</strong>g-through-a-wall” effect (Fig. 2).<br />
γ γ<br />
WISP<br />
Figure 2: Schematic overview <str<strong>on</strong>g>of</str<strong>on</strong>g> a “light sh<strong>in</strong><strong>in</strong>g through a<br />
wall experiment”. The gray blob <strong>in</strong>dicates the mix<strong>in</strong>g term<br />
between phot<strong>on</strong>s and the WISP.<br />
In the first part <str<strong>on</strong>g>of</str<strong>on</strong>g> such an experiment WISPs are produced<br />
from <strong>in</strong>tense laser light, either by <strong>in</strong>teracti<strong>on</strong> with<br />
a str<strong>on</strong>g magnetic field or by k<strong>in</strong>etic mix<strong>in</strong>g. This first<br />
part is separated by a light-tight wall from the sec<strong>on</strong>d part.<br />
Only WISPs can traverse the wall due to their very low<br />
cross-secti<strong>on</strong>s. Beh<strong>in</strong>d the wall they could c<strong>on</strong>vert back<br />
<strong>in</strong>to phot<strong>on</strong>s with exactly the same properties as the light<br />
generat<strong>in</strong>g the WISPs. This gives the impressi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> “lightsh<strong>in</strong><strong>in</strong>g-through-a-wall”<br />
(LSW).<br />
The c<strong>on</strong>versi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>cident phot<strong>on</strong>s to axi<strong>on</strong>s or<br />
axi<strong>on</strong>-like particles ϕ <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> a magnetic field<br />
is governed by the Primak<str<strong>on</strong>g>of</str<strong>on</strong>g>f effect [22]. Beh<strong>in</strong>d the absorber,<br />
some <str<strong>on</strong>g>of</str<strong>on</strong>g> these ALPs will rec<strong>on</strong>vert via the <strong>in</strong>verse-<br />
Primak<str<strong>on</strong>g>of</str<strong>on</strong>g>f process <strong>in</strong>to phot<strong>on</strong>s with the <strong>in</strong>itial properties.<br />
In a symmetric LSW setup the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the Primak<str<strong>on</strong>g>of</str<strong>on</strong>g>f<br />
transiti<strong>on</strong> Pγ→ϕ is the same as for the <strong>in</strong>verse-Primak<str<strong>on</strong>g>of</str<strong>on</strong>g>f<br />
Pϕ→γ. Therefore the LSW probability is just the square <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Pγ→ϕ = g 2 B 2 E 2 /(4m 4 ϕ ) · s<strong>in</strong>2 (m 2 ϕ<br />
L/(4E)) with B the<br />
magnetic field strength, L the length <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>versi<strong>on</strong> regi<strong>on</strong><br />
and E the phot<strong>on</strong> energy. Mass (mϕ) and two phot<strong>on</strong><br />
coupl<strong>in</strong>g (g) <str<strong>on</strong>g>of</str<strong>on</strong>g> the ALPs are assumed to be uncorrelated.<br />
With βϕ denot<strong>in</strong>g the velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the ALP and q = pγ − pϕ<br />
<strong>on</strong>e achieves:<br />
Pγ→ϕ→γ = 1<br />
16β2 (gBL)<br />
ϕ<br />
4<br />
( s<strong>in</strong>(qL/2)<br />
qL/2<br />
For qL
Figure 3: Schematic view <str<strong>on</strong>g>of</str<strong>on</strong>g> the ALPS LSW experiment. PD denotes various photo detectors, CM the coupl<strong>in</strong>g mirror<br />
and EM the end mirror <str<strong>on</strong>g>of</str<strong>on</strong>g> the res<strong>on</strong>ant cavity. See the text and [23] for a descripti<strong>on</strong>.<br />
ti<strong>on</strong>, cf. Fig. 3. ALPS is the first experiment which successfully<br />
exploits a large-scale optical res<strong>on</strong>ator for WISP<br />
searches. The ma<strong>in</strong> parts and their basic functi<strong>on</strong>ality are<br />
expla<strong>in</strong>ed <strong>in</strong> [1] and reference there<strong>in</strong>. Dur<strong>in</strong>g the measurement<br />
period <strong>in</strong> the year 2009 a c<strong>on</strong>t<strong>in</strong>uously circulat<strong>in</strong>g<br />
power <strong>in</strong>side the ALPS producti<strong>on</strong> regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> around 1.2 kW<br />
at 532 nm was achieved [23].<br />
ALPS also exploits successfully a new method to cover<br />
the gaps <strong>in</strong> the sensitivity for higher masses, where the ALP<br />
wave runs out <str<strong>on</strong>g>of</str<strong>on</strong>g> phase w.r.t. the phase <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser beam,<br />
cf. Fig. 4. Introduc<strong>in</strong>g Ar gas at a pressure <str<strong>on</strong>g>of</str<strong>on</strong>g> 0.18 mbar<br />
changes the phot<strong>on</strong> momentum and tunes therefore the refracti<strong>on</strong><br />
<strong>in</strong>dex. In the ALPS setup the γ−ALP relative<br />
phase velocity <strong>in</strong>creases thereby to have an extra half oscillati<strong>on</strong><br />
length. Even if the sensitivity is lowered compared<br />
to vacuum c<strong>on</strong>diti<strong>on</strong>s this helps to cover the higher mass<br />
gaps.<br />
ALPS RESULT<br />
ALPS took around 50 data sets (1 h frames) under different<br />
experimental c<strong>on</strong>diti<strong>on</strong>s: with magnet <strong>on</strong> or <str<strong>on</strong>g>of</str<strong>on</strong>g>f, laser<br />
polarizati<strong>on</strong> parallel or perpendicular to the magnetic field<br />
and different gas pressures. Details <strong>on</strong> the methodology<br />
and analysis are described <strong>in</strong> [1, 23]. From the n<strong>on</strong> observati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> any WISP signal a 95 % c<strong>on</strong>fidence level <strong>on</strong><br />
the c<strong>on</strong>versi<strong>on</strong> probabilty was obta<strong>in</strong>ed, rang<strong>in</strong>g between<br />
Pγ→ϕ→γ = 1 − 10 × 10 −25 for the different experimental<br />
setups. Fig. 4 shows the ALPS results for pseudoscalar and<br />
scalar axi<strong>on</strong>-like particles together with the results obta<strong>in</strong>ed<br />
from BMV [25], BFRT [26], GammeV [27], LIPSS [28]<br />
and OSQAR [29]. The gaps at higher masses are covered<br />
by the ALPS gas runs as described above. ALPS provide<br />
now the most str<strong>in</strong>gent laboratory bounds <strong>on</strong> ALPs <strong>in</strong> the<br />
sub-eV mass range.<br />
Also for hidden phot<strong>on</strong> and m<strong>in</strong>icharged particle<br />
searches ALPS has achieved the highest sensitivity <strong>in</strong> the<br />
laboratory, cf. Fig. 5. The ALPS LSW results <strong>on</strong> hidden<br />
phot<strong>on</strong> search fills the gap between lab searches for deviati<strong>on</strong>s<br />
from Coulomb’s law and astrophysical bounds.<br />
<br />
Remarkable, with the achieved sensitivity ALPS almost<br />
completely rules out the h<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> WMAP and large-scalestructure<br />
probes with n<strong>on</strong>-standard radiati<strong>on</strong> density c<strong>on</strong>tributi<strong>on</strong><br />
due to hidden phot<strong>on</strong>s, cf. [23] and references<br />
there<strong>in</strong>.<br />
OUTLOOK<br />
The present generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> laboratory experiments<br />
search<strong>in</strong>g for WISPs has not found any evidence for those<br />
elusive particles. It’s worth menti<strong>on</strong><strong>in</strong>g that this probes<br />
for new physics (if ALPs exist) at the 100 TeV scale 1 already.<br />
The physics case for WISPs is still strengthen<strong>in</strong>g<br />
due to <strong>on</strong>go<strong>in</strong>g theoretical studies and puzzl<strong>in</strong>g observati<strong>on</strong>s<br />
<strong>in</strong> astrophysics as menti<strong>on</strong>ed above. The QCD axi<strong>on</strong><br />
itself rema<strong>in</strong>s a “holy grail” for particle physics. To solve<br />
the CP problem <strong>in</strong> QCD and understand Dark Matter with<br />
<strong>on</strong>e stroke is very allur<strong>in</strong>g. Perhaps even Dark Energy will<br />
f<strong>in</strong>d its explanati<strong>on</strong> <strong>in</strong> the WISP sector via the detecti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> “chamele<strong>on</strong>s” [31]. Due to these prospects attempts are<br />
<strong>on</strong>go<strong>in</strong>g to largely improve all comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> LSW experiments.<br />
Usually LSW experiments sh<strong>in</strong>e laser light through l<strong>on</strong>g<br />
and tight magnet bores. With the help <str<strong>on</strong>g>of</str<strong>on</strong>g> res<strong>on</strong>ant optical<br />
cavities effective laser light powers around 100 kW might<br />
be possible <strong>in</strong> future. The ALPS collaborati<strong>on</strong> is presently<br />
prepar<strong>in</strong>g such a setup.<br />
ALPS has used <strong>on</strong>e spare HERA dipole magnet to<br />
achieve the results menti<strong>on</strong>ed above. At DESY we study<br />
now the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> an experiment with up to 20+20<br />
HERA dipoles.<br />
Most <str<strong>on</strong>g>of</str<strong>on</strong>g> the present-day LSW experiments use commercial<br />
CCD cameras to search for rec<strong>on</strong>verted phot<strong>on</strong>s from<br />
WISPs beh<strong>in</strong>d the wall. In the future the detecti<strong>on</strong> sensitivity<br />
might be enhanced c<strong>on</strong>siderably by us<strong>in</strong>g transiti<strong>on</strong><br />
edge sensors (TES) [32, 33]. Here a sensor is cooled down<br />
to about 100 mK and operated <strong>in</strong> the transiti<strong>on</strong> regi<strong>on</strong> be-<br />
1 The axi<strong>on</strong>-to-phot<strong>on</strong> coupl<strong>in</strong>g g is given by g = α · gγ/πfα, where<br />
fα denotes the new energy scale and gγ a factor derived from theory<br />
expected to vary by about an order <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude for the QCD axi<strong>on</strong>.
Figure 4: Exclusi<strong>on</strong> limit (95% C.L.) for pseudoscalar<br />
(top) and scalar (bottom) axi<strong>on</strong>-like particles obta<strong>in</strong>ed by<br />
the ALPS experiment from vacumm and gas runs together<br />
with the results from various other LSW experiments [23],<br />
see the text for details. Dashed and dotted l<strong>in</strong>es show the<br />
bounds derived from the PVLAS measurement <strong>on</strong> ALP <strong>in</strong>duced<br />
dichroism and birefr<strong>in</strong>gence [30].<br />
tween a superc<strong>on</strong>duct<strong>in</strong>g and normal c<strong>on</strong>duct<strong>in</strong>g state. Due<br />
to the very low heat capacity <str<strong>on</strong>g>of</str<strong>on</strong>g> such a state the energy deposit<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a s<strong>in</strong>gle phot<strong>on</strong> results <strong>in</strong> a significant temperature<br />
rise and is well measurable. TES detectors allow for essentially<br />
background-free count<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>dividual phot<strong>on</strong>s,<br />
register their arrival times and allow to estimate their energies.<br />
A new technology will be exploited to boost the sensitivity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> LSW experiments even further: the challenge is<br />
to realize a sec<strong>on</strong>d res<strong>on</strong>ant optical cavity <strong>in</strong> the part <str<strong>on</strong>g>of</str<strong>on</strong>g> an<br />
experiment beh<strong>in</strong>d the wall. The idea <str<strong>on</strong>g>of</str<strong>on</strong>g> such a res<strong>on</strong>antly<br />
enhanced axi<strong>on</strong> phot<strong>on</strong> regenerati<strong>on</strong> was put forward first<br />
<strong>in</strong> 1993 by F. Hoogeveen and T. Ziegenhagen [34] and <strong>in</strong>dependently<br />
rediscovered <strong>in</strong> 2007 by P. Sikivie, D.B. Tanner<br />
and K. van Bibber [35]. The basic idea is to set up an optical<br />
res<strong>on</strong>ator also <strong>in</strong> the regenerati<strong>on</strong> part <str<strong>on</strong>g>of</str<strong>on</strong>g> a LSW experiment<br />
very similar to the optical res<strong>on</strong>ator <strong>in</strong> the first part.<br />
The sec<strong>on</strong>d res<strong>on</strong>ator effectively <strong>in</strong>creases the c<strong>on</strong>versi<strong>on</strong><br />
probability <str<strong>on</strong>g>of</str<strong>on</strong>g> a WISP <strong>in</strong>to a phot<strong>on</strong>. To understand this <strong>on</strong>e<br />
Figure 5: ALPS exclusi<strong>on</strong> limit (95% C.L.) for hidden phot<strong>on</strong>s<br />
(top) and m<strong>in</strong>icharged particles (bottom) together with<br />
the results from various other experiments [23].<br />
has to c<strong>on</strong>sider that the freely propagat<strong>in</strong>g WISP-related<br />
wave beh<strong>in</strong>d the wall <str<strong>on</strong>g>of</str<strong>on</strong>g> the LSW experiment comprises a<br />
very t<strong>in</strong>y electromagnetic phot<strong>on</strong> comp<strong>on</strong>ent. Due to this<br />
small comp<strong>on</strong>ent the WISP might c<strong>on</strong>vert <strong>in</strong>to a real phot<strong>on</strong>.<br />
An optical res<strong>on</strong>ator enhances this small comp<strong>on</strong>ent<br />
<strong>in</strong> the same way as the wave amplitude for real phot<strong>on</strong>s<br />
is <strong>in</strong>creased. Hence the transiti<strong>on</strong> probability <str<strong>on</strong>g>of</str<strong>on</strong>g> WISPs to<br />
phot<strong>on</strong>s rises with the power amplificati<strong>on</strong> factor <str<strong>on</strong>g>of</str<strong>on</strong>g> an optical<br />
res<strong>on</strong>ator <strong>in</strong> the sec<strong>on</strong>d part <str<strong>on</strong>g>of</str<strong>on</strong>g> the LSW experiment<br />
beh<strong>in</strong>d the wall. C<strong>on</strong>sequently the sensitivity <str<strong>on</strong>g>of</str<strong>on</strong>g> such a setup<br />
for the coupl<strong>in</strong>g c<strong>on</strong>stant g improves with the square<br />
root <str<strong>on</strong>g>of</str<strong>on</strong>g> this factor. The technical challenge is to lock the<br />
sec<strong>on</strong>d cavity to exactly the same frequency and the same<br />
mode as the first cavity (used to enhance the effective laser<br />
phot<strong>on</strong> flux) while keep<strong>in</strong>g it dark <strong>in</strong> the light <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s<br />
searched for as WISP footpr<strong>in</strong>ts. At ALPS we study different<br />
approaches to realize a regenerati<strong>on</strong> cavity.<br />
Comb<strong>in</strong><strong>in</strong>g all the improvements menti<strong>on</strong>ed above would<br />
result <strong>in</strong> a greatly improved sensitivity allow<strong>in</strong>g LSW experiments<br />
to surpass present day <strong>in</strong>direct limits from astrophysics<br />
(g = 10 −10 GeV −1 ) and touch the parameter<br />
regi<strong>on</strong> for ALPS <strong>in</strong>dicated by astrophysics phenomena
Figure 6: Estimates <str<strong>on</strong>g>of</str<strong>on</strong>g> sensitivities for the search <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
axi<strong>on</strong>-like particles for different dipole magnet types (taken<br />
from [36])<br />
.<br />
g < 10 −11 GeV −1 ) as shown <strong>in</strong> Fig. 6. A detailed analysis<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> future possibilities can be found <strong>in</strong> [36]. Other excit<strong>in</strong>g<br />
possibilities for future WISP searches have been presented<br />
at this workshop (see K. Homma’s and T. Tomaru’s c<strong>on</strong>tributi<strong>on</strong>s<br />
to the proceed<strong>in</strong>gs).<br />
ACKNOWLEDGEMENTS<br />
I am very grateful to the organizers <str<strong>on</strong>g>of</str<strong>on</strong>g> PIF 2010 for giv<strong>in</strong>g<br />
me the opportunity to participate <strong>in</strong> and to c<strong>on</strong>tribute<br />
to this very excit<strong>in</strong>g meet<strong>in</strong>g with a broad physics scope! I<br />
thank my colleagues <str<strong>on</strong>g>of</str<strong>on</strong>g> the ALPS collaborati<strong>on</strong> for stimulat<strong>in</strong>g<br />
and fruitful discussi<strong>on</strong>s as well as for the fun work<strong>in</strong>g<br />
with experts <strong>in</strong> very different scientific discipl<strong>in</strong>es.<br />
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Prob<strong>in</strong>g extremely light fields via res<strong>on</strong>ance scatter<strong>in</strong>g<br />
by focus<strong>in</strong>g <strong>in</strong>tense laser ∗<br />
Kensuke Homma<br />
Graduate School <str<strong>on</strong>g>of</str<strong>on</strong>g> Science, Hiroshima Univ., Hiroshima 739-8526, Japan, and<br />
Fakultät für Physik, Ludwig Maximilians Universität München, D-85748 Garch<strong>in</strong>g, Germany<br />
Abstract<br />
Recent astr<strong>on</strong>omical observati<strong>on</strong>s suggest that there are<br />
unknown c<strong>on</strong>stituents <str<strong>on</strong>g>of</str<strong>on</strong>g> the universe such as dark energy<br />
and dark matter. They may be undiscovered extremely light<br />
fields <strong>in</strong> the vacuum which <strong>on</strong>ly weakly couple to matter.<br />
We suggest a method to observe phot<strong>on</strong>-phot<strong>on</strong> scatter<strong>in</strong>g<br />
via res<strong>on</strong>ance states <str<strong>on</strong>g>of</str<strong>on</strong>g> these fields by focus<strong>in</strong>g <strong>in</strong>tense laser<br />
fields <strong>in</strong> laboratory experiments.<br />
INTRODUCTION<br />
Dark energy and dark matter may be attributable to<br />
undiscovered fields <str<strong>on</strong>g>of</str<strong>on</strong>g> small mass below 1eV [1, 2, 3].<br />
These fields are thought to evade our effort to detect them<br />
<strong>in</strong> laboratory, because they are supposed to weakly couple<br />
to matter. Am<strong>on</strong>g them the most challeng<strong>in</strong>g puzzle is the<br />
dark energy problem. The observed dark energy density<br />
is too small to expla<strong>in</strong> it by a simple field theoretical view<br />
po<strong>in</strong>t [4]. In order to approach to this problem, let us start<br />
by rais<strong>in</strong>g follow<strong>in</strong>g view po<strong>in</strong>ts.<br />
The first po<strong>in</strong>t is that the m<strong>in</strong>imum energy state may be<br />
different depend<strong>in</strong>g <strong>on</strong> the order parameter to def<strong>in</strong>e the energy<br />
state. The sec<strong>on</strong>d po<strong>in</strong>t is that the space-time scales<br />
are totally different between particle physics and cosmology.<br />
In particle physics or field theory we <strong>in</strong>troduce local<br />
fields to def<strong>in</strong>e the energy state. We then discuss the energy<br />
state <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum through the order parameter such<br />
as particle masses represented by the square term <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
polynomial <str<strong>on</strong>g>of</str<strong>on</strong>g> the field <strong>in</strong> the Lagrangian. On the other<br />
hand, <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> cosmology the observable is geometry.<br />
We relate curvature <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum with the gravitati<strong>on</strong>al<br />
sources which provide the <strong>in</strong>formati<strong>on</strong> <strong>on</strong> the energy density<br />
and the pressure. For <strong>in</strong>stance, it is not surpris<strong>in</strong>g at all,<br />
even if the states <str<strong>on</strong>g>of</str<strong>on</strong>g> soundlessness and darkness, which are<br />
different states <str<strong>on</strong>g>of</str<strong>on</strong>g> sensory organs, do not po<strong>in</strong>t an identical<br />
state <str<strong>on</strong>g>of</str<strong>on</strong>g> nature. Similarly it might not be a real problem,<br />
even though we cannot immediately accommodate the c<strong>on</strong>necti<strong>on</strong><br />
<strong>on</strong> the energy state <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum between particle<br />
physics and cosmology as represented by the dark energy<br />
problem. Before claim<strong>in</strong>g the problem, we should admit<br />
that we have probed the vacuum <strong>on</strong>ly a little with very different<br />
scales. Thus if we could <strong>in</strong>troduce a new k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
observables or order parameters to def<strong>in</strong>e the state <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
vacuum <strong>in</strong> different scales, it would lead us to the deeper<br />
understand<strong>in</strong>g <strong>on</strong> the state <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum. Therefore, it is<br />
∗ Work supported by the Grant-<strong>in</strong>-Aid for Scientific Research<br />
no.21654035<br />
important for experiments to collect <strong>in</strong>formati<strong>on</strong> <strong>in</strong> different<br />
space-time scales as much as possible.<br />
In spite <str<strong>on</strong>g>of</str<strong>on</strong>g> the fact that the vacuum is not a normal<br />
medium, if we could view the vacuum as if it is a medium,<br />
at least, we can obta<strong>in</strong> some h<strong>in</strong>ts <strong>on</strong> observables which<br />
have been neither applied to particle physics nor cosmology<br />
so far. Of course, we should remember that we cannot<br />
associate noti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> to this ether like view po<strong>in</strong>t. Let<br />
us c<strong>on</strong>sider laser-matter <strong>in</strong>teracti<strong>on</strong>s where we <strong>in</strong>troduce<br />
polarizati<strong>on</strong>s as an order parameter with respect to external<br />
electric fields. All laser physicists are familiar with birefr<strong>in</strong>gence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> crystals and higher harm<strong>on</strong>ic generati<strong>on</strong> from<br />
them as the n<strong>on</strong>l<strong>in</strong>ear resp<strong>on</strong>se <str<strong>on</strong>g>of</str<strong>on</strong>g> atoms to external electric<br />
fields. We may try to apply these observables to <strong>in</strong>vestigate<br />
the state <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum. We now replace the role <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
crystal with the vacuum by <strong>in</strong>troduc<strong>in</strong>g noti<strong>on</strong> that the vacuum<br />
is also a special k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> medium under <strong>in</strong>fluence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
external fields. In this case the <strong>in</strong>vestigati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the high <strong>in</strong>tense<br />
laser-laser <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> the vacuum is a quite natural<br />
approach to probe the medium-like nature. Therefore, we<br />
may foresee the applicability <str<strong>on</strong>g>of</str<strong>on</strong>g> these known observables<br />
developed for laser-matter <strong>in</strong>teracti<strong>on</strong>.<br />
However, the dynamics <strong>in</strong> the vacuum is different from<br />
that <str<strong>on</strong>g>of</str<strong>on</strong>g> atoms <strong>in</strong> matter. First <str<strong>on</strong>g>of</str<strong>on</strong>g> all, we need discuss the<br />
scale dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> the dynamics <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum.<br />
In this respect phot<strong>on</strong> is a valuable probe to see natures<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum, s<strong>in</strong>ce we can <strong>in</strong>troduce the different<br />
frequencies by many orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude <strong>in</strong> laboratory experiments.<br />
In higher energy side, e.g. higgs particle supposed<br />
to act as a fricti<strong>on</strong> to give masses to particles becomes<br />
important. In order to produce this, we would need<br />
∼ 100 GeV as the beam energy. This is <strong>on</strong>ly possible<br />
by <strong>in</strong>troduc<strong>in</strong>g higher beam momentum such as high energy<br />
colliders. What does happen <strong>in</strong> 1 eV-100 MeV range?<br />
Here the virtual vacuum polarizati<strong>on</strong>s by quantum electrodynamics<br />
and possibly quantum chromodynamics become<br />
the source <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>-phot<strong>on</strong> <strong>in</strong>teracti<strong>on</strong>s for an <strong>in</strong>stant moment.<br />
Below 1eV, the nature <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum resp<strong>on</strong>se is<br />
not well known. In this energy range exchanged field between<br />
phot<strong>on</strong>s become very light. Therefore, they can be<br />
candidates <str<strong>on</strong>g>of</str<strong>on</strong>g> dark matter and/or dark energy. We note<br />
all these <strong>in</strong>teracti<strong>on</strong>s are characterized by masses <str<strong>on</strong>g>of</str<strong>on</strong>g> exchanged<br />
fields and the coupl<strong>in</strong>g strength to phot<strong>on</strong>s.<br />
As the most ambitious motivati<strong>on</strong> to search for a lowmass<br />
field, let us <strong>in</strong>troduce a scalar field <strong>in</strong> the c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
dark energy. Scalar-Tensor-Theory with Λ (ST T Λ) [2] is<br />
<strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> many dark energy models <strong>in</strong> the market [1]. This
theory provides a natural explanati<strong>on</strong> why the observed<br />
dark energy is so small without any f<strong>in</strong>e tun<strong>in</strong>g. This predicts<br />
decay<strong>in</strong>g Λ with t −2 as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the age <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
universe t. The present age <str<strong>on</strong>g>of</str<strong>on</strong>g> the universe is t = 10 60 <strong>in</strong><br />
Planckian unit. Thus it naturally expla<strong>in</strong>s why Λ is 10 −120<br />
at present <strong>in</strong> the same unit. The uniqueness <str<strong>on</strong>g>of</str<strong>on</strong>g> the theory<br />
from an experimental po<strong>in</strong>t view is that it can give us a<br />
testable predicti<strong>on</strong> with the explicit allowed mass and coupl<strong>in</strong>g<br />
strength. As a result <str<strong>on</strong>g>of</str<strong>on</strong>g> the theory we expect a scalar<br />
field with mass <str<strong>on</strong>g>of</str<strong>on</strong>g> neV range by allow<strong>in</strong>g a few orders <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
latitudes which couples to matter via gravitati<strong>on</strong>al strength.<br />
In past experiments prob<strong>in</strong>g a deviati<strong>on</strong> from Newt<strong>on</strong>ian<br />
potential form, massive and huge bodies were used as a test<br />
probes [5]. However, when they measure the gravitati<strong>on</strong>al<br />
effects <strong>in</strong> a short distance, they must suffer from the background<br />
physical process like Coulomb force. In c<strong>on</strong>trast, if<br />
we could use phot<strong>on</strong>-phot<strong>on</strong> scatter<strong>in</strong>g as a probe for such<br />
a new k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> force, the experiment would be free from the<br />
background physical process, s<strong>in</strong>ce the total phot<strong>on</strong>-phot<strong>on</strong><br />
elastic cross secti<strong>on</strong> <strong>in</strong> the optical energy is <strong>on</strong>ly 10 −42 b at<br />
most [6].<br />
However, the biggest issue appears, because the huge<br />
and massive probes were actually demanded to have a sensitivity<br />
to the gravitati<strong>on</strong>al coupl<strong>in</strong>g strength. Nevertheless,<br />
if we could overcome this drawback to use phot<strong>on</strong>s<br />
as the test probe, the method opens up a new w<strong>in</strong>dow to<br />
probe weakly coupl<strong>in</strong>g and low-mass fields (f<strong>in</strong>ite l<strong>on</strong>g<br />
range force). As l<strong>on</strong>g as the field has a f<strong>in</strong>ite mass and coupl<strong>in</strong>g<br />
to matter, we can directly produce low-mass fields as<br />
res<strong>on</strong>ance states such as higgs particle producti<strong>on</strong> <strong>in</strong> high<br />
energy colliders. The producti<strong>on</strong> cross secti<strong>on</strong> is <strong>in</strong> pr<strong>in</strong>ciple<br />
free from the c<strong>on</strong>stra<strong>in</strong>ts by the weak coupl<strong>in</strong>g, if the<br />
center <str<strong>on</strong>g>of</str<strong>on</strong>g> mass system energy Ecms <str<strong>on</strong>g>of</str<strong>on</strong>g> two collid<strong>in</strong>g phot<strong>on</strong>s<br />
is adjusted to the top <str<strong>on</strong>g>of</str<strong>on</strong>g> the res<strong>on</strong>ance po<strong>in</strong>t. In the<br />
follow<strong>in</strong>g secti<strong>on</strong>s we discuss how to realize the phot<strong>on</strong>phot<strong>on</strong><br />
collisi<strong>on</strong> system and the sensitivity to the coupl<strong>in</strong>g<br />
as weak as gravitati<strong>on</strong>al coupl<strong>in</strong>g for the mass range well<br />
below optical frequency 1 eV.<br />
QUASI-PARALLEL SYSTEM OF<br />
PHOTON-PHOTON COLLISIONS<br />
If we are <strong>in</strong>terested <strong>in</strong> extremely low-mass ranges even<br />
below meV, we have to reduce the CMS energy <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>phot<strong>on</strong><br />
collisi<strong>on</strong>s compared to the <strong>in</strong>cident phot<strong>on</strong> energy if<br />
optical laser fields are assumed. As l<strong>on</strong>g as the res<strong>on</strong>ance is<br />
allowed to decay <strong>in</strong>to <strong>on</strong>ly two phot<strong>on</strong>s, the scatter<strong>in</strong>g process<br />
looks like elastic scatter<strong>in</strong>g even if a low-mass field<br />
is exchanged via the res<strong>on</strong>ance state <strong>in</strong> CMS. Thus there is<br />
no frequency shift <strong>in</strong> the f<strong>in</strong>al state <strong>in</strong> CMS. However, if we<br />
boost this system to the directi<strong>on</strong> perpendicular to the collid<strong>in</strong>g<br />
axis, the frequency shift takes place al<strong>on</strong>g that boost<br />
axis. In the forward directi<strong>on</strong> <strong>on</strong> the boost axis we expect a<br />
frequency up shift to close to the double <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>cident frequency,<br />
while a zero frequency phot<strong>on</strong> must be emitted to<br />
the backward directi<strong>on</strong> due to the energy-momentum c<strong>on</strong>servati<strong>on</strong><br />
<strong>in</strong>dependent <str<strong>on</strong>g>of</str<strong>on</strong>g> the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the exchanged<br />
field. We refer this boosted system as the quasi-parallel<br />
system (QPS). This may be <strong>in</strong>terpreted as if the sec<strong>on</strong>d harm<strong>on</strong>ic<br />
phot<strong>on</strong> is generated from the n<strong>on</strong>l<strong>in</strong>ear vacuum resp<strong>on</strong>se.<br />
This could be an <strong>in</strong>terest<strong>in</strong>g analogy to sec<strong>on</strong>d harm<strong>on</strong>ic<br />
generati<strong>on</strong> due to the n<strong>on</strong>l<strong>in</strong>ear resp<strong>on</strong>se <str<strong>on</strong>g>of</str<strong>on</strong>g> a crystal<br />
with a laser <strong>in</strong>jecti<strong>on</strong> which was pi<strong>on</strong>eered by Franken<br />
et al. [7]. Inversely if we realize the QPS as a laboratory<br />
frame, the corresp<strong>on</strong>d<strong>in</strong>g CMS energy can be very much<br />
lowered. The CMS energy with variables <strong>in</strong> QPS can be<br />
def<strong>in</strong>ed as<br />
Ecms ∼ 2ϑω, (1)<br />
where ϑ is def<strong>in</strong>ed as a half <strong>in</strong>cident angle between two<br />
<strong>in</strong>com<strong>in</strong>g phot<strong>on</strong>s with ϑ ≪ 1 and ω is the beam energy<br />
<strong>in</strong> unit <str<strong>on</strong>g>of</str<strong>on</strong>g> ¯h = c = 1. This relati<strong>on</strong> <strong>in</strong>dicates that we have<br />
two experimental handles to adjust Ecms. If we take the<br />
head-<strong>on</strong> collisi<strong>on</strong> geometry, we have to <strong>in</strong>troduce very l<strong>on</strong>g<br />
wavelength as the <strong>in</strong>cident phot<strong>on</strong>s. However, it is not too<br />
difficult to <strong>in</strong>troduce the very small <strong>in</strong>cident angle. In such<br />
a case Ecms can be lowered by keep<strong>in</strong>g ω c<strong>on</strong>stant. We<br />
also know that the cross secti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> photo-phot<strong>on</strong> scatter<strong>in</strong>g<br />
<strong>in</strong> QED process σqed <strong>in</strong> QPS is quite suppressed due to the<br />
fourth power dependence <strong>on</strong> the <strong>in</strong>cident angle which is<br />
expressed as σqed ∼ (α 2 /m 4 e) 2 ω 6 ϑ 4 with the f<strong>in</strong>e structure<br />
c<strong>on</strong>stant α and electr<strong>on</strong> mass me [8]. Therefore, the low<br />
frequency phot<strong>on</strong>s <strong>in</strong> QPS is the best system to probe such<br />
a low-mass field.<br />
QPS BY FOCUSING WITH SINGLE<br />
GAUSSIAN LASER BEAM<br />
However, it is difficult to <strong>in</strong>troduce two collid<strong>in</strong>g phot<strong>on</strong><br />
beams which satisfy the small <strong>in</strong>cident angle based <strong>on</strong> the<br />
simple geometrical optics due to the wavy nature <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s<br />
<strong>in</strong> the diffracti<strong>on</strong> limit <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s. Below meV range<br />
we are naturally led to <strong>in</strong>troduce a geometry by focus<strong>in</strong>g a<br />
s<strong>in</strong>gle laser beam as illustrated <strong>in</strong> Fig.1. What is important<br />
here is that <strong>in</strong> the diffracti<strong>on</strong> limit there are uncerta<strong>in</strong>ties <strong>on</strong><br />
the <strong>in</strong>cident momentum due to the uncerta<strong>in</strong>ty pr<strong>in</strong>ciple, <strong>in</strong><br />
other words, there are uncerta<strong>in</strong>ties <strong>on</strong> the <strong>in</strong>cident angles<br />
between two phot<strong>on</strong>s am<strong>on</strong>g the s<strong>in</strong>gle beam, even though<br />
phot<strong>on</strong>s are <strong>in</strong> the degenerated state at the output <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
laser crystal. This should be c<strong>on</strong>trasted to the case <str<strong>on</strong>g>of</str<strong>on</strong>g> high<br />
energy collider where the momentum spread <str<strong>on</strong>g>of</str<strong>on</strong>g> each collid<strong>in</strong>g<br />
particle or the uncerta<strong>in</strong>ty based <strong>on</strong> the de Broglie<br />
length is negligibly small compared to the magnitudes <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
relevant momentum exchanges they are <strong>in</strong>terested <strong>in</strong>. This<br />
different <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong> becomes critically important for<br />
the follow<strong>in</strong>g discussi<strong>on</strong>s.<br />
DYNAMICS OF PHOTON-PHOTON<br />
SCATTERING<br />
As the simplest coupl<strong>in</strong>g between two phot<strong>on</strong>s and a<br />
low-mass field we focus <strong>on</strong> the quantum anomaly type coupl<strong>in</strong>g<br />
g 2 /M which <strong>in</strong>cludes square <str<strong>on</strong>g>of</str<strong>on</strong>g> electric charge g to<br />
couple virtual fermi<strong>on</strong> loops to two external phot<strong>on</strong>s and a<br />
dimensi<strong>on</strong>al coupl<strong>in</strong>g 1/M to low-mass neutral fields [9].
Figure 1: Sec<strong>on</strong>d harm<strong>on</strong>ic generati<strong>on</strong> <strong>in</strong> Quasi Parallel<br />
System by focus<strong>in</strong>g a s<strong>in</strong>gle Gaussian laser beam.<br />
If M is Planckian mass scale <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 27 eV, the coupl<strong>in</strong>g expresses<br />
gravitati<strong>on</strong>al <strong>on</strong>e. We may discuss possibilities to<br />
exchange scalar and pseudoscalar type <str<strong>on</strong>g>of</str<strong>on</strong>g> fields by requir<strong>in</strong>g<br />
comb<strong>in</strong>ati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> polarizati<strong>on</strong>s <strong>in</strong> the <strong>in</strong>itial and<br />
f<strong>in</strong>al states [10]. The virtue <str<strong>on</strong>g>of</str<strong>on</strong>g> laser experiments is <strong>in</strong> the<br />
specificati<strong>on</strong>s <strong>on</strong> the all phot<strong>on</strong> sp<strong>in</strong> states both <strong>in</strong> the <strong>in</strong>itial<br />
and f<strong>in</strong>al states <strong>in</strong> the two body phot<strong>on</strong>-phot<strong>on</strong> <strong>in</strong>teracti<strong>on</strong>.<br />
This allows us to discuss types <str<strong>on</strong>g>of</str<strong>on</strong>g> exchanged fields <strong>in</strong> general.<br />
HOW TO OVERCOME THE EXTREMELY<br />
NARROW RESONANCE<br />
The exact res<strong>on</strong>ance c<strong>on</strong>diti<strong>on</strong> is the requirement <str<strong>on</strong>g>of</str<strong>on</strong>g> m =<br />
Ecms where m is the mass <str<strong>on</strong>g>of</str<strong>on</strong>g> exchanged field. The squire<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the scatter<strong>in</strong>g amplitude A can be expressed as Breit-<br />
Wigner(BW) res<strong>on</strong>ance functi<strong>on</strong> [11]<br />
|A| 2 = (4π) 2<br />
a 2<br />
χ2 , (2)<br />
(ϑ) + a2 where χ and width a are def<strong>in</strong>ed as χ(ϑ) ≡ ω 2 −ω 2 r(ϑ) and<br />
a ≡ (ω 2 r/16π)(g 2 m/M) 2 , respectively. The energy ωr<br />
satisfy<strong>in</strong>g the res<strong>on</strong>ance c<strong>on</strong>diti<strong>on</strong> can be def<strong>in</strong>ed as ωr ≡<br />
m 2 /(1 − cos 2ϑr) [10]. If we take Planckian mass as M,<br />
the width a becomes extremely small. This implies that the<br />
res<strong>on</strong>ance width is too small to hit the peak positi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
res<strong>on</strong>ance functi<strong>on</strong>. How can we overcome this problem?<br />
We take a unique approach to this situati<strong>on</strong>. Let us rem<strong>in</strong>d<br />
you <str<strong>on</strong>g>of</str<strong>on</strong>g> higgs hunt<strong>in</strong>g by high energy colliders as an<br />
example to hit the top <str<strong>on</strong>g>of</str<strong>on</strong>g> the res<strong>on</strong>ance. In high energy<br />
colliders the spread <str<strong>on</strong>g>of</str<strong>on</strong>g> the beam energy is much smaller<br />
than the width <str<strong>on</strong>g>of</str<strong>on</strong>g> the res<strong>on</strong>ance functi<strong>on</strong> which they try to<br />
probe. On the hand, <strong>in</strong> collisi<strong>on</strong>s at the diffracti<strong>on</strong> limit <strong>in</strong><br />
QPS, the res<strong>on</strong>ance width looks almost like delta-functi<strong>on</strong><br />
and the uncerta<strong>in</strong>ty <strong>on</strong> the Ecms is much wider than the<br />
res<strong>on</strong>ance width as we discussed above. We may use the<br />
follow<strong>in</strong>g feature <str<strong>on</strong>g>of</str<strong>on</strong>g> delta-functi<strong>on</strong>. Although the width <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the delta-functi<strong>on</strong> is <strong>in</strong>f<strong>in</strong>itesimal, as far as it is <strong>in</strong>tegrated<br />
over ±∞, the value <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tegral becomes order <str<strong>on</strong>g>of</str<strong>on</strong>g> unity. In<br />
the case <str<strong>on</strong>g>of</str<strong>on</strong>g> BW functi<strong>on</strong>, even if we <strong>in</strong>tegrate it over ±a,<br />
the value <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tegral is just a half <str<strong>on</strong>g>of</str<strong>on</strong>g> the value <strong>in</strong>tegrated<br />
over ±∞. This implies that as l<strong>on</strong>g as we capture the res<strong>on</strong>ance<br />
peak with<strong>in</strong> a f<strong>in</strong>ite range bey<strong>on</strong>d ±a, the value<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tegral becomes proporti<strong>on</strong>al to a but not a 2 . Actually<br />
<strong>in</strong> the diffracti<strong>on</strong> limit <strong>in</strong>cident angles <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>com<strong>in</strong>g<br />
phot<strong>on</strong>s are <strong>in</strong> pr<strong>in</strong>ciple uncerta<strong>in</strong>. Thus we must use the<br />
averaged cross secti<strong>on</strong> by <strong>in</strong>tegrat<strong>in</strong>g the square <str<strong>on</strong>g>of</str<strong>on</strong>g> the scatter<strong>in</strong>g<br />
amplitude over a possible range <strong>on</strong> Ecms determ<strong>in</strong>ed<br />
by uncerta<strong>in</strong>ties <strong>on</strong> <strong>in</strong>cident angles.<br />
Let us reflect this feature <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> s<strong>in</strong>gle beam focus<strong>in</strong>g<br />
experiment. We can <strong>in</strong>troduce a probability distributi<strong>on</strong><br />
functi<strong>on</strong> <strong>on</strong> the possible <strong>in</strong>cident angles between randomly<br />
selected phot<strong>on</strong> pairs am<strong>on</strong>g the <strong>in</strong>cident s<strong>in</strong>gle laser<br />
beam. We then def<strong>in</strong>e the range <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tegral <strong>on</strong> the <strong>in</strong>cident<br />
angel as ∆ϑ ∼ d/(2f) with the beam diameter d and the<br />
focal length f. If this range does not c<strong>on</strong>ta<strong>in</strong> the res<strong>on</strong>ance<br />
angle ϑr ≡ m/2ω, that is, ϑr > ∆ϑ, the averaged squared<br />
amplitude |A| 2 becomes proporti<strong>on</strong>al to a 2 which <strong>in</strong>dicates<br />
the suppressi<strong>on</strong> by M −4 . On the other hand, if ϑr < ∆ϑ is<br />
satisfied, we can obta<strong>in</strong> the proporti<strong>on</strong>ality to a, namely, a<br />
sensitivity enhancement by M 2 compared to the case without<br />
res<strong>on</strong>ance. Thus various focus<strong>in</strong>g parameters to adjust<br />
∆ϑ by c<strong>on</strong>troll<strong>in</strong>g the beam diameter and focal length can<br />
<strong>in</strong>troduce a sharp cut<str<strong>on</strong>g>of</str<strong>on</strong>g>f <strong>on</strong> the cross secti<strong>on</strong> which eventually<br />
c<strong>on</strong>trols sensitive mass ranges <str<strong>on</strong>g>of</str<strong>on</strong>g> this method through<br />
the relati<strong>on</strong> m < 2ω∆ϑ.<br />
SENSITIVITY TO GRAVITATIONAL<br />
COUPLING STRENGTH<br />
Let us now discuss how much <strong>in</strong>tense laser fields are<br />
necessary to have a sensitivity to gravitati<strong>on</strong>al coupl<strong>in</strong>g<br />
strength based <strong>on</strong> the c<strong>on</strong>cept <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>-phot<strong>on</strong> scatter<strong>in</strong>g<br />
<strong>in</strong> QPS. As the most challeng<strong>in</strong>g case let us assume<br />
m ∼ 10 −9 eV which can be a candidate <str<strong>on</strong>g>of</str<strong>on</strong>g> dark energy if<br />
it could couple to matter as weak as gravitati<strong>on</strong>al coupl<strong>in</strong>g<br />
strength 1/MP ∼ 10 −27 eV −1 [2]. The yield <str<strong>on</strong>g>of</str<strong>on</strong>g> harm<strong>on</strong>ic<br />
generati<strong>on</strong> Y <strong>in</strong>tegrated over the solid angle which satisfies<br />
the c<strong>on</strong>diti<strong>on</strong> that <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> f<strong>in</strong>al state phot<strong>on</strong>s has the<br />
frequency close to 2ω, can be expressed as<br />
Y ∼ Kexp(g 2 m/M) 2 s<strong>in</strong> 2 ϑL, (3)<br />
where Kexp is a total experimental factor depend<strong>in</strong>g <strong>on</strong><br />
the the durati<strong>on</strong> time <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse, the energy selecti<strong>on</strong><br />
and the focus<strong>in</strong>g parameter result<strong>in</strong>g <strong>in</strong> ϑr/∆ϑ relevant<br />
for the normalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the probability <strong>on</strong> <strong>in</strong>cident<br />
angles, (g 2 m/M) 2 ∼ (αm/M) 2 ∼ 10 −76 is the factor<br />
after averag<strong>in</strong>g <strong>on</strong> BW res<strong>on</strong>ance functi<strong>on</strong>, s<strong>in</strong> −2 ϑ is the<br />
factor related with the phase volume <strong>in</strong>tegral and the phot<strong>on</strong><br />
flux factor after <strong>in</strong>tegrated over the solid angle, and<br />
L is a lum<strong>in</strong>osity-like factor <strong>in</strong> the collider c<strong>on</strong>cept. The<br />
significant difference <str<strong>on</strong>g>of</str<strong>on</strong>g> L from that <str<strong>on</strong>g>of</str<strong>on</strong>g> collider c<strong>on</strong>cept is<br />
that phot<strong>on</strong>s are annihilated and created from degenerated
states rather than from the vacuum state |0 >. This implies<br />
that annihilati<strong>on</strong> and creati<strong>on</strong> operators cause √ ¯ N <strong>in</strong><br />
each vertex <str<strong>on</strong>g>of</str<strong>on</strong>g> the Feynman diagram with the mean number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s ¯ N <strong>in</strong> a laser field by assum<strong>in</strong>g ¯ N is large<br />
enough [12]. On the producti<strong>on</strong> vertex <str<strong>on</strong>g>of</str<strong>on</strong>g> the low-mass<br />
field, we expect a factor <str<strong>on</strong>g>of</str<strong>on</strong>g> √ 2<br />
N¯ at the amplitude level due<br />
to two phot<strong>on</strong> annihilati<strong>on</strong> <strong>in</strong> the degenerated number state.<br />
Tak<strong>in</strong>g square <str<strong>on</strong>g>of</str<strong>on</strong>g> this factor gives a similar factor to collider<br />
lum<strong>in</strong>osity which <strong>in</strong>cludes a factor proport<strong>in</strong>al to n 2 where<br />
n is the number <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles per beam bunch.<br />
In order to have <strong>on</strong>e sec<strong>on</strong>d harm<strong>on</strong>ic phot<strong>on</strong> per laser<br />
focus<strong>in</strong>g, we would need 10 34 optical phot<strong>on</strong>s corresp<strong>on</strong>d<strong>in</strong>g<br />
to ∼ 10 16 J with Kexp ∼ 10 −10 for the focal length<br />
f ∼ 1000 m, the beam diameter d ∼ 2 m and the pulse<br />
durati<strong>on</strong> τ ∼ 10 fs. Furthermore, we may also <strong>in</strong>duce decays<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the produced res<strong>on</strong>ances by add<strong>in</strong>g a degenerated<br />
phot<strong>on</strong> states with different frequency to let the <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> two<br />
phot<strong>on</strong>s from the res<strong>on</strong>ance decay <strong>in</strong>to the prepared degenerated<br />
state, where creati<strong>on</strong> operator causes the additi<strong>on</strong>al<br />
factor <str<strong>on</strong>g>of</str<strong>on</strong>g> √ ¯ N at the scatter<strong>in</strong>g amplitude level. Therefore,<br />
if we mix two frequencies <strong>in</strong> advance with equal <strong>in</strong>tensity<br />
¯N, we may expect the <strong>in</strong>crease <str<strong>on</strong>g>of</str<strong>on</strong>g> the lum<strong>in</strong>osity-like factor<br />
L ∼ ¯ N 3 . In this case the necessary phot<strong>on</strong> <strong>in</strong>tensity<br />
may be very much reduced to order <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 22 optical phot<strong>on</strong>s<br />
corresp<strong>on</strong>d<strong>in</strong>g to several kJ with the same Kexp factor. We<br />
note that the phot<strong>on</strong> frequency to be observed changes from<br />
the sec<strong>on</strong>d harm<strong>on</strong>ic <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>cident phot<strong>on</strong>s depend<strong>in</strong>g <strong>on</strong><br />
the frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> the field to <strong>in</strong>duce decays.<br />
SUMMARY<br />
Higher harm<strong>on</strong>ic generati<strong>on</strong> <strong>in</strong> Quasi-Parallel-System<br />
can be a novel probe to discuss weakly coupl<strong>in</strong>g low-mass<br />
fields. Experimental realizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QPS as parallel as possible<br />
is a challenge for future experiments. The dom<strong>in</strong>ant<br />
enhancement mechanism is orig<strong>in</strong>ated by c<strong>on</strong>ta<strong>in</strong><strong>in</strong>g the<br />
low-mass res<strong>on</strong>ant peak with<strong>in</strong> the uncerta<strong>in</strong>ty <strong>on</strong> the center<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> mass energy <strong>in</strong> the diffracti<strong>on</strong> limit <str<strong>on</strong>g>of</str<strong>on</strong>g> the focused<br />
laser beam. A degenerated field to <strong>in</strong>duce decay <str<strong>on</strong>g>of</str<strong>on</strong>g> res<strong>on</strong>ance<br />
is important to probe coupl<strong>in</strong>g as weak as gravity<br />
though higher harm<strong>on</strong>ic generati<strong>on</strong>. Given high <strong>in</strong>tensity<br />
laser fields ∼ 2 kJ per pulse available <strong>in</strong> Extreme Light<br />
Infrastructure [13], we foresee the breakthrough <strong>on</strong> the exist<strong>in</strong>g<br />
sensitivity to coupl<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> low-mass fields to phot<strong>on</strong>s<br />
based <strong>on</strong> this novel idea.<br />
ACKNOWLEDGMENTS<br />
This work is based <strong>on</strong> <strong>in</strong>tensive discussi<strong>on</strong>s with Y. Fujii,<br />
D. Habs and T. Tajima under support by the DFG Cluster<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Excellence MAP (Munich-Center for Advanced Phot<strong>on</strong>ics).<br />
REFERENCES<br />
[1] Y. F. Cai, E. N. Saridakis, M. R. Setare and J. Q. Xia,<br />
arXiv:0909.2776 [hep-th]; S. Tsujikawa, arXiv:1004.1493<br />
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[2] Y. Fujii and K. Maeda, The Scalar-Tensor Theory <str<strong>on</strong>g>of</str<strong>on</strong>g> Gravitati<strong>on</strong><br />
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[3] For example, see Figure 2 <strong>in</strong> J. Jaeckel and A. R<strong>in</strong>gwald,<br />
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[4] V. Sahni and A. A. Starob<strong>in</strong>sky, Int. J. Mod. Phys. D 9, 373<br />
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[5] See, for example, Figures; 2.13, 4.16-17 <strong>in</strong> E. Fischbach and<br />
C. Talmadge, The Search for N<strong>on</strong>-Newt<strong>on</strong>ian Gravity (AIP<br />
Press, Spr<strong>in</strong>ger-Verlag, N.Y., 1998).<br />
[6] B. De Tollis, Nuovo Cimento 32 757 (1964); B. De Tollis,<br />
Nouvo Cimento 35 1182 (1965).<br />
[7] P. Franken, A. E. Hill, C. W. Peters, and G. We<strong>in</strong>reich, Phys.<br />
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[8] See p.183 <strong>in</strong> W. Dittrich and H. Gies, Prob<strong>in</strong>g the Quantum<br />
Vacuum (Spr<strong>in</strong>ger, Berl<strong>in</strong>, 2007).<br />
[9] Y. Fujii and K. Homma, arXiv:1006.1762 [gr-qc].<br />
[10] K. Homma, D. Habs and T. Tajima, arXiv:1006.4533<br />
[quant-ph].<br />
[11] For example, see secti<strong>on</strong> for Cross-secti<strong>on</strong> formulae for specific<br />
processes <strong>in</strong> C. Amsler et al. (Particle Data Group),<br />
Phy. Lett. B667, 1 (2008) and 2009 partial update for the<br />
2010 editi<strong>on</strong>.<br />
[12] For example see Rodney Loud<strong>on</strong>, The Quantum Theory<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Light 3rd editi<strong>on</strong> (Oxford University Press, New York,<br />
2000).<br />
[13] http://www.extreme-light-<strong>in</strong>frastructure.eu/.
Abstract<br />
Dynamical view <str<strong>on</strong>g>of</str<strong>on</strong>g> pair creati<strong>on</strong> via the Schw<strong>in</strong>ger mechanism ∗<br />
Naoto Tanji †<br />
Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> physics, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo, Komaba, Tokyo 153-8902, Japan<br />
Particle producti<strong>on</strong> via the Schw<strong>in</strong>ger mechanism has<br />
been studied as a mechanism <str<strong>on</strong>g>of</str<strong>on</strong>g> matter formati<strong>on</strong> <strong>in</strong> the<br />
c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g> heavy-i<strong>on</strong> collisi<strong>on</strong>s. We describe the particle<br />
pair creati<strong>on</strong> <strong>in</strong> a str<strong>on</strong>g electric field focus<strong>in</strong>g <strong>on</strong> its realtime<br />
dynamics. Motivated by the Color Glass C<strong>on</strong>densate<br />
framework, we <strong>in</strong>vestigate the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> a magnetic field<br />
which is parallel to the electric field, and show that the<br />
magnetic field enhances quark producti<strong>on</strong>. Also the pair<br />
creati<strong>on</strong> <strong>in</strong> a boost-<strong>in</strong>variantly expand<strong>in</strong>g electric field is<br />
discussed.<br />
INTRODUCTION<br />
Particle pair creati<strong>on</strong> from vacuum <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
a str<strong>on</strong>g electric field, which is known as the Schw<strong>in</strong>ger<br />
mechanism [1], is <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the most remarkable c<strong>on</strong>sequences<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quantum field theory. This particle producti<strong>on</strong><br />
mechanism has attracted c<strong>on</strong>siderable theoretical<br />
and experimental <strong>in</strong>terests, because it c<strong>on</strong>cerns the n<strong>on</strong>perturbative<br />
aspects <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum field theory. However, no<br />
direct observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger particle producti<strong>on</strong> has<br />
ever been obta<strong>in</strong>ed because it requires very str<strong>on</strong>g electric<br />
fields above the critical strength Ec = m 2 e/e ∼ 10 16 V/cm,<br />
which is bey<strong>on</strong>d current technological capabilities.<br />
Although sufficient field strength has not yet been realized<br />
<strong>in</strong> laboratories as a QED electric field, it may be atta<strong>in</strong>able<br />
as a QCD color field because <str<strong>on</strong>g>of</str<strong>on</strong>g> its str<strong>on</strong>g nature.<br />
Color electric fields are expected to be generated <strong>in</strong> highenergy<br />
particle collisi<strong>on</strong> experiments, such as relativistic<br />
heavy-i<strong>on</strong> collisi<strong>on</strong>s. High energy particle collisi<strong>on</strong> experiments<br />
may be a promis<strong>in</strong>g playground for the str<strong>on</strong>g field<br />
physics.<br />
Figure 1: A schematic <str<strong>on</strong>g>of</str<strong>on</strong>g> the l<strong>on</strong>gitud<strong>in</strong>al evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> relativistic<br />
heavy-i<strong>on</strong> collisi<strong>on</strong>s.<br />
∗ Work supported by the Japan Society for the Promoti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Science<br />
for Young Scientists<br />
† tanji@nt1.c.u-tokyo.ac.jp<br />
Relativistic heavy-i<strong>on</strong> collisi<strong>on</strong> experiments have been<br />
d<strong>on</strong>e at Relativistic Heavy I<strong>on</strong> Collider (RHIC) and are<br />
start<strong>in</strong>g at Large Hadr<strong>on</strong> Collider (LHC) to explore the nature<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> matter <strong>in</strong> extreme c<strong>on</strong>diti<strong>on</strong>. A space-time picture<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> relativistic heavy-i<strong>on</strong> collisi<strong>on</strong>s is illustrated <strong>in</strong> Fig. 1.<br />
The formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> quark-glu<strong>on</strong> plasma (QGP), which c<strong>on</strong>sists<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quarks and glu<strong>on</strong>s liberated from c<strong>on</strong>f<strong>in</strong>ement, is<br />
expected there. In the descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP, hydrodynamic<br />
simulati<strong>on</strong>s suppos<strong>in</strong>g the local thermalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the system<br />
have achieved great successes (see e.g. [2]). However,<br />
a full understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> how high-energy matter is created<br />
and how local thermalizati<strong>on</strong> is realized before the hydrodynamic<br />
evoluti<strong>on</strong> is still lack<strong>in</strong>g.<br />
One <str<strong>on</strong>g>of</str<strong>on</strong>g> the scenarios <str<strong>on</strong>g>of</str<strong>on</strong>g> matter formati<strong>on</strong> <strong>in</strong> heavy-i<strong>on</strong><br />
collisi<strong>on</strong>s is based <strong>on</strong> the flux-tube model [3]. When two<br />
Lorentz-c<strong>on</strong>tracted disks <str<strong>on</strong>g>of</str<strong>on</strong>g> nuclei collide and pass through<br />
each other, they exchange color charges (glu<strong>on</strong>s), and after<br />
the collisi<strong>on</strong>, many flux tubes are generated between the<br />
two capacitor plates (Fig. 2(a)). These flux tubes decay<br />
<strong>in</strong>to quarks and glu<strong>on</strong>s by the Schw<strong>in</strong>ger pair creati<strong>on</strong> [4].<br />
Recently also the Color Glass C<strong>on</strong>densate (CGC) model,<br />
which is an effective theory to describe high-energy nuclei<br />
<strong>in</strong> saturated regi<strong>on</strong> (see e.g. [5] for review), has predicted<br />
the formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> color electric fields <strong>in</strong> the l<strong>on</strong>gitud<strong>in</strong>al<br />
beam directi<strong>on</strong> [6]. The field strength predicted by<br />
the CGC is gE ∼ 1 GeV 2 at RHIC energy. This is str<strong>on</strong>g<br />
enough to cause the Schw<strong>in</strong>ger particle producti<strong>on</strong>. Therefore<br />
the Schw<strong>in</strong>ger mechanism attracts renewed <strong>in</strong>terest <strong>in</strong><br />
the c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g> the CGC [7]. One <str<strong>on</strong>g>of</str<strong>on</strong>g> the remarkable differences<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the color flux tube given by the CGC from that <strong>in</strong><br />
the orig<strong>in</strong>al flux-tube model is the existence <str<strong>on</strong>g>of</str<strong>on</strong>g> the l<strong>on</strong>gitud<strong>in</strong>al<br />
color magnetic fields <strong>in</strong> additi<strong>on</strong> to the electric fields<br />
[6].<br />
In this paper, we <strong>in</strong>vestigate the Schw<strong>in</strong>ger mechanism<br />
focus<strong>in</strong>g <strong>on</strong> the follow<strong>in</strong>g issues.<br />
(a) flux tubes<br />
(b) the boost-<strong>in</strong>variant c<strong>on</strong>figurati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electric fields<br />
Figure 2: Color electric fields expected <strong>in</strong> the early stage <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
heavy-i<strong>on</strong> collisi<strong>on</strong>s.
Real-time descripti<strong>on</strong> Because the system <str<strong>on</strong>g>of</str<strong>on</strong>g> heavy-i<strong>on</strong><br />
collisi<strong>on</strong>s is a dynamic <strong>on</strong>e, we study the n<strong>on</strong>equilibrium<br />
dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair creati<strong>on</strong>.<br />
Effects <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic fields The CGC predicts the generati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>on</strong>gitud<strong>in</strong>al color-magnetic fields as well as<br />
color-electric fields just after a collisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> heavy-nuclei,<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> which state is called Glasma. We exam<strong>in</strong>e the effects <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
such magnetic fields <strong>on</strong> the pair creati<strong>on</strong>.<br />
Expand<strong>in</strong>g geometry <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric fields Electric fields<br />
expected to be generated <strong>in</strong> the early stage <str<strong>on</strong>g>of</str<strong>on</strong>g> heavy-i<strong>on</strong> collisi<strong>on</strong>s<br />
exist <strong>on</strong>ly between two nuclei reced<strong>in</strong>g from each<br />
other at a velocity close to the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light (Fig. 2(b)).<br />
This situati<strong>on</strong> is quite different from that <str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger’s<br />
orig<strong>in</strong>al work, <strong>in</strong> which spatially uniform fields are treated.<br />
We study how the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> particle producti<strong>on</strong> is modified<br />
<strong>in</strong> this field c<strong>on</strong>figurati<strong>on</strong>.<br />
REAL-TIME DESCRIPTION OF PAIR<br />
CREATION<br />
Can<strong>on</strong>ical quantizati<strong>on</strong> <strong>in</strong> background fields<br />
To get a real-time descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pair creati<strong>on</strong>, we <strong>in</strong>troduce<br />
an <strong>in</strong>stantaneous particle def<strong>in</strong>iti<strong>on</strong> by an explicit<br />
quantizati<strong>on</strong> procedure under a background electric field<br />
[8]. The time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the quantum state <str<strong>on</strong>g>of</str<strong>on</strong>g> charged<br />
particles is described by the variati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> that particle def<strong>in</strong>iti<strong>on</strong>.<br />
As an illustrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> it, we first study the QED particle<br />
producti<strong>on</strong> <strong>in</strong> a spatially uniform electric field. Extensi<strong>on</strong><br />
to the quark producti<strong>on</strong> under a n<strong>on</strong>-Abelian colorelectric<br />
field is possible [9]. We suppose the electric field<br />
E = (0, 0, Ez(t)) is vanish<strong>in</strong>g at t < 0 and is turned<br />
<strong>on</strong> at time t = 0. The gauge A 0 = 0 is chosen so that<br />
A 1 = A 2 = 0 and Ez(t) = − d<br />
dt A3 (t). The charged sp<strong>in</strong>or<br />
field ψ obeys the Dirac equati<strong>on</strong>:<br />
[iγ µ (∂µ + ieAµ) − m] ψ(t, x) = 0. (1)<br />
The field quantizati<strong>on</strong> is accomplished by impos<strong>in</strong>g the<br />
can<strong>on</strong>ical anti-commutati<strong>on</strong> relati<strong>on</strong> {ψ(t, x), π(t, x ′ )} =<br />
iδ(x − x ′ ), where π(t, x) = iψ † (t, x) is can<strong>on</strong>ical c<strong>on</strong>jugate<br />
momentum. The quantized quark field may be expanded<br />
as<br />
ψ(x) = ∑<br />
s=↑,↓<br />
∫<br />
d 3 [<br />
p +ψ <strong>in</strong> ps(x)a <strong>in</strong> p,s + −ψ <strong>in</strong> ps(x)b <strong>in</strong>†<br />
]<br />
−p,s ,<br />
(2)<br />
where a <strong>in</strong> p,s [b <strong>in</strong> p,s] is the annihilati<strong>on</strong> operator <str<strong>on</strong>g>of</str<strong>on</strong>g> a particle<br />
[antiparticle] with momentum p and sp<strong>in</strong> s, and ±ψ <strong>in</strong> ps(x)<br />
are classical soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the Dirac equati<strong>on</strong> (1). The superscript<br />
‘<strong>in</strong>’ dist<strong>in</strong>guishes the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong> for ±ψ <strong>in</strong> ps(x): at<br />
t < 0, +ψ <strong>in</strong> ps(x) [−ψ <strong>in</strong> ps(x)] is identical to the positive [negative]<br />
energy soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the free Dirac equati<strong>on</strong>. We set the<br />
state to be <strong>in</strong>-vacuum |0, <strong>in</strong>⟩, where no particle exists <strong>in</strong>itially<br />
and which is def<strong>in</strong>ed by a <strong>in</strong> p,s|0, <strong>in</strong>⟩ = b <strong>in</strong> p,s|0, <strong>in</strong>⟩ = 0.<br />
At t > 0, ±ψ <strong>in</strong> ps(x) evolves under the <strong>in</strong>fluence <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electric field and becomes superpositi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a positive and<br />
negative energy (frequency) state. To describe the pair creati<strong>on</strong><br />
dynamically, we <strong>in</strong>troduce a time-dependent particle<br />
def<strong>in</strong>iti<strong>on</strong> by decompos<strong>in</strong>g the field operator ψ(x) <strong>in</strong>to positive<br />
and negative frequency <strong>in</strong>stantaneously:<br />
ψ(t0, x) = ∑<br />
s=↑,↓<br />
∫<br />
d 3 [<br />
p +ψ (t0)<br />
ps (x)ap,s(t0)<br />
+−ψ (t0)<br />
ps (x)b †<br />
−p,s(t0)<br />
]<br />
,<br />
where +ψ (t0)<br />
ps (x) [−ψ (t0)<br />
ps (x)] is a positive [negative] frequency<br />
soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Dirac equati<strong>on</strong> under the pure gauge<br />
A 3 = A 3 (t = t0). Instantaneous particle picture is def<strong>in</strong>ed<br />
by ap,s(t) and bp,s(t). Of course, ap,s(t) and bp,s(t) agree<br />
with a <strong>in</strong> p,s and b <strong>in</strong> p,s at t < 0, respectively. The particle def<strong>in</strong>iti<strong>on</strong><br />
at time t and that <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>-state are related by the<br />
time-dependent Bogoliubov transformati<strong>on</strong>:<br />
ap,s(t) = αp,s(t)a <strong>in</strong> p+eA(t),s<br />
+ βp,s(t)b <strong>in</strong> †<br />
−p−eA(t),s ,<br />
(3)<br />
b †<br />
−p,s(t) = α ∗ <strong>in</strong> †<br />
p,s(t)b−p−eA(t),s − β∗ p,s(t)a <strong>in</strong> (4)<br />
p+eA(t),s ,<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> which coefficients satisfy<br />
|αps(t)| 2 + |βps(t)| 2 = 1. (5)<br />
Because the creati<strong>on</strong> and annihilati<strong>on</strong> operators are<br />
mixed by the Bogoliubov transformati<strong>on</strong>, the vacuum<br />
expectati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the number operator may be n<strong>on</strong>zero:<br />
⟨0, <strong>in</strong>|a † 2 V<br />
ps(t)aps(t)|0, <strong>in</strong>⟩ = |βps(t)| (2π) 3 , where V is the<br />
volume <str<strong>on</strong>g>of</str<strong>on</strong>g> the system. This means that particle creati<strong>on</strong><br />
happens. We def<strong>in</strong>e a particle pair distributi<strong>on</strong> functi<strong>on</strong> as<br />
fps(t) = ⟨0, <strong>in</strong>|a † ps(t)aps(t)|0, <strong>in</strong>⟩ (2π)3<br />
V<br />
= ⟨0, <strong>in</strong>|b †<br />
−ps(t)b−ps(t)|0, <strong>in</strong>⟩ (2π)3<br />
V<br />
= |βps(t)| 2 .<br />
Because <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge and the momentum c<strong>on</strong>servati<strong>on</strong>,<br />
antiparticles have always opposite momentum to particles.<br />
The distributi<strong>on</strong> functi<strong>on</strong> can not exceed unity: fp,s(t) =<br />
|βp,s| 2 = 1 − |αp,s| 2 ≤ 1, because <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>stra<strong>in</strong>t (5).<br />
This is a manifestati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Pauli’s exclusi<strong>on</strong> pr<strong>in</strong>ciple.<br />
Pair creati<strong>on</strong> <strong>in</strong> c<strong>on</strong>stant electric fields<br />
In Fig. 3, the time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the momentum distributi<strong>on</strong><br />
functi<strong>on</strong> (6) under a c<strong>on</strong>stant electric field Ez(t) =<br />
E is plotted. Hereafter, all figures are shown <strong>in</strong> the<br />
dimensi<strong>on</strong>-less unit scaled by √ eE [or √ eE0 <strong>in</strong> a n<strong>on</strong>steady<br />
field case, where E0 is an <strong>in</strong>itial field strength].<br />
After the switch-<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field, particles are created<br />
with approximately vanish<strong>in</strong>g l<strong>on</strong>gitud<strong>in</strong>al momenta.<br />
Their occupati<strong>on</strong> numbers, <strong>in</strong> other words, the ( heights <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the distributi<strong>on</strong>s can be approximated by exp − πm2<br />
)<br />
⊥<br />
eE ,<br />
where m⊥ is the transverse mass def<strong>in</strong>ed by m 2 ⊥ ≡ m2 +p 2 ⊥ .<br />
(6)
(a) l<strong>on</strong>gitud<strong>in</strong>al momentum distributi<strong>on</strong> with fixed<br />
transverse momentum p⊥ = 0<br />
(b) transverse momentum distributi<strong>on</strong> with fixed<br />
l<strong>on</strong>gitud<strong>in</strong>al momentum √ eEpz = 1<br />
Figure 3: Time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the momentum distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> created particles under the c<strong>on</strong>stant electric field. m2<br />
2eE = 0.1.<br />
These features are c<strong>on</strong>sistent with the semi-classical tunnel<strong>in</strong>g<br />
descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pair creati<strong>on</strong> [10]. However, the l<strong>on</strong>gitud<strong>in</strong>al<br />
momenta that particles br<strong>in</strong>g when they are created<br />
are not exactly zero but broadened around zero due<br />
to quantum fluctuati<strong>on</strong>. After created, particles are accelerated<br />
by the electric field accord<strong>in</strong>g to the classical equati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong>: pz = eEt + c<strong>on</strong>st. In c<strong>on</strong>trast, there is no<br />
accelerati<strong>on</strong> <strong>in</strong> the transverse directi<strong>on</strong>s and the transverse<br />
momentum distributi<strong>on</strong>s exhibits a Gaussian-like form.<br />
Back reacti<strong>on</strong><br />
Previously, we have treated the pair creati<strong>on</strong> <strong>in</strong> the c<strong>on</strong>stant<br />
electric field. However, if charged particles are created,<br />
they generate further electromagnetic fields and the<br />
orig<strong>in</strong>al field should be modified. This effect, called back<br />
reacti<strong>on</strong>, is not negligible under a str<strong>on</strong>g field where pair<br />
creati<strong>on</strong> happens <strong>in</strong>tensively. To take account <str<strong>on</strong>g>of</str<strong>on</strong>g> the back<br />
reacti<strong>on</strong>, we treat an electromagnetic field as a dynamical<br />
variable obey<strong>in</strong>g the Maxwell equati<strong>on</strong>s. The uniformity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the system reduces the Maxwell equati<strong>on</strong>s <strong>in</strong>to the s<strong>in</strong>gle<br />
equati<strong>on</strong><br />
dEz<br />
dt = −d2 A 3<br />
dt 2 = −jz, (7)<br />
Figure 4: Time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the l<strong>on</strong>gitud<strong>in</strong>al momentum<br />
distributi<strong>on</strong>. The effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the back reacti<strong>on</strong> is taken <strong>in</strong>to<br />
account. m2 = 0.01 and e = 1.<br />
2eE0<br />
where jz is the charge current generated by created particles<br />
and antiparticles.<br />
We have numerically solved the coupled equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> (1)<br />
and (7). The results are shown <strong>in</strong> Figs. 4 and 5. The l<strong>on</strong>gitud<strong>in</strong>al<br />
momentum distributi<strong>on</strong> (Fig. 4) oscillates <strong>in</strong> momentum<br />
space, and also the current and the electric field<br />
(Fig. 5) show oscillati<strong>on</strong>s. These oscillati<strong>on</strong>s can be understood<br />
as usual plasma oscillati<strong>on</strong>.<br />
Other than the plasma oscillati<strong>on</strong>, which is a classical<br />
dynamics, also quantum effects such as the Pauli block<strong>in</strong>g<br />
and <strong>in</strong>terference between matter fields play remarkable<br />
roles <strong>in</strong> the time evoluti<strong>on</strong>. Because <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference, the<br />
momentum distributi<strong>on</strong> shows f<strong>in</strong>e oscillati<strong>on</strong>s at later time<br />
(Fig. 4).<br />
EFFECTS OF MAGNETIC FIELDS<br />
S<strong>in</strong>ce the formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>on</strong>gitud<strong>in</strong>al color-magnetic fields<br />
as well as color-electric fields is predicted by the framework<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the CGC [6], we study the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> a l<strong>on</strong>gitud<strong>in</strong>al<br />
magnetic field <strong>on</strong> the pair creati<strong>on</strong>. As illustrated <strong>in</strong> Fig. 6,<br />
under the l<strong>on</strong>gitud<strong>in</strong>al magnetic field, transverse momentum<br />
p⊥ is discretized <strong>in</strong>to the Landau levels as<br />
p 2 ⊥ −→ (2n + 1)eB (n = 0, 1, 2, . . . ). (8)<br />
Notice that even the lowest Landau level depends <strong>on</strong> the<br />
magnetic field strength B for the case <str<strong>on</strong>g>of</str<strong>on</strong>g> scalar particles.<br />
S<strong>in</strong>ce the transverse momentum acts as an effective mass<br />
m2 ⊥ = m2 + p2 ⊥ , scalar particles become effectively heavy<br />
under a str<strong>on</strong>g magnetic field, so that their pair producti<strong>on</strong><br />
(a) current density (b) electric field<br />
Figure 5: Time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge current density and<br />
the electric field. e = 1. (a = m2<br />
2eE0 ).
Figure 6: A schematic <str<strong>on</strong>g>of</str<strong>on</strong>g> the Landau levels and the sp<strong>in</strong>magnetic<br />
field <strong>in</strong>teracti<strong>on</strong>.<br />
Figure 7: Magnetic field dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge current<br />
density. m2<br />
2gE0<br />
= 0.01 and g = 1.<br />
is str<strong>on</strong>gly suppressed. In c<strong>on</strong>trast, ow<strong>in</strong>g to sp<strong>in</strong>-magnetic<br />
field <strong>in</strong>teracti<strong>on</strong>, the lowest level <str<strong>on</strong>g>of</str<strong>on</strong>g> sp<strong>in</strong>or particles is zero<br />
and thereby <strong>in</strong>dependent <str<strong>on</strong>g>of</str<strong>on</strong>g> B. Therefore, the producti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> sp<strong>in</strong>or particles <strong>in</strong> that level is not at all suppressed. Not<br />
<strong>on</strong>ly the creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the lowest level is not suppressed, the<br />
magnetic field enhances field quantities such as the current<br />
and the total particle number. That is because the number <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
modes degenerat<strong>in</strong>g <strong>in</strong> a unit transverse area is proporti<strong>on</strong>al<br />
to B.<br />
In Fig. 7, the magnetic field strength dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
charge current density is exhibited. The current density is<br />
<strong>in</strong>deed enhanced by the magnetic field. Furthermore, the<br />
enhanced current makes the time scale <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma oscillati<strong>on</strong><br />
shorter through the back reacti<strong>on</strong>. This mechanism<br />
may have significance <strong>in</strong> the time evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Glasma.<br />
PARTICLE PRODUCTION IN<br />
EXPANDING ELECTRIC FIELDS<br />
So far, we have <strong>in</strong>vestigated the pair creati<strong>on</strong> <strong>in</strong> the spatially<br />
uniform electric fields. However, <strong>in</strong> a system <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
heavy-i<strong>on</strong> collisi<strong>on</strong>s, color electric fields are expected to<br />
be generated between two color-charged nuclei reced<strong>in</strong>g<br />
from each other at a velocity close to the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light<br />
(Fig. 2(b)). A characteristic <str<strong>on</strong>g>of</str<strong>on</strong>g> this electric field is the <strong>in</strong>variance<br />
under the Lorentz-boost <strong>in</strong> the l<strong>on</strong>gitud<strong>in</strong>al directi<strong>on</strong>.<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> spatially homogeneous fields, particles<br />
and antiparticles with (approximately) vanish<strong>in</strong>g l<strong>on</strong>gitud<strong>in</strong>al<br />
momentum are created. If this is true also <strong>in</strong> the boost<strong>in</strong>variant<br />
electric field c<strong>on</strong>f<strong>in</strong>ed <strong>in</strong> the forward light c<strong>on</strong>e,<br />
these particle-antiparticle pairs clearly break the boost <strong>in</strong>variance<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the background field by <strong>in</strong>troduc<strong>in</strong>g <strong>on</strong>e spe-<br />
cific frame, namely their center-<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass frame. How is this<br />
situati<strong>on</strong> modified if an electric field exists <strong>on</strong>ly <strong>in</strong>side the<br />
forward light c<strong>on</strong>e? Actually, under the boost-<strong>in</strong>variantly<br />
expand<strong>in</strong>g electric field, particles are created not as an<br />
eigenstate <str<strong>on</strong>g>of</str<strong>on</strong>g> the l<strong>on</strong>gitud<strong>in</strong>al momentum, which violates<br />
the boost-symmetry, but as a superpositi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> several momentum<br />
modes preserv<strong>in</strong>g the boost-symmetry [11]. The<br />
particles have the scal<strong>in</strong>g velocity distributi<strong>on</strong> vz(t, z) =<br />
z/t from the first <strong>in</strong>stance they are created. This velocity<br />
distributi<strong>on</strong> is the same as the flow velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the boost<strong>in</strong>variantly<br />
expand<strong>in</strong>g fluid <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP, i.e. the Bjorken flow<br />
[12]. Therefore, our result would narrow the gap between<br />
the pre-equilibrium stage <str<strong>on</strong>g>of</str<strong>on</strong>g> heavy-i<strong>on</strong> collisi<strong>on</strong>s and the<br />
state <str<strong>on</strong>g>of</str<strong>on</strong>g> the boost-<strong>in</strong>variantly expand<strong>in</strong>g QGP.<br />
SUMMARY<br />
In this paper, we have <strong>in</strong>vestigated the real-time dynamics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger particle producti<strong>on</strong> and its phenomenological<br />
applicati<strong>on</strong>s to matter formati<strong>on</strong> <strong>in</strong> relativistic<br />
heavy-i<strong>on</strong> collisi<strong>on</strong>s.<br />
Because <str<strong>on</strong>g>of</str<strong>on</strong>g> the back reacti<strong>on</strong>, the electric field varies <strong>in</strong><br />
time. Us<strong>in</strong>g the <strong>in</strong>itial field strength gE0 ∼ 1 GeV 2 , which<br />
is expected <strong>in</strong> the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> the CGC, <strong>on</strong>e can estimate<br />
the time scale <str<strong>on</strong>g>of</str<strong>on</strong>g> this field variati<strong>on</strong> to be a few fm/c. This<br />
time scale is the same order as that <str<strong>on</strong>g>of</str<strong>on</strong>g> the pre-equilibrium<br />
stage <str<strong>on</strong>g>of</str<strong>on</strong>g> heavy-i<strong>on</strong> collisi<strong>on</strong>s ∼ 1 fm/c. This result would<br />
verify the importance <str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger mechanism at the<br />
<strong>in</strong>itial stage <str<strong>on</strong>g>of</str<strong>on</strong>g> heavy-i<strong>on</strong> collisi<strong>on</strong>s.<br />
We have also discussed the enhancement <str<strong>on</strong>g>of</str<strong>on</strong>g> the quark<br />
producti<strong>on</strong> by the l<strong>on</strong>gitud<strong>in</strong>al magnetic field, and the<br />
emergence <str<strong>on</strong>g>of</str<strong>on</strong>g> the boost-<strong>in</strong>variant velocity distributi<strong>on</strong> from<br />
the expand<strong>in</strong>g electric field. These results would have a<br />
significance to understand the formati<strong>on</strong> process <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP.<br />
REFERENCES<br />
[1] J. S. Schw<strong>in</strong>ger, Phys. Rev. 82 (1951) 664.<br />
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2003.<br />
[3] F. E. Low, Phys. Rev. D12 (1975) 163; S. Nuss<strong>in</strong>ov, Phys.<br />
Rev. Lett. 34 (1975) 1286.<br />
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T. Matsui, Phys. Lett. B164 (1985) 373; G. Gat<str<strong>on</strong>g>of</str<strong>on</strong>g>f, A. K.<br />
Kerman, T. Matsui, Phys. Rev. D36 (1987) 114.<br />
[5] E. Iancu, R. Venugopalan, arXiv:hep-ph/0303204.<br />
[6] T. Lappi, L. McLerran, Nucl. Phys. A772 (2006) 200.<br />
[7] D. Kharzeev, E. Lev<strong>in</strong>, K. Tuch<strong>in</strong>, Phys. Rev. C75 (2007)<br />
044903; P. Castor<strong>in</strong>a, D. Kharzeev, H. Satz, Eur. Phys. J.<br />
C52 (2007) 187; K. Fukushima, F. Gelis, T. Lappi, Nucl.<br />
Phys. A831 (2009) 184.<br />
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[10] V. Popov, Sov. Phys. JETP 34 (1972) 709; A. Casher,<br />
H. Neuberger, S. Nuss<strong>in</strong>ov, Phys. Rev. D20 (1979) 179.<br />
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Exact soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> pair producti<strong>on</strong>s <strong>in</strong> str<strong>on</strong>g electric field with f<strong>in</strong>ite width<br />
Abstract<br />
A. Iwazaki, Nishogakusha University, Chiyoda-ku 3-6-16, Tokyo, 102-8336, Japan<br />
We show that chiral anomaly is a very useful tool for discuss<strong>in</strong>g<br />
Schw<strong>in</strong>ger mechanism when coll<strong>in</strong>ear str<strong>on</strong>g electric<br />
and magnetic fields are present. We can obta<strong>in</strong> number<br />
densities <str<strong>on</strong>g>of</str<strong>on</strong>g> particles without calculat<strong>in</strong>g their wave functi<strong>on</strong>s.<br />
By tak<strong>in</strong>g <strong>in</strong>to account back reacti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particles<br />
we can explicitly show soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field even<br />
when it has f<strong>in</strong>ite size or when the pair creati<strong>on</strong>s occur <strong>in</strong><br />
heat bath.<br />
CHIRAL ANOMALY AND SCHWINGER<br />
MECHANISM<br />
The Schw<strong>in</strong>ger mechanism[1] is a n<strong>on</strong>-perturbative phenomena<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> particle pair producti<strong>on</strong>s under str<strong>on</strong>g electric<br />
field. Namely, when the unifrom electric field E is present,<br />
the potential eEx <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles has no lower bound<br />
and becomes negative <strong>in</strong>f<strong>in</strong>ity as x → −∞. Then, it is<br />
energetically favorable that for example, electr<strong>on</strong>-positr<strong>on</strong><br />
pairs are sp<strong>on</strong>taneously produced and they partially screen<br />
the electric field. S<strong>in</strong>ce such a producti<strong>on</strong> is suppressed by<br />
the factor exp(−m 2 /eE) where m(e > 0) is the electr<strong>on</strong><br />
mass (charge), the phenomena is n<strong>on</strong>-perturbative; we can<br />
not expand it <strong>in</strong> the power series <str<strong>on</strong>g>of</str<strong>on</strong>g> eE.<br />
In general, <strong>in</strong> order to discuss such n<strong>on</strong>-perturbative<br />
phenomena we need to obta<strong>in</strong> wave functi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> charged<br />
particles under electric fields[2]. When the electric field<br />
has spatially n<strong>on</strong>trivial c<strong>on</strong>figulati<strong>on</strong> or complicated time<br />
dependence, it is quite difficult to obta<strong>in</strong> the wave functi<strong>on</strong>s.<br />
Furthermore, if there is a magnetic field <strong>in</strong> additi<strong>on</strong><br />
to the electric field, Dirac equati<strong>on</strong> becomes too complicated.<br />
On the other hand, when the magnetic field is sufficiently<br />
str<strong>on</strong>g for the particles to occupy <strong>on</strong>ly the lowest<br />
Landau level, the phenomena are simplified. This is because<br />
<strong>on</strong>ly the moti<strong>on</strong>s <strong>in</strong> the l<strong>on</strong>gitud<strong>in</strong>al directi<strong>on</strong> parallel<br />
to the magnetic field are allowed; the moti<strong>on</strong>s <strong>in</strong> the<br />
transvers directi<strong>on</strong>s are frozen ow<strong>in</strong>g to the magnetic field.<br />
Thus, the phenomena occur <strong>in</strong> spatially <strong>on</strong>e dimensi<strong>on</strong>.<br />
In this report we expla<strong>in</strong> the utility[3] <str<strong>on</strong>g>of</str<strong>on</strong>g> chiral anomaly<br />
for the discussi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Schw<strong>in</strong>ger mechanism when such a<br />
str<strong>on</strong>g magnetic field is present. In particular, without solv<strong>in</strong>g<br />
the complicated field equati<strong>on</strong>s, we can f<strong>in</strong>d physically<br />
mean<strong>in</strong>gful quantities associated with the pair producti<strong>on</strong><br />
simply by us<strong>in</strong>g the chiral anomaly,<br />
∂t(nR − nL) = e2<br />
4π 2 ⃗ E · ⃗ B, (1)<br />
where nR and nL denote the number density <str<strong>on</strong>g>of</str<strong>on</strong>g> right and<br />
left chiral fermi<strong>on</strong>s (hereafter we c<strong>on</strong>sider electr<strong>on</strong>s and<br />
positr<strong>on</strong>s). We have assumed [3] that both <str<strong>on</strong>g>of</str<strong>on</strong>g> electric and<br />
magnetic fields are much larger than the mass <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s<br />
and that the particles occupy <strong>on</strong>ly the lowest Lnadau level.<br />
The anomaly equati<strong>on</strong> implies that chirality is not c<strong>on</strong>served<br />
when both electric (E) and magnetic fields (B) are<br />
present. The chirality is equal to the helicity <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle<br />
when particle mass is negligible. Thus, the anomaly implies<br />
that the temporal evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the difference, nL−nR,<br />
is given by ⃗ E · ⃗ B.<br />
Here we should note that electr<strong>on</strong>s (positr<strong>on</strong>s) under<br />
the str<strong>on</strong>g magnetic field have sp<strong>in</strong>s anti-parallel (parallel)<br />
to the magnetic field. On the other hand, when the<br />
pair producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> and positr<strong>on</strong>s arises, electr<strong>on</strong>s<br />
(positr<strong>on</strong>s) are accelerated to the directi<strong>on</strong> anti-parallel<br />
(parallel) to the electric field. Therefore, when the electric<br />
field and magnetic field are parallel to each other, their sp<strong>in</strong><br />
and momentum are po<strong>in</strong>ted to the same directi<strong>on</strong>. Hence,<br />
all <str<strong>on</strong>g>of</str<strong>on</strong>g> produced particles are right handed; nR ̸= 0 and<br />
nL = 0. Obviously their number densities are equal to nR.<br />
Thus, the producti<strong>on</strong> rate <str<strong>on</strong>g>of</str<strong>on</strong>g> the particles is governed by the<br />
anomaly equati<strong>on</strong>. This is the reas<strong>on</strong> why the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the chiral anomaly describes the pair producti<strong>on</strong> when both<br />
str<strong>on</strong>g electric and magnetic fields are present. We expla<strong>in</strong><br />
it <strong>in</strong> several cases.<br />
We should stress that all <str<strong>on</strong>g>of</str<strong>on</strong>g> quatum effects <strong>on</strong> Schw<strong>in</strong>ger<br />
mechanism are <strong>in</strong>volved <strong>in</strong> the anomaly equati<strong>on</strong>. Hence,<br />
by simply solv<strong>in</strong>g classical equati<strong>on</strong>s al<strong>on</strong>g with the<br />
anomaly equati<strong>on</strong> we can obta<strong>in</strong> quantities <strong>in</strong>clud<strong>in</strong>g quatum<br />
effects.<br />
Time dependent homogeneus electric field<br />
Assum<strong>in</strong>g that there are no particles before t = 0 and<br />
the uniform electric field is switched <strong>on</strong> at t = 0, the<br />
pair producti<strong>on</strong> starts to occur and the number densities <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
electr<strong>on</strong>s ne and positr<strong>on</strong>s np(= ne) <strong>in</strong>crease with time;<br />
2ne = nR. It is given [4] from the anomaly equati<strong>on</strong> as<br />
ne(t) =<br />
∫ t<br />
0<br />
′ e2<br />
dt<br />
8π2 E(t′ )B (2)<br />
where the dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> E <strong>on</strong> t is arbitrary. In this calculati<strong>on</strong><br />
we have not taken <strong>in</strong>to account back reacti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> produced<br />
particles. We can treat the back reacti<strong>on</strong> by solv<strong>in</strong>g<br />
a Maxwell equati<strong>on</strong> ∂tE = −J with the electric current<br />
J = 2ene <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and positr<strong>on</strong>s. (The velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
particles is the light velocity <strong>in</strong> vacuum so that J is given as<br />
J =charge×velocity×number density.) Then, ne satisfies<br />
the equati<strong>on</strong>,<br />
(<br />
∂ 2 t + e3B 4π2 )<br />
ne = 0. (3)<br />
Soluti<strong>on</strong> can be found with the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong>s ne(t =<br />
0) = 0 and ∂tne(t = 0) = e 2 E(t = 0)B/8π 2 where
E(t = 0) is the electric field switched <strong>on</strong> at t = 0.<br />
In this way we can easily obta<strong>in</strong> the number density <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
produced electr<strong>on</strong>s when homogeneous electric and magnetic<br />
field are sufficiently str<strong>on</strong>g so as to neglect the mass<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s. This is a simple example[2] <str<strong>on</strong>g>of</str<strong>on</strong>g> the utility <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the anomaly equati<strong>on</strong>.<br />
Electric flux tube<br />
It is remarkable that we can explicitly obta<strong>in</strong> the number<br />
density <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s even if the electric field is tube<br />
like and back reacti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the produced particles are important.<br />
The back reacti<strong>on</strong>s are reduc<strong>in</strong>g the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electric field by the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles and <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
azimuthal magnetic field, which is <strong>in</strong>duced by the electric<br />
current <str<strong>on</strong>g>of</str<strong>on</strong>g> the particles.<br />
Next we expla<strong>in</strong> how the number density is obta<strong>in</strong>ed by<br />
us<strong>in</strong>g the chiral anomaly <strong>in</strong> such a complicated situati<strong>on</strong>[3].<br />
We assume an axial symmetric electric field E(r, t)<br />
switched <strong>on</strong> at t = 0, where r denotes cyl<strong>in</strong>drical radial<br />
coord<strong>in</strong>ate. Then, electr<strong>on</strong>s and positr<strong>on</strong>s are produced<br />
and their axial symmetric electric current generates azimuthal<br />
magnetic field Bθ(r, t). We also assume that the<br />
back ground str<strong>on</strong>g magnetic field B is uniform and static.<br />
These quantities are governed by Maxwell equati<strong>on</strong>s,<br />
∂tBθ(r, t) = ∂rE(r, t)<br />
∂tE(r, t) = ∂r(rBθ(r, t))<br />
− J(r, t), (4)<br />
r<br />
with the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong>s Bθ(r, t = 0) = 0 and E(r, t =<br />
0) = E0 exp(−r 2 /R 2 ), where R denotes the width <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electric field and J(r, t) is the electric current carried by<br />
electr<strong>on</strong>s and positr<strong>on</strong>s, given by J = 2ene.<br />
We should make a comment that the form <str<strong>on</strong>g>of</str<strong>on</strong>g> the current<br />
J = 2ene may be obta<strong>in</strong>ed by impos<strong>in</strong>g the c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the energy c<strong>on</strong>servati<strong>on</strong>,<br />
∫<br />
∂t d 3 {<br />
1<br />
x<br />
2 (E2 + B 2 }<br />
θ) + ϵ = 0 (5)<br />
where ϵ denotes the energy<br />
∫<br />
density <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and<br />
t<br />
positr<strong>on</strong>s; ϵ = nepF = ne 0 dt′ eE(t ′ ). That is, positr<strong>on</strong>s<br />
and electr<strong>on</strong>s are produced with momentum p = 0 and accerelated<br />
by the electric field so that their Fermi momentum<br />
pF is equal to ± ∫ t<br />
0 dt′ eE(t ′ ). Thus, the Fermi distributi<strong>on</strong><br />
at zero temperature is formed. Obviously, the energy density<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> each particle is given by 1<br />
2ne ∫ t<br />
0 dt′ eE(t ′ ).<br />
We rewrite the c<strong>on</strong>diti<strong>on</strong> <strong>in</strong> the follow<strong>in</strong>g,<br />
∫<br />
d 3 x(E∂tE + Bθ∂tBθ + ∂tϵ) (6)<br />
=<br />
∫<br />
d 3 {<br />
x E ( ∂tE − 1<br />
r ∂r(rBθ) ) ∫<br />
}<br />
+ ∂tϵ (7)<br />
= d 3 ∫<br />
x(−JE + ∂tϵ) (8)<br />
= d 3 xE(−J + 2ene) = 0 (9)<br />
where we have used the Maxwell equati<strong>on</strong>s and the equa-<br />
ti<strong>on</strong> ∂tnepF = γBE ∫ t<br />
0 dt′ eE = E ∫ t<br />
0 dt′ ∂tne = neE<br />
with γ ≡ e2<br />
8π 2 . S<strong>in</strong>ce E(t = 0) can be taken arbitrary, we<br />
f<strong>in</strong>d that the current J is given by 2ene.<br />
Us<strong>in</strong>g eqs. (1) and (4), we derive the equati<strong>on</strong> for the<br />
electric field E under the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> the back reacti<strong>on</strong>,<br />
∂ 2 t E(r, t) =<br />
(<br />
∂ 2 r + ∂r<br />
r<br />
)<br />
e3<br />
− B E(r, t). (10)<br />
4π2 We see that the effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the back reacti<strong>on</strong> gives rise to an<br />
effective mass term m 2 ≡ e3<br />
4π 2 B <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field.<br />
By solv<strong>in</strong>g eq. (10) and us<strong>in</strong>g the chiral anomaly (1), we<br />
explicitly obta<strong>in</strong> the number density <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s,<br />
ne(r, t)<br />
= e2 E0R 2<br />
16π 2<br />
∫<br />
∞<br />
<br />
<br />
and the azimuthal magnetic field<br />
Bθ(r, t)<br />
2 E0R<br />
= −<br />
2<br />
0<br />
∫ ∞<br />
0<br />
kdk s<strong>in</strong>(t√ k 2 + m 2 )<br />
√ k 2 + m 2<br />
J0(kr)e −k2R 2 <br />
/4<br />
,<br />
kdk s<strong>in</strong>(t√k2 + m2 )<br />
√ J1(kr)e<br />
k2 + m2 −k2R 2 /4<br />
,<br />
where J0 and J1 denote Bessel functi<strong>on</strong>s. We can see that<br />
these soluti<strong>on</strong>s are reduced to the <strong>on</strong>es <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
homogeneous electric field as R → ∞.<br />
In Figs. 1 and 2, we show the spatial and temporal behaviors<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> these soluti<strong>on</strong>s with R = 10 and t <strong>in</strong> unit <str<strong>on</strong>g>of</str<strong>on</strong>g> R.<br />
nr<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
0 5 10 15<br />
r<br />
20 25 30<br />
Figure 1: number densities ne(r) at t = 0.2 (small dots)<br />
and at t = 0.6 (large dots)<br />
Br<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
0 5 10 15<br />
r<br />
20 25 30<br />
Figure 2: azimuthal magnetic fields Bθ(r) at t = 0.2 (small<br />
dots) and at t = 0.6 (large dots)
Pair producti<strong>on</strong> <strong>in</strong> heat bath<br />
F<strong>in</strong>ally, we give another example for the utility <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
chiral anomaly, Schw<strong>in</strong>ger mechanism <strong>in</strong> heat bath. That<br />
is, the pair producti<strong>on</strong> arises <strong>in</strong> heat bath and the produced<br />
particles are thermalized immediately after their producti<strong>on</strong>.<br />
Thus, the distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particles is given by a<br />
Fermi distributi<strong>on</strong> with f<strong>in</strong>ite temperature.<br />
For simplicity, we assume that the electric field is uniform<br />
and we take <strong>in</strong>to account <strong>on</strong>ly back reacti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> reduc<strong>in</strong>g<br />
the electric field energy by pair producti<strong>on</strong>s. In<br />
this case we need electric current <strong>in</strong> the heat bath when<br />
we solve a Maxwell equati<strong>on</strong> ∂tE = −J. As expla<strong>in</strong>ed<br />
above, we impose the c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the energy c<strong>on</strong>servati<strong>on</strong>,<br />
∫<br />
3 1<br />
∂t d x( 2E2 + ϵ) = ∫ d3x(−EJ + ∂tϵ) = 0 <strong>in</strong> order to<br />
f<strong>in</strong>d J. Here the energy density is given by<br />
∫ ∞<br />
p<br />
ϵ = γ dp<br />
(11)<br />
1 + exp(p − µ(t))β<br />
0<br />
with β = 1/T where T is the temperature, where µ(t) is<br />
the chemical potential which depends <strong>on</strong> the number density<br />
ne(t) through the formula,<br />
∫ ∞<br />
1<br />
ne = γ dp<br />
. (12)<br />
1 + exp(p − µ(t))β<br />
0<br />
Us<strong>in</strong>g the formulae ∂tϵ = ∂nϵ∂tne = ∂nϵγEB <strong>in</strong> the c<strong>on</strong>diti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the energy c<strong>on</strong>servati<strong>on</strong>, we f<strong>in</strong>d that J = γB∂nϵ.<br />
Therefore, us<strong>in</strong>g the anomaly equati<strong>on</strong> and Maxwell<br />
equati<strong>on</strong>, we obta<strong>in</strong> the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> ne,<br />
∂ 2 t ne + 2γeB ne exp(neβ/n0)<br />
= 0, (13)<br />
exp(neβ/n0) − 1<br />
with n0 ≡ eB/8π 2 . It is easy to see that the formula <strong>in</strong><br />
the heat bath is reduced to the <strong>on</strong>e <strong>in</strong> vaccum when we<br />
take β → ∞; effective mass becomes m = √ e 3 B/4π 2<br />
as we have shown above; the mass means the frequency <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
ne(t) ∝ s<strong>in</strong>(mt).<br />
Although we can not analytically solve the equati<strong>on</strong>, numerical<br />
soluti<strong>on</strong>s are available. We have shown the temporal<br />
behaviors <str<strong>on</strong>g>of</str<strong>on</strong>g> the number density (Fig. 3) and the electric<br />
field (Fig. 4) <strong>in</strong> both vacuum and heat bath. We can see<br />
that the electric field decays more rapidly <strong>in</strong> the heat bath<br />
than <strong>in</strong> vacuum. Similarly, we see that the number density<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and positr<strong>on</strong>s is smaller <strong>in</strong> the heat bath<br />
than <strong>in</strong> vacuum. This is caused by the fact that accord<strong>in</strong>g<br />
to the Fermi distributi<strong>on</strong>, each electr<strong>on</strong> and positr<strong>on</strong> can<br />
have much larger energies <strong>in</strong> the heat bath with f<strong>in</strong>ite β<br />
than <strong>in</strong> vacuum with β = ∞ when it is produced. S<strong>in</strong>ce a<br />
pair producti<strong>on</strong> <strong>in</strong> the heat bath decreases the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electric field more than <strong>in</strong> vacuum, the electric field decays<br />
more rapidly <strong>in</strong> the heat bath than <strong>in</strong> vacuum. Therefore,<br />
we f<strong>in</strong>d that Schw<strong>in</strong>ger muchanism proceeds more effectively<br />
<strong>in</strong> heat bath than <strong>in</strong> vacuum.<br />
CONCLUSION<br />
To summarize, we have shown that simply us<strong>in</strong>g the chiral<br />
anomaly we can obta<strong>in</strong> physically <strong>in</strong>terest<strong>in</strong>g quantities<br />
1.4<br />
1.2<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0.5 1.0 1.5 2.0 2.5<br />
Figure 3: number densities ne(t) with arbitrary scale <strong>in</strong><br />
vacuum (dash) and <strong>in</strong> heat bath (l<strong>in</strong>e)<br />
1.0<br />
0.5<br />
0.5<br />
0.5 1.0 1.5 2.0 2.5<br />
Figure 4: electric fields E(t) with arbitrary scale <strong>in</strong> vacuum<br />
(dash) and <strong>in</strong> heat bath (l<strong>in</strong>e)<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Schw<strong>in</strong>ger mechanism without calculat<strong>in</strong>g wave functi<strong>on</strong>s.<br />
Thus, we can discuss pair producti<strong>on</strong>s under electric<br />
flux tube <strong>in</strong> vacuum or homogeneous electric field <strong>in</strong><br />
heat bath, which could not be obta<strong>in</strong>ed with the method <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the evaluati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> wave functi<strong>on</strong>s. The simplificati<strong>on</strong> comes<br />
from the fact that the problem <strong>in</strong> Schw<strong>in</strong>ger mechanism<br />
is reduced to <strong>on</strong>e dimenti<strong>on</strong>al <strong>on</strong>e when str<strong>on</strong>g magnetic<br />
field is present. Such a coll<strong>in</strong>ear str<strong>on</strong>g magnetic field and<br />
electric field (B, E ≫ (electr<strong>on</strong> mass or quark mass) 2 )<br />
are produced by high-energy heavy-i<strong>on</strong> collisi<strong>on</strong>s [5]. In<br />
the collisi<strong>on</strong>s, corresp<strong>on</strong>d<strong>in</strong>g str<strong>on</strong>g fields are color electric<br />
and magnetic fields. Thus, they decay <strong>in</strong>to quarks by<br />
Schw<strong>in</strong>ger mechanism very rapidly (< 1fm/c).<br />
REFERENCES<br />
[1] J. Schw<strong>in</strong>ger, Phys. Rev. 82 (1951) 664.<br />
[2] N. Tanji, Ann. Phys. 324 (2009) 1691; see references there<strong>in</strong>.<br />
[3] A. Iwazaki, Phys. Rev. C80 (2009) 052202.<br />
[4] S.P. Gavrilov and D.M. Gitman, Phys. Rev. D53 (1996) 7162.<br />
[5] E. Iancu, A. Le<strong>on</strong>idov and L. McLerran, hep-ph/0202270.
Abstract<br />
Str<strong>on</strong>g field physics <strong>in</strong> c<strong>on</strong>densed matter ∗<br />
There are deep similarities between n<strong>on</strong>-l<strong>in</strong>ear QFT<br />
studied <strong>in</strong> high-energy and n<strong>on</strong>-equilibrium physics <strong>in</strong> c<strong>on</strong>densed<br />
matter. Ideas such as the Schw<strong>in</strong>ger mechanism and<br />
the Volkov state are deeply related to n<strong>on</strong>-l<strong>in</strong>ear transport<br />
and photovoltaic Hall effect <strong>in</strong> c<strong>on</strong>densed matter. Here, we<br />
give a review <strong>on</strong> these relati<strong>on</strong>s.<br />
INTRODUCTION<br />
In str<strong>on</strong>g field physics, researchers are <strong>in</strong>terested <strong>in</strong> the<br />
change <str<strong>on</strong>g>of</str<strong>on</strong>g> the “quantum vacuum” due to str<strong>on</strong>g external<br />
fields. A typical example is the decay <str<strong>on</strong>g>of</str<strong>on</strong>g> the QED vacuum<br />
<strong>in</strong> str<strong>on</strong>g electric fields due to the Schw<strong>in</strong>ger mechanism<br />
[1]. When a str<strong>on</strong>g enough electric field is applied<br />
to the vacuum, pair creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and positr<strong>on</strong>s<br />
takes place and the <strong>in</strong>sulati<strong>on</strong> breaks down. The threshold<br />
for this phenomena is known as Schw<strong>in</strong>ger’s critical<br />
field and is given by Eth = m 2 e/e = 1.3 × 10 16 V/cm.<br />
S<strong>in</strong>ce the critical field is extremely str<strong>on</strong>g, direct experimental<br />
verificati<strong>on</strong> is still a challenge. On the other<br />
hand, <strong>in</strong> the c<strong>on</strong>densed matter community, there is an <strong>in</strong>creas<strong>in</strong>g<br />
<strong>in</strong>terest <strong>in</strong> n<strong>on</strong>-equilibrium phase transiti<strong>on</strong>s and<br />
n<strong>on</strong>-l<strong>in</strong>ear transport <strong>in</strong> str<strong>on</strong>gly correlated electr<strong>on</strong> systems<br />
(Fig. 1)[2, 3, 4, 5, 6, 7]. In the experiments, <strong>on</strong>e also applies<br />
str<strong>on</strong>g electric fields and the orig<strong>in</strong>al <strong>in</strong>sulat<strong>in</strong>g phase is destroyed.<br />
However, the threshold for dielctric breakdown<br />
is orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude smaller than the Schw<strong>in</strong>ger mechanism<br />
<strong>in</strong> QED s<strong>in</strong>ce the excitati<strong>on</strong> gap is far smaller. This<br />
makes c<strong>on</strong>densed matter systems to be an idealistic play-<br />
Field strength<br />
E<br />
pair creati<strong>on</strong><br />
Schw<strong>in</strong>ger limit<br />
~1 eV/a<br />
<strong>in</strong>duced-polarizati<strong>on</strong><br />
V<br />
n<strong>on</strong>l<strong>in</strong>ear transport<br />
dielectric breakdown<br />
photovoltaic Hall effect<br />
(n<strong>on</strong>l<strong>in</strong>ear)-optics<br />
~1 eV Phot<strong>on</strong> energyΩ<br />
T. Oka † , University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo, Japan<br />
“Keldysh l<strong>in</strong>e”<br />
ξ E ~ Ω<br />
photo-<strong>in</strong>duced<br />
phase transiti<strong>on</strong><br />
Figure 1: Several phenomena <strong>in</strong> c<strong>on</strong>densed matter physics<br />
<strong>in</strong> str<strong>on</strong>g electric fields plotted <strong>in</strong> the (E, Ω)-space.<br />
∗ Work supported by Grant-<strong>in</strong>-Aid for Scientific Research <strong>on</strong> Priority<br />
Area “New Fr<strong>on</strong>tier <str<strong>on</strong>g>of</str<strong>on</strong>g> Materials Science Opened by Molecular Degrees<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Freedom”.<br />
† oka@cms.phys.s.u-tokyo.ac.jp<br />
ground to test and develop theoretical ideas <strong>in</strong> n<strong>on</strong>-l<strong>in</strong>ear<br />
QFT. N<strong>on</strong>-l<strong>in</strong>ear physics has been studied rather <strong>in</strong>dependently<br />
<strong>in</strong> the two fields, high energy and c<strong>on</strong>densed matter,<br />
dur<strong>in</strong>g the past few decades, and several parallel ideas were<br />
developed. The aim <str<strong>on</strong>g>of</str<strong>on</strong>g> this article is to expla<strong>in</strong> some <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
corresp<strong>on</strong>dences (Table 1).<br />
PAIR CREATION IN STRONG ELECTRIC<br />
FIELDS<br />
Hesenberg-Euler’s effective acti<strong>on</strong> and the n<strong>on</strong>l<strong>in</strong>ear<br />
extensi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Berry’s phase approach to<br />
polarizati<strong>on</strong> [4]:<br />
We study lattice electr<strong>on</strong>s <strong>in</strong> homogenious electric fields.<br />
In the time-dependent gauge, this can be realized by add<strong>in</strong>g<br />
a time dependent phase to the hopp<strong>in</strong>g term <strong>in</strong> the lattice<br />
Hamilt<strong>on</strong>ian. For example, for a <strong>on</strong>e-dimensi<strong>on</strong>al model, a<br />
typical Hamilt<strong>on</strong>ian is given by<br />
H(Φ) = −<br />
L∑ ∑<br />
i=1<br />
σ<br />
(e iΦ c †<br />
i+1σ ciσ + e −iΦ c †<br />
iσ ci+1σ)(1)<br />
+U ∑<br />
ni↑ni↓ + ∑<br />
V<strong>in</strong>i.<br />
i<br />
We impose periodic boundary c<strong>on</strong>diti<strong>on</strong> and the phase Φ is<br />
proporti<strong>on</strong>al to the magnetic flux through the r<strong>in</strong>g (L number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> sites). The time derivative <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic flux is related<br />
to the applied electric field through F (t) = eaE(t) =<br />
dΦ(t)/dt, where e is the charge quantum and a the lattice<br />
c<strong>on</strong>stant. U represents <strong>on</strong>-site Coulomb repulsi<strong>on</strong> and Vi<br />
the local potential. The hopp<strong>in</strong>g term is set to unity. The<br />
Hubbard model (U > 0, Vi = 0) at half-fill<strong>in</strong>g is <strong>in</strong> the<br />
Mott <strong>in</strong>sulat<strong>in</strong>g phase for positive U <strong>in</strong> <strong>on</strong>e dimensi<strong>on</strong>.<br />
Here, we study what happens to the an <strong>in</strong>sulator when we<br />
apply str<strong>on</strong>g electric fields. We denote the eigenstates <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the Hamilt<strong>on</strong>ian H(Φ) by |ψn(Φ)⟩, n = 0, 1, . . . and study<br />
the time evoluti<strong>on</strong> start<strong>in</strong>g from the groundstate |ψ0(Φ)⟩.<br />
The groundstate-to-groundstate amplitude def<strong>in</strong>ed by<br />
Ξ(t) ≡ ⟨ψ0(Φ(t))|e −i<br />
∫ t<br />
0 H(Φ(s))ds |ψ0(0)⟩e i<br />
∫ t<br />
0 E0(Φ(s))ds<br />
is <str<strong>on</strong>g>of</str<strong>on</strong>g> central importance. In the l<strong>on</strong>g time limit, an asymptotic<br />
behavior (d is dimensi<strong>on</strong>) Ξ(t) ∼ e itLd L is expected<br />
to take place where L is the c<strong>on</strong>densed matter versi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the Heisenberg-Euler effective Lagrangian. The imag<strong>in</strong>ary<br />
part describes quantum tunnel<strong>in</strong>g where Γ(F )/L d ≡<br />
2Im L(F ) gives the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> the exp<strong>on</strong>ential decay <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
vacuum (groundstate). This quantity is proporti<strong>on</strong>al to the<br />
decay rate <str<strong>on</strong>g>of</str<strong>on</strong>g> the Loschmidt Echo L(t) = |Ξ(t)| 2 . The<br />
i<br />
(2)
Table 1: Related ideas <strong>in</strong> str<strong>on</strong>g field physics<br />
High Energy C<strong>on</strong>densed Matter<br />
Schw<strong>in</strong>ger mechanism <strong>in</strong> QED Landau-Zener tunnel<strong>in</strong>g <strong>in</strong> band <strong>in</strong>sulators<br />
Heisenberg-Euler effective Lagragian N<strong>on</strong>-adiabatic geometric phase, Loschmidt Echo<br />
Vacuum polarizati<strong>on</strong> Extended Berry’s phase theory <str<strong>on</strong>g>of</str<strong>on</strong>g> polarizati<strong>on</strong><br />
Pair creati<strong>on</strong> <strong>in</strong> <strong>in</strong>teract<strong>in</strong>g systems (e.g. QCD) Many-body Schw<strong>in</strong>ger-Landau-Zener mechanism<br />
<strong>in</strong> str<strong>on</strong>gly correlated system<br />
Dirac particles <strong>in</strong> circularly polarized light Photovoltaic Hall effect<br />
Furry picture Floquet picture<br />
real part ReL is written <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> a n<strong>on</strong>-adiabatic phase<br />
called the Ahar<strong>on</strong>ov-Anandan phase (which we denote γ)<br />
that the wave functi<strong>on</strong> acquires dur<strong>in</strong>g the time-evoluti<strong>on</strong>.<br />
For band <strong>in</strong>sulators (U = 0) <strong>in</strong> dc-electric fields, the effective<br />
Lagrangian becomes [4]<br />
∫<br />
dk<br />
Re L(F ) = −F<br />
BZ (2π) d<br />
γ(k)<br />
, (3)<br />
2π<br />
∫<br />
dk<br />
Im L(F ) = −F<br />
(2π) d<br />
1<br />
ln [1 − p(k)] , (4)<br />
4π<br />
BZ<br />
where the momemtum <strong>in</strong>tegral is over the Brillou<strong>in</strong> Z<strong>on</strong>e<br />
(BZ). There is an <strong>in</strong>terest<strong>in</strong>g parallel theory developeded<br />
<strong>in</strong> the c<strong>on</strong>densed matter comunity. This is known as the<br />
Berry’s phase theory <str<strong>on</strong>g>of</str<strong>on</strong>g> polarizati<strong>on</strong>[8, 9, 10, 11, 12], where<br />
the ground-state expectati<strong>on</strong> value <str<strong>on</strong>g>of</str<strong>on</strong>g> the twist operator<br />
2π −i e L ˆ X , which shifts the phase <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> wave functi<strong>on</strong>s<br />
<strong>on</strong> site j by − 2π<br />
L j, plays a crucial role. It was revealed that<br />
the real part <str<strong>on</strong>g>of</str<strong>on</strong>g> a quantity<br />
w = −i<br />
2π<br />
ln⟨0|e−i L<br />
2π ˆ X<br />
|0⟩ (5)<br />
gives the electric polarizati<strong>on</strong> Pel = −Rew [10] while its<br />
imag<strong>in</strong>ary part gives a criteri<strong>on</strong> for metal-<strong>in</strong>sulator transiti<strong>on</strong>,<br />
i.e., D = 4πImw is f<strong>in</strong>ite <strong>in</strong> <strong>in</strong>sulators and divergent<br />
<strong>in</strong> metals [11]. The present effective Lagrangian can be<br />
regarded as a n<strong>on</strong>-adiabatic (f<strong>in</strong>ite electric field) extensi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> w. To give a more accurate argument, we rewrite the<br />
effective Lagrangian <strong>in</strong> the time-<strong>in</strong>dependent gauge<br />
L(F ) ∼ −i¯h<br />
τL ln<br />
(<br />
i −<br />
⟨0|e ¯h τ(H+F ˆ i X)<br />
|0⟩e ¯h τE0<br />
)<br />
(6)<br />
for d = 1. Let us set τ = h/LF and c<strong>on</strong>sider the<br />
small F limit. For <strong>in</strong>sulators we can replace H with the<br />
groundstate energy E0 to have L(F ) ∼ wF <strong>in</strong> the l<strong>in</strong>earresp<strong>on</strong>se<br />
regime. Thus the real part <str<strong>on</strong>g>of</str<strong>on</strong>g> Heisenberg-Euler’s<br />
expressi<strong>on</strong>[13] for the n<strong>on</strong>-l<strong>in</strong>ear polarizati<strong>on</strong> P (F ) =<br />
−∂L(F )/∂F naturally reduces to the Berry’s phase formula<br />
Pel <strong>in</strong> the small field limit F → 0. Its imag<strong>in</strong>ary part<br />
gives the criteri<strong>on</strong> for photo-<strong>in</strong>duced metal-<strong>in</strong>sulator transiti<strong>on</strong>,<br />
orig<strong>in</strong>ally proposed for the zero field case.<br />
Many-body Schw<strong>in</strong>ger-Landau-Zener mechanism<br />
<strong>in</strong> st<strong>on</strong>gly correlated <strong>in</strong>sulators:<br />
Next, let us c<strong>on</strong>sider dielectric breakdown <strong>in</strong> a str<strong>on</strong>gly<br />
correlated system. In the <strong>on</strong>e-dimensi<strong>on</strong>al Mott <strong>in</strong>sulator<br />
where the groundstate is a state with <strong>on</strong>e electr<strong>on</strong> per site,<br />
the relevant charge excitati<strong>on</strong>s are doubl<strong>on</strong>s, i.e,. doubly<br />
occupied sites, and holes, i.e., sites with no electr<strong>on</strong>. Pairs<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> doubl<strong>on</strong>s and holes play a similar role as the pair <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s<br />
and positr<strong>on</strong>s <strong>in</strong> the Schw<strong>in</strong>ger mechanism. Indeed,<br />
it has been shown that dielectric breakdown <strong>in</strong> Mott <strong>in</strong>sulators<br />
takes place due to pair producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charge excitati<strong>on</strong>s<br />
through quntum tunnel<strong>in</strong>g, which is called the many-<br />
(a)<br />
Im<br />
Φ<br />
Re E<br />
imag<strong>in</strong>ary time path<br />
(b)<br />
35<br />
30<br />
25<br />
20<br />
Fth 15<br />
10<br />
5<br />
Φ(t ) *<br />
n=1<br />
n=0<br />
0 2π/L 4π/L<br />
Φ(t 0)<br />
LZ<br />
Fth ITM<br />
Fth 0<br />
0 5 10 15 20<br />
U<br />
Φ = Φ(t)<br />
Re Φ<br />
Figure 2: (a) Many-body energy levels aga<strong>in</strong>st the complex<br />
AB flux Φ for a f<strong>in</strong>ite, half-filled 1D Hubbard model<br />
(L = 10, N↑ = N↓ = 5, U = 0.5). Only charge excitati<strong>on</strong>s<br />
are plotted. Quantum tunnel<strong>in</strong>g occurs between the<br />
groundstate (labeled as n = 0) and a low-ly<strong>in</strong>g excited state<br />
(n = 1) as the flux Φ(t) = F t <strong>in</strong>creases <strong>on</strong> the real axis,<br />
while the tunnel<strong>in</strong>g is absent for the states plotted as dashed<br />
l<strong>in</strong>es. The wavy l<strong>in</strong>es start<strong>in</strong>g from the s<strong>in</strong>gular po<strong>in</strong>ts (×)<br />
at Φ(t ∗ ) represent the branch cuts for different Riemann<br />
surfaces, al<strong>on</strong>g which the soluti<strong>on</strong>s n = 0 and n = 1<br />
are c<strong>on</strong>nected. In the DDP approach, the tunnel<strong>in</strong>g factor<br />
is calculated from the dynamical phase associated with<br />
adiabatic time evoluti<strong>on</strong> (DDP path) that encircles a gapclos<strong>in</strong>g<br />
po<strong>in</strong>t at Φ(t ∗ ) <strong>on</strong> the complex Φ plane. (b) Threshold<br />
electric field obta<strong>in</strong>ed by the imag<strong>in</strong>ary time method<br />
(solid) and the naive Landau-Zener formula (dashed).
IV-characteristics c<strong>on</strong>ductance<br />
<br />
<br />
<br />
Figure 4: (upper) DC current. (lower left) IVcharacteristics.<br />
(lower right) C<strong>on</strong>ductance.<br />
Especially, <strong>in</strong> a circularly polarized light, a gap opens at<br />
the Dirac po<strong>in</strong>t [7]. This has an important physical c<strong>on</strong>sequence<br />
s<strong>in</strong>ce a gap <str<strong>on</strong>g>of</str<strong>on</strong>g> a 2+1 dimensi<strong>on</strong>al Dirac electr<strong>on</strong><br />
is related to parity anomaly and is detectable through transport<br />
measurements, i.e., the Hall effect. In 2+1 dimensi<strong>on</strong>s,<br />
the Hall c<strong>on</strong>ductivity can be written as a momentum <strong>in</strong>tegral<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Berry curvature (∼ Chern density) over the<br />
Brillou<strong>in</strong> z<strong>on</strong>e. This is known as the TKNN formula[32],<br />
and is know extended to ac-driven transport via the Floquet<br />
picture (∼ Furry picture) [7]<br />
σxy(Aac) = e 2<br />
∫<br />
dk<br />
(2π) d<br />
∑<br />
fα(k) [ ∇k × Aα(k) ]<br />
. (10)<br />
z<br />
α<br />
Here, Aα(k) ≡ −i⟨⟨Φα(k)|∇ k |Φα(k)⟩⟩ is the photo<strong>in</strong>duced<br />
artificial gauge field. In the Floquet picture, the<br />
Green’s functi<strong>on</strong> <strong>in</strong>corporates the effect <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> absorpti<strong>on</strong><br />
and emissi<strong>on</strong> (Fig. 3 (a)), and Hall c<strong>on</strong>ductivity is<br />
given by the bubble diagram <strong>in</strong> the n<strong>on</strong>-<strong>in</strong>teract<strong>in</strong>g case,<br />
which is noth<strong>in</strong>g but the parity anomaly diagram. The<br />
photo-<strong>in</strong>duced Berry curvature shown <strong>in</strong> Fig. 3 (c) acts as<br />
an artificial magnetic field and becomes f<strong>in</strong>ite when the circularly<br />
poralized light is <strong>in</strong>troduced.<br />
The current <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> circularly poralized light<br />
<strong>in</strong> a graphene ribb<strong>on</strong> attached to two electrodes is plotted<br />
<strong>in</strong> Fig. 4. The calculati<strong>on</strong> has been d<strong>on</strong>e by comb<strong>in</strong><strong>in</strong>g the<br />
Keldysh green’s functi<strong>on</strong> method with the Floquet picture.<br />
The Hall current, which is orig<strong>in</strong>ally absent, <strong>in</strong>creases as<br />
the strength <str<strong>on</strong>g>of</str<strong>on</strong>g> light becomes str<strong>on</strong>ger. The numerical result<br />
supports our understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the photovotaic Hall effect<br />
obta<strong>in</strong> by the extended TKNN formula (eqn.(10)).<br />
We would like to acknowledge Naoto Tsuji, Mart<strong>in</strong> Eckste<strong>in</strong><br />
and Philipp Werner for enlight<strong>in</strong>g discussi<strong>on</strong>s. It is<br />
a pleasure to thank Gerald Dunne for illum<strong>in</strong>at<strong>in</strong>g discussi<strong>on</strong>s<br />
dur<strong>in</strong>g PIF2010.<br />
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Abstract<br />
N<strong>on</strong>-l<strong>in</strong>ear charge transport <strong>in</strong> plasma under str<strong>on</strong>g field ∗<br />
S. Nakamura † , Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Kyoto University, Kyoto 606-8502, JAPAN<br />
We study n<strong>on</strong>l<strong>in</strong>ear charge transport <strong>in</strong> a str<strong>on</strong>gly <strong>in</strong>teract<strong>in</strong>g<br />
system <str<strong>on</strong>g>of</str<strong>on</strong>g> charges under the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> external<br />
electric field, by us<strong>in</strong>g the AdS/CFT corresp<strong>on</strong>dence. We<br />
show that the pair-creati<strong>on</strong> process assisted by the external<br />
field can cause negative differential resistivity.<br />
INTRODUCTION<br />
Recent development <str<strong>on</strong>g>of</str<strong>on</strong>g> high-<strong>in</strong>tensity lasers is open<strong>in</strong>g a<br />
new w<strong>in</strong>dow for studies <strong>on</strong> str<strong>on</strong>g-field dynamics. One <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the ma<strong>in</strong> subjects <strong>in</strong> the str<strong>on</strong>g-field physics is pair creati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles assisted by the str<strong>on</strong>g external field. In<br />
this talk, we c<strong>on</strong>sider a pair-creati<strong>on</strong> process <strong>in</strong> a str<strong>on</strong>gly<br />
<strong>in</strong>teract<strong>in</strong>g system under the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> an external field<br />
and its relati<strong>on</strong>ship to the n<strong>on</strong>l<strong>in</strong>ear charge transport. We<br />
shall show that the pair-creati<strong>on</strong> process affects the n<strong>on</strong>l<strong>in</strong>ear<br />
charge transport <strong>in</strong> such a way that the negative differential<br />
resistivity (NDR) 1 can be realized.<br />
NDR is a n<strong>on</strong>l<strong>in</strong>ear phenomen<strong>on</strong> <strong>in</strong> charge transport<br />
where the electric field (E) decreases with <strong>in</strong>creas<strong>in</strong>g current<br />
density (J), and vice versa (Fig. 1). This has been observed<br />
<strong>in</strong> various materials and devices[2]. S<strong>in</strong>ce electr<strong>on</strong>ic<br />
devices that exhibit NDR are useful <strong>in</strong> electric circuits, the<br />
understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> NDR is important from the viewpo<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<strong>in</strong>dustrial applicati<strong>on</strong>s, as well. However, theoretical study<br />
<strong>on</strong> NDR has difficulties com<strong>in</strong>g from the follow<strong>in</strong>g facts:<br />
• NDR is a n<strong>on</strong>l<strong>in</strong>ear phenomen<strong>on</strong> where we need to go<br />
bey<strong>on</strong>d the l<strong>in</strong>ear resp<strong>on</strong>se theory.<br />
• The system is far from equilibrium ow<strong>in</strong>g to the dissipati<strong>on</strong><br />
caused by the f<strong>in</strong>ite current.<br />
• N<strong>on</strong>perturbative analysis is necessary if the NDR is<br />
associated with the transiti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum such as<br />
the metal-<strong>in</strong>sulator transiti<strong>on</strong>.<br />
We employ the AdS/CFT corresp<strong>on</strong>dence [3, 4, 5] to overcome<br />
these difficulties.<br />
The typical J-E characteristics we obta<strong>in</strong> falls <strong>in</strong>to the<br />
category <str<strong>on</strong>g>of</str<strong>on</strong>g> the S-shaped NDR 2 which is sketched at Fig. 1.<br />
NDR is realized between B and C <strong>in</strong> Fig. 1. J(E) is a<br />
∗ Talk based <strong>on</strong> the orig<strong>in</strong>al work <strong>in</strong> Ref. [1]. The present work was<br />
supported by MEXT KAKENHI (21105006), Grant-<strong>in</strong>-Aid for Scientific<br />
Research <strong>on</strong> Innovative Areas “Elucidati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> New Hadr<strong>on</strong>s with a Variety<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Flavors,” and by the Grant-<strong>in</strong>-Aid for the Global COE Program<br />
“The Next Generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Spun from Universality and Emergence”<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> MEXT <str<strong>on</strong>g>of</str<strong>on</strong>g> Japan.<br />
† nakamura@ruby.scphys.kyoto-u.ac.jp<br />
1 This may also be referred to as negative differential c<strong>on</strong>ductivity<br />
(NDC) <strong>in</strong> some literature.<br />
2 This corresp<strong>on</strong>ds to the SNDC <strong>in</strong> Ref. [2]<br />
J<br />
A<br />
C<br />
Figure 1: Schematic J-E characteristics with NDR.<br />
multivalued functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> E. Experimentally, the multivalued<br />
behavior is obta<strong>in</strong>ed by measur<strong>in</strong>g E as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the c<strong>on</strong>trolled current density J; note that the J-E curve is<br />
“N-shaped” if the axes are swapped. The functi<strong>on</strong> E <str<strong>on</strong>g>of</str<strong>on</strong>g> J is<br />
still a s<strong>in</strong>gle-valued functi<strong>on</strong>, and the NDR is well-def<strong>in</strong>ed<br />
if J is c<strong>on</strong>trolled. If we c<strong>on</strong>trol E <strong>in</strong>stead, the NDR branch<br />
is unstable and the hysteresis is observed. In this talk, we<br />
regard J as a c<strong>on</strong>trol parameter and E is determ<strong>in</strong>ed as a<br />
result <str<strong>on</strong>g>of</str<strong>on</strong>g> dynamics.<br />
B<br />
ADS/CFT CORRESPONDENCE<br />
The AdS/CFT corresp<strong>on</strong>dence is a c<strong>on</strong>jectured equivalence<br />
between str<strong>on</strong>gly-<strong>in</strong>teract<strong>in</strong>g n<strong>on</strong>-abelian quantum<br />
gauge theories and higher-dimensi<strong>on</strong>al classical gravitati<strong>on</strong>al<br />
theories. This enables us to analyze the n<strong>on</strong>perturbative<br />
nature <str<strong>on</strong>g>of</str<strong>on</strong>g> the gauge theories <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> general relativity.<br />
The corresp<strong>on</strong>dence is at the level <str<strong>on</strong>g>of</str<strong>on</strong>g> the microscopic<br />
theory, and it is potentially possible to describe n<strong>on</strong>equilibrium<br />
process. What is surpris<strong>in</strong>g is that the noti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> equilibrium<br />
appears <strong>in</strong> the gravity side <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> black holes.<br />
The coarse gra<strong>in</strong><strong>in</strong>g is automatically performed when we<br />
solve the E<strong>in</strong>ste<strong>in</strong>’s equati<strong>on</strong> to obta<strong>in</strong> the black hole soluti<strong>on</strong>.<br />
These features <str<strong>on</strong>g>of</str<strong>on</strong>g> the AdS/CFT corresp<strong>on</strong>dence<br />
tempts us to apply it to n<strong>on</strong>l<strong>in</strong>ear n<strong>on</strong>equilibrium physics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>gly-<strong>in</strong>teract<strong>in</strong>g gauge theories. Indeed, the n<strong>on</strong>l<strong>in</strong>ear<br />
c<strong>on</strong>ductivity <str<strong>on</strong>g>of</str<strong>on</strong>g> a global charge <strong>in</strong> some system has<br />
already been computed by us<strong>in</strong>g the AdS/CFT corresp<strong>on</strong>dence<br />
[6]. 3<br />
The AdS/CFT corresp<strong>on</strong>dence has also disadvantages.<br />
One <str<strong>on</strong>g>of</str<strong>on</strong>g> them is that we cannot analyze U(1) gauge theory<br />
like QED. The c<strong>on</strong>venti<strong>on</strong>al analysis <strong>in</strong> the AdS/CFT<br />
corresp<strong>on</strong>dence is restricted to SU(Nc) gauge theories at<br />
the large-Nc limit. However, we can still <strong>in</strong>troduce n<strong>on</strong>dynamical<br />
external fields <str<strong>on</strong>g>of</str<strong>on</strong>g> the U(1) gauge theory. The<br />
SU(Nc) gauge theory can also be idealized <strong>in</strong> such a way<br />
that the <strong>in</strong>teracti<strong>on</strong> am<strong>on</strong>g the charges is <str<strong>on</strong>g>of</str<strong>on</strong>g> Coulomb type.<br />
3 See also, for example, Refs. [7, 8].<br />
D<br />
E
What we shall do <strong>in</strong> the present work is to c<strong>on</strong>sider an idealized<br />
gauge theory that has the Coulomb type <strong>in</strong>teracti<strong>on</strong><br />
am<strong>on</strong>g the charges <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> the U(1) external<br />
electric field.<br />
MICROSCOPIC THEORY<br />
Our idealized gauge theory is the (3+1)-dimensi<strong>on</strong>al<br />
SU(Nc) N =4 super-symmetric Yang-Mills (SYM) theory<br />
with Nf flavors <str<strong>on</strong>g>of</str<strong>on</strong>g> fundamental N =2 hypermultiplets. This<br />
is a supersymmetric cous<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Quantum Chromodynamics<br />
(QCD), but the <strong>in</strong>ter-quark potential is <str<strong>on</strong>g>of</str<strong>on</strong>g> Coulomb type at<br />
zero temperature with <strong>in</strong>f<strong>in</strong>ite current quark mass due to<br />
the c<strong>on</strong>formal nature <str<strong>on</strong>g>of</str<strong>on</strong>g> N = 4 SYM. The supersymmetry<br />
is broken at f<strong>in</strong>ite temperatures. The number <str<strong>on</strong>g>of</str<strong>on</strong>g> colors Nc<br />
is taken to be <strong>in</strong>f<strong>in</strong>ity with the ’t Ho<str<strong>on</strong>g>of</str<strong>on</strong>g>t coupl<strong>in</strong>g gYMN 2 c<br />
kept fixed, where gYM is the Yang-Mills coupl<strong>in</strong>g c<strong>on</strong>stant.<br />
We def<strong>in</strong>e λ ≡ 2gYMN 2 c <strong>in</strong> this article, and take the str<strong>on</strong>gcoupl<strong>in</strong>g<br />
limit λ ≫ 1. In our setup, the quarks and the antiquarks<br />
are str<strong>on</strong>gly correlated ow<strong>in</strong>g to the large λ. The<br />
quarks carry the global U(1) bary<strong>on</strong> (U(1)B) charge (or the<br />
quark charge), and we analyze the c<strong>on</strong>ductivity associated<br />
with this charge. The above setup is employed to make the<br />
AdS/CFT corresp<strong>on</strong>dence applicable.<br />
Ignor<strong>in</strong>g the super-partners, the theory c<strong>on</strong>ta<strong>in</strong>s SU(Nc)<br />
adjo<strong>in</strong>t glu<strong>on</strong>s and the Nf species <str<strong>on</strong>g>of</str<strong>on</strong>g> SU(Nc) fundamental<br />
quarks (and the anti-fundamental antiquarks). The glu<strong>on</strong>s<br />
and the quarks play the roles <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s and the electr<strong>on</strong>s<br />
<strong>in</strong> QED, respectively. The antiquark may be regarded as a<br />
counterpart <str<strong>on</strong>g>of</str<strong>on</strong>g> positr<strong>on</strong>s or holes, depend<strong>in</strong>g <strong>on</strong> the system<br />
we make an analogy.<br />
The system is <strong>in</strong> the dec<strong>on</strong>f<strong>in</strong>ement phase <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s<br />
(which means that the degree <str<strong>on</strong>g>of</str<strong>on</strong>g> freedom <str<strong>on</strong>g>of</str<strong>on</strong>g> the glu<strong>on</strong>ic<br />
sector is O(N 2 c )) at n<strong>on</strong>zero temperatures, but the quark<br />
and antiquark may still form the bound states depend<strong>in</strong>g <strong>on</strong><br />
the parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> the theory. If they form the bound states,<br />
the system is an <strong>in</strong>sulator s<strong>in</strong>ce the bound states are neutral.<br />
The system becomes a c<strong>on</strong>ductor if the bound states<br />
are unstable and the charge carriers are liberated. This system<br />
shares several features similar to those <str<strong>on</strong>g>of</str<strong>on</strong>g> the excit<strong>on</strong>ic<br />
<strong>in</strong>sulators [9, 10] or sQGP [11]. We f<strong>in</strong>d that the system<br />
shows NDR due to the pair-creati<strong>on</strong> process <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge<br />
carriers. Our result suggests a possibility to observe NDR<br />
<strong>in</strong> some excit<strong>on</strong>ic <strong>in</strong>sulators or <strong>in</strong> some quark-hadr<strong>on</strong> systems,<br />
as we shall discuss later.<br />
Let us c<strong>on</strong>sider how to realize a n<strong>on</strong>equilibrium steady<br />
state (NESS) with a c<strong>on</strong>stant current <strong>in</strong> the c<strong>on</strong>ductor<br />
phase. S<strong>in</strong>ce our quarks/antiquarks <strong>in</strong>teract str<strong>on</strong>gly with<br />
the glu<strong>on</strong>s, the k<strong>in</strong>etic energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the quarks/antiquarks will<br />
be dissipated. Because <str<strong>on</strong>g>of</str<strong>on</strong>g> the dissipati<strong>on</strong>, the system will<br />
be heated up if we ma<strong>in</strong>ta<strong>in</strong> a c<strong>on</strong>stant current. However,<br />
we can realize a steady state with a c<strong>on</strong>stant current by tak<strong>in</strong>g<br />
the probe limit Nc ≫ Nf . The degree <str<strong>on</strong>g>of</str<strong>on</strong>g> freedom <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the glu<strong>on</strong>ic sector is O(N 2 c ), whereas that <str<strong>on</strong>g>of</str<strong>on</strong>g> the flavor sector<br />
(the quark/antiquark sector) is O(NcNf ); the glu<strong>on</strong>ic<br />
sector has <strong>in</strong>f<strong>in</strong>itely large degrees <str<strong>on</strong>g>of</str<strong>on</strong>g> freedom <strong>in</strong> comparis<strong>on</strong><br />
with the flavor sector at this limit. As a result, the<br />
dissipated energy from the flavor sector is absorbed <strong>in</strong>to an<br />
<strong>in</strong>f<strong>in</strong>itely large reservoir <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong>s and the system is wellapproximated<br />
as a NESS for the time period shorter than<br />
O(Nc) [12]. The glu<strong>on</strong>ic sector acts as a “heat bath” for<br />
the flavor sector <strong>in</strong> this sense. Note that the <strong>in</strong>teracti<strong>on</strong> between<br />
the charge carriers and the “heat bath” is taken <strong>in</strong>to<br />
account <strong>in</strong> our setup.<br />
GRAVITY DUAL<br />
The gravity dual <str<strong>on</strong>g>of</str<strong>on</strong>g> the forego<strong>in</strong>g microscopic theory is<br />
the so-called D3-D7 system [13], where the Nf D7-branes<br />
are embedded <strong>in</strong> the background geometry given by a direct<br />
product <str<strong>on</strong>g>of</str<strong>on</strong>g> a 5-dimensi<strong>on</strong>al AdS-Schwarzschild black hole<br />
(AdS-BH) and S 5 . The flavor sector is governed by the<br />
dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the D7-branes, whereas the glu<strong>on</strong>ic sector is<br />
described by the AdS-BH. We take the str<strong>in</strong>g tensi<strong>on</strong> to be<br />
1 for simplicity, namely, 2πl 2 s = 1, where ls is the str<strong>in</strong>g<br />
length. The metric <str<strong>on</strong>g>of</str<strong>on</strong>g> the AdS-BH part is given by<br />
ds 2 = − 1<br />
z 2<br />
(1 − z 4 /z 4 H )2<br />
1 + z 4 /z 4 H<br />
dt 2 + 1 + z4 /z 4 H<br />
z 2<br />
d⃗x 2 + dz2<br />
, (1)<br />
z2 where z is the radial coord<strong>in</strong>ate <str<strong>on</strong>g>of</str<strong>on</strong>g> the black hole. The horiz<strong>on</strong><br />
is located at z = zH and the boundary is at z = 0.<br />
The Hawk<strong>in</strong>g temperature that corresp<strong>on</strong>ds to the temperature<br />
√ <str<strong>on</strong>g>of</str<strong>on</strong>g> the glu<strong>on</strong>ic sector (heat bath) is given by T =<br />
2/(πzH). ⃗x denotes the 3-dimensi<strong>on</strong>al spatial directi<strong>on</strong>s.<br />
The S5 metric is dΩ2 5 = dθ2 + s<strong>in</strong> 2 θdψ2 + cos2 θdΩ2 3,<br />
where 0 ≤ θ ≤ π/2, and dΩd is the volume element <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the unit d-dimensi<strong>on</strong>al sphere. The radius <str<strong>on</strong>g>of</str<strong>on</strong>g> the S5 has<br />
been taken to be 1, which is equivalent to the choice <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
λ = (2π) 2 .<br />
The D7-branes are wrapped <strong>on</strong> an S3 part <str<strong>on</strong>g>of</str<strong>on</strong>g> the S5 . We<br />
choose our space-time coord<strong>in</strong>ates <strong>in</strong> such a way that the<br />
S3 is located at ψ = 0. Let us choose the worldvolume<br />
coord<strong>in</strong>ates <str<strong>on</strong>g>of</str<strong>on</strong>g> the D7-brane to be the same as the spacetime<br />
coord<strong>in</strong>ates. We also assume that the external U(1)B<br />
electric field E is applied al<strong>on</strong>g the x directi<strong>on</strong>. We have<br />
a U(1) gauge field Aµ <strong>on</strong> the D7-branes, which couples to<br />
the U(1)B current. The relati<strong>on</strong>ship between the external<br />
field E and the result<strong>in</strong>g current J al<strong>on</strong>g the x directi<strong>on</strong><br />
is given by the GKP-Witten prescripti<strong>on</strong> [4, 5] as (see also<br />
J<br />
2 N z2 +O(z 4 ), where<br />
we have employed the gauge ∂xAt = 0. N is given by<br />
N = Nf TD7(2π2 ), where TD7 is the D7-brane tensi<strong>on</strong>. In<br />
our choice <str<strong>on</strong>g>of</str<strong>on</strong>g> λ = (2π) 2 and 2πl2 s = 1, N = NcNf /(2π) 2 .<br />
We c<strong>on</strong>sider the vanish<strong>in</strong>g quark-charge density <strong>in</strong> most<br />
cases and we set the other comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> the vector potential<br />
to be zero unless specified.<br />
The D7-brane acti<strong>on</strong> with the present setup is explicitly<br />
Ref. [6]) Ax(z, t) = −Et+c<strong>on</strong>st.+ 1<br />
written as<br />
∫<br />
SD7 = −N<br />
dtd 3 xdz cos 3 [<br />
θ |gtt|gxxgzz<br />
(<br />
− gzz( ˙ Ax) 2 − |gtt|(A ′ x) 2) ] 1/2<br />
, (2)<br />
where the prime (the dot) denotes the differentiati<strong>on</strong> with<br />
respect to z (t). We have already <strong>in</strong>tegrated the S 3 part
under the assumpti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the symmetry al<strong>on</strong>g it. gtt, gxx<br />
and gzz are the <strong>in</strong>duced world-volume metric, and they are<br />
equal to the background metric (1) except for gzz = 1/z 2 +<br />
θ ′ (z) 2 .<br />
NONLINEAR CONDUCTIVITY<br />
It was found [6] that the <strong>on</strong>-shell D7-brane acti<strong>on</strong> becomes<br />
complex unless we choose a specific comb<strong>in</strong>ati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
J and E; the relati<strong>on</strong>ship between J and E is determ<strong>in</strong>ed<br />
by the reality c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>on</strong>-shell acti<strong>on</strong>, hence, J is<br />
obta<strong>in</strong>ed as a n<strong>on</strong>l<strong>in</strong>ear functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> E. The <strong>on</strong>-shell acti<strong>on</strong><br />
is given by [6] ¯ SD7 = −N ∫ dzdtd 3 x √ ¯gzz|gtt| −1√ F1F2<br />
with F1 = |gtt|gxx − E 2 and F2 = |gtt|g 2 xx cos 6 ¯ θ −<br />
gxxJ 2 /N 2 , where ¯gzz is the <strong>in</strong>duced metric given by ¯ θ,<br />
which is the <strong>on</strong>-shell c<strong>on</strong>figurati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> θ(z). S<strong>in</strong>ce both F1<br />
and F2 cross zero somewhere between the boundary and<br />
the horiz<strong>on</strong>, the <strong>on</strong>ly way to make ¯ SD7 real is to choose<br />
J and E so that F1 and F2 cross zero at the same po<strong>in</strong>t<br />
z = z∗. The hypersurface given by z = z∗ is <str<strong>on</strong>g>of</str<strong>on</strong>g>ten<br />
called the “s<strong>in</strong>gular shell”. Then, the reality c<strong>on</strong>diti<strong>on</strong><br />
F1(z∗) = F2(z∗) = 0 gives us the relati<strong>on</strong>ship between<br />
J and E <strong>in</strong> the form <str<strong>on</strong>g>of</str<strong>on</strong>g> J = σ0E [6], where<br />
σ0 = N T (e 2 + 1) 1/4 cos 3 ¯ θ(z∗). (3)<br />
Our task is to solve the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> (EOM) for θ<br />
to obta<strong>in</strong> the explicit representati<strong>on</strong>. ¯ θ(z) can be expanded<br />
as ¯ θ(z) = mqz + O(z 3 ), where mq is the current quark<br />
mass [13], which is a parameter <str<strong>on</strong>g>of</str<strong>on</strong>g> the microscopic theory.<br />
mq is related to the gap <strong>in</strong> c<strong>on</strong>densed matters.<br />
Let us choose the temperature to be T = √ 2/π so that<br />
zH = 1, e = E/2 and z∗ = √ E 2 /4 + 1 − E/2. 4 We<br />
further fix NcNf = 40. NcNf governs the pair-creati<strong>on</strong><br />
rate <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge carriers as we shall expla<strong>in</strong> later. We<br />
need to solve the EOM for θ numerically. The boundary<br />
c<strong>on</strong>diti<strong>on</strong> we employ is θ(z)/z|z=0 = mq and we request<br />
the absence <str<strong>on</strong>g>of</str<strong>on</strong>g> s<strong>in</strong>gularity <strong>in</strong> the D7-brane c<strong>on</strong>figurati<strong>on</strong>.<br />
For earlier studies <strong>on</strong> the n<strong>on</strong>l<strong>in</strong>ear c<strong>on</strong>ductivity by us<strong>in</strong>g<br />
this method, see for example, Refs. [7, 8].<br />
RESULTS<br />
Examples <str<strong>on</strong>g>of</str<strong>on</strong>g> J-mq curves at several values <str<strong>on</strong>g>of</str<strong>on</strong>g> E are<br />
shown <strong>in</strong> Fig. 2. Of course, mq has a unique value at a<br />
given model and we need to choose some particular value<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> mq. We f<strong>in</strong>d that there are two different possible values<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> J at given mq and given E <strong>in</strong> some parameter regi<strong>on</strong>,<br />
which <strong>in</strong>dicate the multi-valued nature <str<strong>on</strong>g>of</str<strong>on</strong>g> J(E). Furthermore,<br />
if we <strong>in</strong>crease E al<strong>on</strong>g the given mq, the smaller<br />
J decreases while the larger J <strong>in</strong>creases; the smaller-J<br />
branch shows NDR, whereas the larger-J branch has a positive<br />
differential resistivity. Note that the smaller-J branch<br />
4 In this article, we have employed the natural units c = ¯h = kB = 1.<br />
If our scale unit is meV (mili eV), T ∼ 5 K. If we identify the unit<br />
quark charge with the unit charge <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s, the effective f<strong>in</strong>e-structure<br />
c<strong>on</strong>stant read from the Coulomb <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> the <strong>in</strong>ter-quark potential is<br />
∼ 1.<br />
mq<br />
1.34<br />
1.33<br />
1.32<br />
1.31<br />
1.30<br />
1.29<br />
E0.12<br />
E0.20<br />
E0.15<br />
1.28<br />
0.000 0.005 0.010 0.015 0.020 0.025<br />
Figure 2: J-mq curves at E = 0.12, 0.15, and 0.20. mq is<br />
maximum at a n<strong>on</strong>zero but small value <str<strong>on</strong>g>of</str<strong>on</strong>g> J.<br />
E<br />
0.25<br />
0.20<br />
0.15<br />
0.10<br />
0.05<br />
0.00<br />
0.000 0.005 0.010 0.015 0.020 0.025<br />
Figure 3: J-E curve at mq = 1.315. Ec = 0.11 <strong>in</strong> this<br />
case. NDR appears <strong>in</strong> J ≤ 0.0031 and is absent for E ≥<br />
0.19.<br />
is a very narrow w<strong>in</strong>dow <strong>in</strong> the full part <str<strong>on</strong>g>of</str<strong>on</strong>g> the J-mq curve.<br />
For example, the J-mq curve at E = 0.2 extends until<br />
J = 0.288, and the width <str<strong>on</strong>g>of</str<strong>on</strong>g> the smaller-J branch al<strong>on</strong>g the<br />
J axes is less than 2% <str<strong>on</strong>g>of</str<strong>on</strong>g> the full part. The detailed analysis<br />
shows that the highest value <str<strong>on</strong>g>of</str<strong>on</strong>g> mq approaches around<br />
1.310 at the E → +0 limit, suggest<strong>in</strong>g that Ec = 0 if<br />
mq < 1.310. This is c<strong>on</strong>sistent with the fact that the system<br />
is a c<strong>on</strong>ductor at sufficiently small mq <strong>in</strong> comparis<strong>on</strong><br />
with T (or sufficiently high T <strong>in</strong> comparis<strong>on</strong> with mq).<br />
An example <str<strong>on</strong>g>of</str<strong>on</strong>g> J-E relati<strong>on</strong> at mq = 1.315 is given <strong>in</strong><br />
Fig. 3. 5 The system is an <strong>in</strong>sulator for E < Ec = 0.11. If<br />
E ≥ Ec, the <strong>in</strong>sulati<strong>on</strong> is broken and we observe a current.<br />
NDR is realized <strong>in</strong> the smaller-J regi<strong>on</strong>. We always have<br />
the J = 0 branch <strong>on</strong> top <str<strong>on</strong>g>of</str<strong>on</strong>g> the vertical axis. Therefore,<br />
our <strong>in</strong>terpretati<strong>on</strong> is that Fig. 3 shows the B-C-D regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Fig. 1 with the axes swapped; our NDR falls with<strong>in</strong> the Sshaped<br />
NDR. There may be a small tunnel<strong>in</strong>g current that<br />
almost overlaps with the vertical axis, but we could not detect<br />
it with<strong>in</strong> our numerical precisi<strong>on</strong>. We leave the detailed<br />
analysis <strong>on</strong> the tunnel<strong>in</strong>g current <strong>in</strong> a future work.<br />
It is important to clarify what is the physically essential<br />
process <strong>in</strong> our NDR. Let us c<strong>on</strong>sider the doped cases.<br />
We can also “dope” the system by <strong>in</strong>troduc<strong>in</strong>g f<strong>in</strong>ite quarkcharge<br />
density [14, 15]. In this case, the system is always a<br />
5 If we choose our scale unit to be meV, the critical electric field Ec<br />
<strong>in</strong> Fig. 3 is Ec ∼ 5 × 10 −1 V/m, and the current density realized at<br />
E = Ec is J ∼ 1 × 10 −4 mA/mm 2 .<br />
J<br />
J
c<strong>on</strong>ductor. The current is given by [6]<br />
√<br />
J = σ2 0 + d2 /(e2 + 1) E, (4)<br />
where d is related to the quark-charge density ρ through<br />
d = ρ/( π<br />
√<br />
2<br />
2 λT ). Ow<strong>in</strong>g to the doped charges, any small<br />
E causes a current and we observe Ohm’s law <strong>in</strong> the small-<br />
J regi<strong>on</strong>. If we raise J, we may aga<strong>in</strong> observe NDR ow<strong>in</strong>g<br />
to the n<strong>on</strong>trivial behavior <str<strong>on</strong>g>of</str<strong>on</strong>g> σ0. It is <strong>in</strong>deed the case if d<br />
is small enough not to smear the c<strong>on</strong>tributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> σ0. In this<br />
case, the curve <strong>in</strong> Fig. 3 will be “N-shaped”, (S-shaped <strong>in</strong><br />
the sense <str<strong>on</strong>g>of</str<strong>on</strong>g> Fig. 1) start<strong>in</strong>g at the orig<strong>in</strong>. The po<strong>in</strong>t is that<br />
the d-dependent term <strong>in</strong> the square root <strong>in</strong> (4) does not have<br />
any structure to produce NDR. Therefore, the σ0-part <strong>in</strong> (4)<br />
is crucial for NDR. It is understood that the current due to<br />
the σ0-part is caused by the pair creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge carriers.<br />
The reas<strong>on</strong>s are as follows: it c<strong>on</strong>tributes the current<br />
with the total system be<strong>in</strong>g kept neutral, and it vanishes if<br />
the mass <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge carriers mq is <strong>in</strong>f<strong>in</strong>ite. [6] 6 As a c<strong>on</strong>clusi<strong>on</strong>,<br />
the pair-creati<strong>on</strong> process is essential for our NDR.<br />
DISCUSSION<br />
We can suggest a phenomenological model <str<strong>on</strong>g>of</str<strong>on</strong>g> NDR. The<br />
phenomenological orig<strong>in</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> NDR are classified <strong>in</strong>to three<br />
types (except for the tunnel effect for some semic<strong>on</strong>ductor<br />
juncti<strong>on</strong>s) <strong>in</strong> Ref. [2]: 1) n<strong>on</strong>l<strong>in</strong>earity <str<strong>on</strong>g>of</str<strong>on</strong>g> mobility, 2)<br />
n<strong>on</strong>l<strong>in</strong>earity <str<strong>on</strong>g>of</str<strong>on</strong>g> carrier density, and 3) n<strong>on</strong>l<strong>in</strong>earity <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electr<strong>on</strong> temperature. 7 We have found that our NDR orig<strong>in</strong>ates<br />
<strong>in</strong> the pair creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the charge carriers but not <strong>in</strong><br />
the normal current <str<strong>on</strong>g>of</str<strong>on</strong>g> the doped charges. This means that<br />
the above feature 2) is crucial <strong>in</strong> our NDR. Although further<br />
study is necessary to reach the f<strong>in</strong>al c<strong>on</strong>clusi<strong>on</strong>, it is<br />
natural to assume that both the normal current and the paircreated<br />
current c<strong>on</strong>tribute 1) and 3) regardless <str<strong>on</strong>g>of</str<strong>on</strong>g> the orig<strong>in</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the charge carriers. If this assumpti<strong>on</strong> is right, 1) and 3)<br />
do not seem to be important <strong>in</strong> our NDR. The behavior <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
our NDR is <strong>in</strong> the category <str<strong>on</strong>g>of</str<strong>on</strong>g> the “SNDC” <strong>in</strong> Ref. [2] and<br />
it may be attributed to the impact i<strong>on</strong>izati<strong>on</strong> expla<strong>in</strong>ed <strong>in</strong><br />
Ref. [2]. The proposal <str<strong>on</strong>g>of</str<strong>on</strong>g> the many-body avalanche model<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> NDR[17, 18] also matches our picture. It is important<br />
to study further the c<strong>on</strong>necti<strong>on</strong> between our results and the<br />
phenomenological models <str<strong>on</strong>g>of</str<strong>on</strong>g> NDR.<br />
We can also see our results from the viewpo<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
quark-hadr<strong>on</strong> physics and that <str<strong>on</strong>g>of</str<strong>on</strong>g> the excit<strong>on</strong>ic <strong>in</strong>sulators.<br />
Let us c<strong>on</strong>sider the sQGP state [11] where the quarkantiquark<br />
bound state exists <strong>in</strong> the dec<strong>on</strong>f<strong>in</strong>ement phase <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
glu<strong>on</strong>s. Our results suggest that the quarks are liberated<br />
at the critical value <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field and their current<br />
may show NDR. We may also have a chance to observe a<br />
qualitatively similar NDR <strong>in</strong> excit<strong>on</strong>ic <strong>in</strong>sulators after the<br />
<strong>in</strong>sulati<strong>on</strong> break<strong>in</strong>g. It is important to study how general<br />
this NDR is, <strong>in</strong> quark/mes<strong>on</strong> systems and <strong>in</strong> the systems <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
6 We also po<strong>in</strong>t out that σ0 is proporti<strong>on</strong>al to NcNf . This suggests<br />
that it may be a <strong>on</strong>e-loop c<strong>on</strong>tributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the quarks, as <strong>in</strong> the perturbative<br />
computati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair-creati<strong>on</strong> rate. Note that the quark loops have been<br />
taken <strong>in</strong>to account to the 1-loop order <strong>in</strong> the probe approximati<strong>on</strong>.<br />
7 See also Ref. [16].<br />
charge-anticharge bound states, <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g<br />
external fields.<br />
We expect that the present system is a good theoretical<br />
playground for studies <strong>on</strong> n<strong>on</strong>l<strong>in</strong>ear charge transport<br />
and n<strong>on</strong>equilibrium steady states. The AdS/CFT corresp<strong>on</strong>dence<br />
can be a new tool for study<strong>in</strong>g n<strong>on</strong>equilibrium<br />
physics as we have dem<strong>on</strong>strated here.<br />
The author thanks the organizers <str<strong>on</strong>g>of</str<strong>on</strong>g> PIF2010 c<strong>on</strong>ference.<br />
Discussi<strong>on</strong>s with the participants <str<strong>on</strong>g>of</str<strong>on</strong>g> various research areas<br />
such as plasma physics, str<strong>on</strong>g-field dynamics and laser<br />
physics are quite fruitful <strong>in</strong> plann<strong>in</strong>g further studies al<strong>on</strong>g<br />
the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the present work.<br />
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043.<br />
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R. M. Thoms<strong>on</strong>, J. High Energy Phys. 0702 (2007), 016.<br />
[16] H. C. Law and K. C. Kao, IEEE Trans. Electr<strong>on</strong> Devices 17<br />
(1970), 562.<br />
[17] Y. Tokura, H. Okamoto, T. Takao, T. Tadaoki and G. Saito,<br />
Phys. Rev. B 38 (1988), 2215.<br />
[18] T. Oka, H. Kishida and H. Aoki, talk given at JPS 2010 Annual<br />
Meet<strong>in</strong>g, March 20th (2010).
Brilliant hardγ-producti<strong>on</strong> ande + e−-creati<strong>on</strong> <strong>in</strong> vacuum with ultra-high<br />
laser fields: Test<strong>in</strong>g theoretical predicti<strong>on</strong>s at ELI-NP<br />
Dietrich Habs, Peter Thirolf, N<strong>in</strong>a Elk<strong>in</strong>a and Hartmut Ruhl<br />
Fakultät für Physik, Ludwig-Maximilians-Universität München, D-85748 Garch<strong>in</strong>g, Germany<br />
Abstract<br />
We want to measure the hard-γ producti<strong>on</strong>, when a<br />
brilliant bunch <str<strong>on</strong>g>of</str<strong>on</strong>g> 600 MeV electr<strong>on</strong>s is <strong>in</strong>jected <strong>in</strong>to<br />
the <strong>in</strong>tense focus <str<strong>on</strong>g>of</str<strong>on</strong>g> two counter-propagat<strong>in</strong>g lasers with<br />
10 24 W/cm 2 <strong>in</strong> vacuum. In a sec<strong>on</strong>d step these hard-γ phot<strong>on</strong>s<br />
can produce e + e − pairs <strong>in</strong> the same laser field <strong>in</strong><br />
vacuum. We describe an experiment planned for ELI-NP,<br />
where we want to test for the first time n<strong>on</strong>-perturbative<br />
high-field QED effects.<br />
INTRODUCTION<br />
One <str<strong>on</strong>g>of</str<strong>on</strong>g> the ma<strong>in</strong> goals <str<strong>on</strong>g>of</str<strong>on</strong>g> the project Extreme Light Infrastructure<br />
(ELI) [1] is to study the “boil<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum”<br />
[2], i.e. the electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong> <strong>in</strong> the vacuum<br />
and phot<strong>on</strong> producti<strong>on</strong> with ultra-high laser fields. At<br />
the Extreme Light Infrastructure - Nuclear <strong>Physics</strong> (ELI-<br />
NP) facility [3] we presently plan for laser <strong>in</strong>tensities <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
1024W/cm2 , equivalent to an electric field strength E =<br />
4.7 · 1015V/m, or normalized vector potentials a = 103 ,<br />
which are still much too small to produce directly pairs<br />
from the vacuum. Here the laser field is characterized by<br />
the Lorentz <strong>in</strong>variant dimensi<strong>on</strong>less normalized vector potential<br />
a:<br />
a = eEλL<br />
me2πc2 = eAµA µ<br />
mec2 , (1)<br />
whereλL is the laser wavelength andAµ is the laser vector<br />
potential. In comparis<strong>on</strong> to the Schw<strong>in</strong>ger <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g>Is =<br />
5·10 29 W/cm 2 , or the Schw<strong>in</strong>ger fieldES:<br />
Es = m2 ec 3<br />
e¯h = 1.3·1018 V/m (2)<br />
the extremely str<strong>on</strong>g suppressi<strong>on</strong> by the exp<strong>on</strong>ential factor<br />
exp[−π ·(Es/E)] ≈ 10 −1000<br />
prevents the observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pair creati<strong>on</strong>, when focus<strong>in</strong>g<br />
the high-power lasers <str<strong>on</strong>g>of</str<strong>on</strong>g> ELI-NP <strong>in</strong>to vacuum. Thus we<br />
need some k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> seed<strong>in</strong>g. The special feature <str<strong>on</strong>g>of</str<strong>on</strong>g> ELI-<br />
NP is, that at the same time a Compt<strong>on</strong>-backscatter<strong>in</strong>g γbeam<br />
facility [5] will be <strong>in</strong>stalled, where a very brilliant,<br />
<strong>in</strong>tense, classical electr<strong>on</strong> beam is produced <strong>in</strong> a warm electr<strong>on</strong><br />
l<strong>in</strong>ac with up to 600 MeV and is used to produce γquanta<br />
with maximum energiesEγ = 19 MeV by backscatter<strong>in</strong>g<br />
laser light. Thisγ-facility can be operated <strong>in</strong> co<strong>in</strong>cidence<br />
with the high-<strong>in</strong>tensity laser pulses with a repetiti<strong>on</strong><br />
rate <str<strong>on</strong>g>of</str<strong>on</strong>g> 1/m<strong>in</strong>. Here <strong>on</strong>e first might th<strong>in</strong>k <str<strong>on</strong>g>of</str<strong>on</strong>g> “dynamically<br />
(3)<br />
assisted pair creati<strong>on</strong>” [6], <strong>in</strong>ject<strong>in</strong>g <strong>in</strong>to the <strong>in</strong>tense lowfrequency<br />
laser focus at the same time the weak-<strong>in</strong>tensity,<br />
high-frequency γ-quanta, result<strong>in</strong>g <strong>in</strong> a new str<strong>on</strong>gly reduced<br />
exp<strong>on</strong>ential suppressi<strong>on</strong> factor:<br />
exp[−(π −2)·(Es/E)] ≈ 10 −350 . (4)<br />
Though this is a largely reduced h<strong>in</strong>drance factor, it is still<br />
much too small to generate for the given repetiti<strong>on</strong> rates and<br />
γ-<strong>in</strong>tensities any observable effects. Narozhnyi [15, 16]<br />
predicted for an additi<strong>on</strong>al <strong>in</strong>jecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> very high-energy<br />
γ-quanta with energyEγ an exp<strong>on</strong>ential h<strong>in</strong>drance factor:<br />
exp[−(8/3)·(Es/E)·(mec 2 /Eγ)] ≈ 10 −1 . (5)<br />
So, if we would <strong>in</strong>ject γ-quanta with much higher energy<br />
Eγ = 1000 · mec 2 =500 MeV, the exp<strong>on</strong>ential suppressi<strong>on</strong><br />
would basically vanish and we could observe pair<br />
creati<strong>on</strong>. However, it would be extremely difficult to produce<br />
such high-energy γ-quanta by Compt<strong>on</strong> backscatter<strong>in</strong>g<br />
with high harm<strong>on</strong>ic laser pulses from the 600 MeV<br />
electr<strong>on</strong> beam with sufficient <strong>in</strong>tensity. Is there another<br />
way to <strong>in</strong>ject γ-quanta <str<strong>on</strong>g>of</str<strong>on</strong>g> up to 500 MeV <strong>in</strong>to the high<strong>in</strong>tensity<br />
laser focus at ELI-NP with sufficient <strong>in</strong>tensity?<br />
If <strong>on</strong>e looks at the Compt<strong>on</strong> backscatter<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> laser phot<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> energy EL and an <strong>in</strong>tensity (characterized by a)<br />
from the classical electr<strong>on</strong> beam with energy Ee = γe ·<br />
mec 2 , <strong>on</strong>e obta<strong>in</strong>s for the γ-energyEγ:<br />
Eγ = n· 4γ2 eEL<br />
1+a 2<br />
with the harm<strong>on</strong>ic number n. While <strong>on</strong>e obta<strong>in</strong>s for the<br />
600 MeV electr<strong>on</strong> beam with γe = 1200 and small values<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the normalized laser vector potential a γ-energies <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
few MeV, <strong>on</strong>e f<strong>in</strong>ds that for <strong>in</strong>creas<strong>in</strong>ga ≈ γe the Doppler<br />
boost for the γ-energy or the factors γ 2 e and a2 cancel out<br />
andEγ ≈ n·4·EL drops to the optical regime. However,<br />
for such large values <str<strong>on</strong>g>of</str<strong>on</strong>g> γe and a, new high-field effects<br />
come <strong>in</strong>to play and Eq. (6) is no l<strong>on</strong>ger valid.<br />
If large external electromagnetic fields are present electr<strong>on</strong>s<br />
can ga<strong>in</strong> large energies. In that case any field <strong>in</strong> their<br />
rest frame can be c<strong>on</strong>sidered as c<strong>on</strong>stant and crossed due<br />
to the transformati<strong>on</strong> properties<br />
E ′<br />
|| = ′<br />
E || , E <br />
⊥ = γ E⊥ +v × <br />
B<br />
B ′<br />
|| = ′<br />
B || , B <br />
⊥ = γ<br />
(6)<br />
, (7)<br />
<br />
B⊥ − 1<br />
c2 v × <br />
E . (8)
This implies that E 2 − c 2 B 2 ≈ 0 and E · B ≈ 0. Any<br />
c<strong>on</strong>stant crossed field can be transformed <strong>in</strong>to a pure magnetic<br />
field <strong>in</strong> an appropriate reference frame. Radiati<strong>on</strong><br />
emissi<strong>on</strong> by electr<strong>on</strong>s or positr<strong>on</strong>s <strong>in</strong> a c<strong>on</strong>stant magnetic<br />
field is naturally c<strong>on</strong>troled by the dimensi<strong>on</strong>less parameter<br />
B⊥ǫ/(mEs), where B⊥ is the magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the external<br />
magnetic field normal to the particle momentum, ǫ is<br />
the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle and m is the mass. This can be<br />
understood by observ<strong>in</strong>g that the peak <str<strong>on</strong>g>of</str<strong>on</strong>g> the emitted radiati<strong>on</strong><br />
energy <str<strong>on</strong>g>of</str<strong>on</strong>g> an energetic electr<strong>on</strong> <strong>in</strong> a c<strong>on</strong>stant magnetic<br />
field versus its k<strong>in</strong>etic energy is approximately given<br />
by (¯hω0/ǫ)(ǫ/m) 3 = B⊥ǫ/(mEs), where ω0 = eB⊥/ǫ.<br />
In the lab frame this parameter becomes the quantum effi-<br />
ciency parametersχ<br />
χ = e¯h<br />
<br />
−(F µν pν) 2<br />
m 3 c 4 , (9)<br />
where pν is the electr<strong>on</strong> or positr<strong>on</strong> 4-momentum. The<br />
total transiti<strong>on</strong> rate for radiati<strong>on</strong> <strong>in</strong> str<strong>on</strong>g external fields<br />
scales likem 2 χ 2/3 /ǫ, where the spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the emittedγradiati<strong>on</strong><br />
now extends to ever larger energies the larger the<br />
parameterχis [7, 9]. If this Lorentz <strong>in</strong>variantχ gets close<br />
to unity, <strong>in</strong>tense γ emissi<strong>on</strong> with 10 8 times higher γ energies<br />
compared to Eq. (6) sets <strong>in</strong> and a significant fracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the electr<strong>on</strong> energy is emitted over small radiati<strong>on</strong> lengths<br />
with high energy γ-quanta [9, 7]. We will focus the 600<br />
MeV classical electr<strong>on</strong> beam directly <strong>in</strong>to the high-power<br />
laser focus and produce an <strong>in</strong>tense, brilliant high-energyγ<br />
beam directly <strong>in</strong>side the high <strong>in</strong>tensity laser focus. Thus at<br />
ELI-NP we will pursue a two-fold strategy: (1) we want<br />
to study the high field processes to produce new brilliant,<br />
high-energyγ-beams as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g>aandγe and then (2)<br />
the new γ-beam will cause pair creati<strong>on</strong> <strong>in</strong> the vacuum <strong>in</strong><br />
the same <strong>in</strong>tense laser focus.<br />
One could also th<strong>in</strong>k <str<strong>on</strong>g>of</str<strong>on</strong>g> produc<strong>in</strong>g the high-energy electr<strong>on</strong><br />
beam by laser accelerati<strong>on</strong>, but c<strong>on</strong>sider<strong>in</strong>g the fluctuati<strong>on</strong>s<br />
<strong>in</strong> presently produced laser-accelerated electr<strong>on</strong><br />
beams (e.g. po<strong>in</strong>t<strong>in</strong>g variati<strong>on</strong>s), it appears much more reliable<br />
to explore these new high-field forces and the producti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> high-energyγ-quanta with a very brilliant, very<br />
stable, classical high-energy electr<strong>on</strong> beam.<br />
With these high-energy quanta γh and the laser phot<strong>on</strong>s<br />
γL even hadr<strong>on</strong>ic QCD processes likeγh+n·γL → π 0 →<br />
γ1+γ2 are energetically allowed, and we may learn someth<strong>in</strong>g<br />
about the hadr<strong>on</strong>ic QCD c<strong>on</strong>tent <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum with<br />
str<strong>on</strong>g laser-dressed processes.<br />
In the follow<strong>in</strong>g we will first decribe the experimental<br />
setup at ELI-NP to explore these new high-field processes,<br />
and then we show the theoretically predicted rates <str<strong>on</strong>g>of</str<strong>on</strong>g>γ producti<strong>on</strong><br />
ande + e − pair creati<strong>on</strong>.<br />
EXPERIMENTAL SETUP AT ELI-NP FOR<br />
DETECTING HARDγ-PRODUCTION<br />
AND PAIR CREATION<br />
In Fig. 1 we show <strong>in</strong> a schematic way the experimental<br />
setup. We focus the electr<strong>on</strong> beam with a triplet lens <strong>in</strong>to<br />
Figure 1: Experimental setup, show<strong>in</strong>g the two focus<strong>in</strong>g<br />
mirrors with the high-field focus and the triplet lense which<br />
focusses the electr<strong>on</strong> beam (red) <strong>in</strong>to the laser focus. Beh<strong>in</strong>d<br />
the laser focus, a dipole magnet deflects the electr<strong>on</strong><br />
beam, while theγ beam (green) c<strong>on</strong>t<strong>in</strong>ues straight <strong>on</strong>.<br />
Figure 2: Extended view <str<strong>on</strong>g>of</str<strong>on</strong>g> the experimental setup, show<strong>in</strong>g<br />
the sampl<strong>in</strong>g measurements <strong>on</strong> the γ beam, where <strong>in</strong><br />
th<strong>in</strong> foilsγ-quanta are c<strong>on</strong>verted toe + e − pairs, which then<br />
are deflected <strong>in</strong> small magnets and measured <strong>in</strong> calorimeters.<br />
the laser focus. The electr<strong>on</strong> bunches have a maximum energy<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> 600 MeV, corresp<strong>on</strong>d<strong>in</strong>g toγe ≈ 1200, a maximum<br />
charge <str<strong>on</strong>g>of</str<strong>on</strong>g> 250 pC (≈ 10 9 e/s) with a typical bunch length<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> 1.5 ps or 450 µm , a diameter <str<strong>on</strong>g>of</str<strong>on</strong>g> 5µm and a normalized<br />
emittance <str<strong>on</strong>g>of</str<strong>on</strong>g> ǫn = 0.2 mm mrad. We use two counterpropagat<strong>in</strong>g<br />
laser pulses from two synchr<strong>on</strong>ized APOL-<br />
LON lasers [11] with 15 fs FWHM, where we can realize<br />
different Lorentz-<strong>in</strong>variant <strong>in</strong>tensity parameters F and G<br />
from the field tensorF µ,ν <strong>in</strong> the focus<br />
µν<br />
FµνF<br />
F = −<br />
2E2 s<br />
G = ǫµνλκF µν F λκ<br />
8E 2 S<br />
= E 2 −c 2 B 2<br />
E 2 S<br />
= c E · B<br />
E 2 S<br />
1 m<br />
(10)<br />
(11)<br />
Especially <strong>in</strong>terest<strong>in</strong>g is a c<strong>on</strong>figurati<strong>on</strong>, where the electric<br />
E fields <str<strong>on</strong>g>of</str<strong>on</strong>g> both oppositely circular polarized lasers are<br />
added <strong>in</strong> the focal plane, while the magnetic B fields cancel
each other. Thus the E field rotates around the laser axis<br />
and its amplitude varies slowly with the envelope <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
laser pulse. The two parabolic mirrors focus the two lasers<br />
to a radius <str<strong>on</strong>g>of</str<strong>on</strong>g> about 1 µm, and we have a maximum field<br />
strength characterized by a normalized vector potential a≈<br />
1000. Thus with<strong>in</strong> the high-field volume <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse<br />
about ≈ 10 5 electr<strong>on</strong>s are c<strong>on</strong>ta<strong>in</strong>ed. In the simulati<strong>on</strong>s<br />
(discussed below) we obta<strong>in</strong>, that each electr<strong>on</strong> <strong>in</strong> the highfield<br />
regime produces about 20 high-energyγ-quanta with<br />
an exp<strong>on</strong>ential spectrum, reach<strong>in</strong>g up to about 600 MeV<br />
with the special quantalγ emissi<strong>on</strong> processes [7]:<br />
e+n·γL → e′+γh<br />
(12)<br />
These γ-quantaγh exhibit the very small open<strong>in</strong>g angle <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the electr<strong>on</strong> beam <str<strong>on</strong>g>of</str<strong>on</strong>g> ≈ 1/(2γ) ≈ 10µrad and a diameter<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser focus <str<strong>on</strong>g>of</str<strong>on</strong>g> about≈ 2µm.<br />
The process is c<strong>on</strong>trolled by the relativistic <strong>in</strong>variant parameters<br />
and<br />
χe = e¯h<br />
m3 ec4 <br />
−(Fµνpν e) 2 = E<br />
·<br />
ES<br />
Ee<br />
mec2 (13)<br />
κγ = e¯h<br />
m3 ec4 <br />
− Fµνkν 2 E<br />
γ = ·<br />
ES<br />
Eγ<br />
mec2 (14)<br />
is the<br />
where Fµν is the four-tensor <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field, pν e<br />
four-momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> the high energy electr<strong>on</strong> and kν γ is the<br />
four-momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> theγ quantum.<br />
The durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the γ-pulse is about 15 fs, like the laser<br />
pulse. In Fig. 1 and Fig. 2 we show how the primary electr<strong>on</strong><br />
bunch is deflected by a dipole magnet beh<strong>in</strong>d the laser<br />
focus and then propagates straight to the electr<strong>on</strong> beam<br />
dump. The high-energy γ-pulse traverses several identical<br />
detector units before it is stopped <strong>in</strong> the γ-beam dump.<br />
Each detector unit c<strong>on</strong>sists <str<strong>on</strong>g>of</str<strong>on</strong>g> a th<strong>in</strong> foil, which c<strong>on</strong>verts<br />
about <strong>on</strong>eγ-quantum <strong>in</strong>to an electr<strong>on</strong>-positr<strong>on</strong> pair. In this<br />
way we can sample the spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tense, 15 fs γbunch<br />
c<strong>on</strong>ta<strong>in</strong><strong>in</strong>g about 106 γ-quanta. The forward-go<strong>in</strong>g<br />
e + e− pairs are opened up by a local magnetic field. Thus<br />
the positr<strong>on</strong> hits the calorimeter array <strong>on</strong> the lift side <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
γ-beam, while the electr<strong>on</strong> hits the calorimeter array <strong>on</strong> the<br />
right side. From the deposited energy, the positi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
shower <strong>in</strong> the calorimeter and the vertex <strong>in</strong> the th<strong>in</strong> foil we<br />
can rec<strong>on</strong>struct the γ-ray spectrum and determ<strong>in</strong>e the spot<br />
size <str<strong>on</strong>g>of</str<strong>on</strong>g> the γ beam. By this sampl<strong>in</strong>g technique we avoid<br />
a pile-up <str<strong>on</strong>g>of</str<strong>on</strong>g> the γ-pulses <strong>in</strong> the detectors. When reduc<strong>in</strong>g<br />
the field strength <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse or reduc<strong>in</strong>g the electr<strong>on</strong><br />
energy, we obta<strong>in</strong> from simulati<strong>on</strong>s a fast decrease <str<strong>on</strong>g>of</str<strong>on</strong>g> these<br />
high field processes. Also a str<strong>on</strong>g dependence <strong>on</strong> the laser<br />
pulse durati<strong>on</strong> is predicted. This will allow us for the first<br />
time to test the assumed physical processes <str<strong>on</strong>g>of</str<strong>on</strong>g> this highfield<br />
regime. At the same time we will be able to predict the<br />
properties <str<strong>on</strong>g>of</str<strong>on</strong>g> the brilliant, <strong>in</strong>tense γ beams for other laser<br />
and electr<strong>on</strong> beam parameters. Theseγ-beams have typical<br />
peak brilliances <str<strong>on</strong>g>of</str<strong>on</strong>g>≈ 1024 /[(mm·mrad) 2 ·s·0.1%BW].<br />
/N 0<br />
N e + e<br />
¢<br />
0 5 10 15<br />
t,<br />
20 25 30<br />
¡1<br />
10<br />
9<br />
a =10<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
3<br />
a =1.2 £10 3<br />
a =1.5 £10 3<br />
Figure 3: Simulated yield <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> positr<strong>on</strong> pairs as a<br />
functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> time for different values <str<strong>on</strong>g>of</str<strong>on</strong>g> the dimensi<strong>on</strong>less<br />
field amplitudea = 10 3 , 1.2×10 3 , 1.5×10 3 . The energy<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the primary electr<strong>on</strong> beam is600MeV .<br />
The polarisati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the brilliant γ-beam is determ<strong>in</strong>ed by<br />
the polarisati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser beams.<br />
Furthermore, we show <strong>in</strong> Fig. 2 the broad-acceptance<br />
magnetic spectrometers with which we want to measure<br />
the energies and spatial distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s and<br />
positr<strong>on</strong>s. We expect for our parameters <strong>on</strong>ly a few e + e −<br />
pairs per shot produced <strong>in</strong> the multiphot<strong>on</strong> Breit-Wheeler<br />
reacti<strong>on</strong><br />
γh +n·γL → e + e −<br />
(15)<br />
For our field strength characterized by a≈1000 and electr<strong>on</strong><br />
energies with γe ≈1200, the pair creati<strong>on</strong> is still<br />
marg<strong>in</strong>al. Exp<strong>on</strong>entially <strong>in</strong>creas<strong>in</strong>g QED cascades [14] are<br />
expected <strong>on</strong>ly for higher laser field strengths. Still these<br />
measurements will present a str<strong>on</strong>g test <str<strong>on</strong>g>of</str<strong>on</strong>g> the new highfield<br />
pair creati<strong>on</strong> simulati<strong>on</strong>s.<br />
The spectra <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s may also be used to align the<br />
two laser pulses and the electr<strong>on</strong> bunch properly <strong>in</strong> space<br />
and time, because the deflecti<strong>on</strong>, accelerati<strong>on</strong> or decelerati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s depend sensitively <strong>on</strong> the comb<strong>in</strong>ed<br />
fields <str<strong>on</strong>g>of</str<strong>on</strong>g> the two lasers. The electr<strong>on</strong>s also probe the outer<br />
fr<strong>in</strong>ge fields <str<strong>on</strong>g>of</str<strong>on</strong>g> the lasers.<br />
Thus we hereby use the high-field laser focus <strong>in</strong> two<br />
functi<strong>on</strong>s at the same time: (1) to generate the high-energy<br />
γ-quanta and (2) to <strong>in</strong>duce their pair decay.<br />
SIMULATIONS<br />
For the f<strong>in</strong>al comparis<strong>on</strong> between experiment and simulati<strong>on</strong><br />
we need the precise parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser focus and<br />
the electr<strong>on</strong> bunch. We not <strong>on</strong>ly have to describe the highfield<br />
laser <strong>in</strong>teracti<strong>on</strong> for the producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> high-energyγquanta<br />
and e + e − pairs properly, but also the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the electr<strong>on</strong>s <strong>in</strong> the laser field like accelerati<strong>on</strong>, decelerati<strong>on</strong><br />
and the moti<strong>on</strong> <strong>in</strong> the p<strong>on</strong>deromotive potential. At
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Figure 4: Simulated spectra <str<strong>on</strong>g>of</str<strong>on</strong>g> generated γ-quanta,<br />
positr<strong>on</strong>s and electr<strong>on</strong>s for a focus with a rotat<strong>in</strong>g E field<br />
and10 24 W/cm 2 and a 600 MeV electr<strong>on</strong> bunch.<br />
present we performed first order calculati<strong>on</strong>s, which give<br />
us the ma<strong>in</strong> spectra <str<strong>on</strong>g>of</str<strong>on</strong>g> the high energy γ-quanta, electr<strong>on</strong>s<br />
and positr<strong>on</strong>s to design the spectrometers. In M<strong>on</strong>te Carlo<br />
computer simulati<strong>on</strong>s, solv<strong>in</strong>g the transport equati<strong>on</strong>s for<br />
the high-field laser <strong>in</strong>teracti<strong>on</strong> us<strong>in</strong>g Landau-Lifshitz-like<br />
forces or quantal <strong>in</strong>teracti<strong>on</strong>s [7, 10, 9], we obta<strong>in</strong>ed our<br />
predicti<strong>on</strong>s. We estimate about 2 electr<strong>on</strong>-positr<strong>on</strong> pairs<br />
per laser shot and about Nγ=20 high-energy phot<strong>on</strong>s per<br />
shot. Fig. 3 shows the str<strong>on</strong>g exp<strong>on</strong>ential <strong>in</strong>crease <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
numbers <str<strong>on</strong>g>of</str<strong>on</strong>g> pairs with the normalized vector potentialaand<br />
the durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse. It dem<strong>on</strong>strates that we are<br />
just start<strong>in</strong>g to see positr<strong>on</strong>s and that a small improvement<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the parameters will lead us <strong>in</strong> the QED cascade regime.<br />
In Fig. 4 <strong>on</strong> sees the exp<strong>on</strong>entially decreas<strong>in</strong>gγ-spectrum,<br />
reach<strong>in</strong>g up to 600 MeV. The high-energy γ-quanta then<br />
lead to the e + e − producti<strong>on</strong>. The electr<strong>on</strong> and positr<strong>on</strong><br />
spectra show, that they are accelerated <strong>in</strong> the rotat<strong>in</strong>g field,<br />
but at the same timeγ-emissi<strong>on</strong> results <strong>in</strong> lower energies.<br />
COMPARISON WITH THE E144 SLAC<br />
EXPERIMENT ON HIGHγ-PRODUCTION<br />
AND PAIR CREATION<br />
The high-field theory will be tested <strong>in</strong> the ELI-NP experiment,<br />
which is very different from the E144 SLAC experiment<br />
[12, 13]. In both experiments the dom<strong>in</strong>ant process<br />
for pair creati<strong>on</strong> is a two-step process, where an electr<strong>on</strong><br />
first produces a high-energy γ-quantum <strong>in</strong> the laser<br />
field and <strong>in</strong> the sec<strong>on</strong>d step via a Breit-Wheeler reacti<strong>on</strong><br />
the high energyγ-quantum is c<strong>on</strong>verted <strong>in</strong>to ane + e − pair.<br />
In Table 1 we compare the parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> the two experiments.<br />
While the laser <strong>in</strong>tensity <strong>in</strong> the SLAC experiment<br />
was 1.3 · 10 18 W/cm 2 , we expect to have ≈ 10 24 W/cm 2<br />
and coresp<strong>on</strong>d<strong>in</strong>gly the normalized vector amplitude <strong>in</strong> the<br />
SLAC experiment was a = 0.36, while we will have a<br />
much larger value <str<strong>on</strong>g>of</str<strong>on</strong>g> a = 1000. Our laser field c<strong>on</strong>fig-<br />
urati<strong>on</strong> can be chosen very flexibly with the two Lorentz<br />
<strong>in</strong>variantsF andG, while <strong>in</strong> the SLAC case <strong>on</strong>ly <strong>on</strong>e laser<br />
beam with crossed E and B fields and F=0 was used. On<br />
the other hand, <strong>in</strong> the SLAC experiment much more highenergetic<br />
electr<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> 46.6 GeV were used, while we will<br />
have <strong>on</strong>ly 0.6 GeV. For many c<strong>on</strong>siderati<strong>on</strong>s aga<strong>in</strong> <strong>on</strong>ly<br />
the Lorentz-<strong>in</strong>variant quantities χe and χγ are important.<br />
While <strong>in</strong> the SLAC experiment the produced high-energy<br />
γ-quanta had about 30 GeV, <strong>in</strong> the ELI-NP case we will<br />
have <strong>on</strong>ly quanta below 600 MeV. Theχvalues <str<strong>on</strong>g>of</str<strong>on</strong>g> the ELI-<br />
NP experiment are larger and thus the creati<strong>on</strong> times for<br />
new particles are shorter. While the number <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s<br />
<strong>in</strong> the accelerated bunch is similar, the number <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s<br />
reach<strong>in</strong>g the high-field laser <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> the ELI-NP experiment<br />
will be a factor <str<strong>on</strong>g>of</str<strong>on</strong>g>10 4 smaller.<br />
Table 1: Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> the ELI-NP and<br />
E144 SLAC experiments [12]<br />
parameter ELI-NP E144 SLAC<br />
norm. vec. potent. a 1000 0.36<br />
laser <strong>in</strong>tensity ≈ 10 24 W/cm 2 1.3·10 18 W/cm 2<br />
laser width 15 fs 1600 fs<br />
σx ≈ 1µm 25µm<br />
σy ≈ 1µm 40µm<br />
Ee 0.6 GeV 46.6 GeV<br />
Ne 1.5·10 9 e 7·10 9<br />
repetiti<strong>on</strong> rate 0.02 Hz 10-20 Hz<br />
χe 1.7 0.3<br />
χγ ≈1 0.2<br />
phot<strong>on</strong>s absorbed<br />
<strong>in</strong> pair cr. npair ≈ 10 9 ≈ 5<br />
CONCLUSIONS AND OUTLOOK<br />
We are prepar<strong>in</strong>g for a very <strong>in</strong>tense laser focus <strong>in</strong> vacuum<br />
at ELI-NP with a large freedom to choose the <strong>in</strong>variant<br />
field parameters. Furthermore, we require very good<br />
vacum <strong>in</strong> the surround<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser focus us<strong>in</strong>g cryopump<strong>in</strong>g.<br />
Thus this setup will represent an ideal laboratory<br />
for prob<strong>in</strong>g the real and imag<strong>in</strong>ary part <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum<br />
<strong>in</strong> high laser fields. In additi<strong>on</strong>, we will add a very brilliant,<br />
<strong>in</strong>tense high-energy electr<strong>on</strong> beam and a brilliant polarized<br />
high-energyγ-beam for seed<strong>in</strong>g the high-field laservacuum<br />
<strong>in</strong>teracti<strong>on</strong>. Thus all comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> QED cascades<br />
[14] can be probed <strong>in</strong>dividually <strong>in</strong> order to test theoretical<br />
predicti<strong>on</strong>s. We may even improve the high-energyγ spectrum<br />
from its exp<strong>on</strong>ential shape to a spectrum with more <strong>in</strong>tensity<br />
at higher energies, by modulat<strong>in</strong>g the electr<strong>on</strong> density<br />
and electr<strong>on</strong> energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the large bunch <strong>in</strong> the 100 fs<br />
range, before the 15 fs high field <strong>in</strong>teracti<strong>on</strong> occurs.
ACKNOWLEDGMENTS<br />
This work is based <strong>on</strong> <strong>in</strong>tensive discussi<strong>on</strong>s with H. Gies,<br />
R. Schützhold, T. He<strong>in</strong>zl, T. Tajima ans Z. Zamfir. It<br />
was supported by the DFG Cluster <str<strong>on</strong>g>of</str<strong>on</strong>g> Excellence MAP<br />
(Munich-Center for Advanced Phot<strong>on</strong>ics).<br />
REFERENCES<br />
[1] http://www.extreme-light-<strong>in</strong>frastructure.eu/.<br />
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[4] J. Schw<strong>in</strong>ger, Phys. Rev. 82, 664 (1951)<br />
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[8] L.D. Landau and E.M. Lifschitz, Klassische Feldtheorie,<br />
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[10] N. Elk<strong>in</strong>a and H. Ruhl, c<strong>on</strong>tributi<strong>on</strong> to the PIF2010 c<strong>on</strong>ference.<br />
[11] J.P. Chambaret, The Extreme Light Infrastructure Project<br />
ELI and its Prototype APOLLON/ILE “the associated laser<br />
bottlenecks”, LEI <str<strong>on</strong>g>C<strong>on</strong>ference</str<strong>on</strong>g>, Brasov, Romania, Oct.16-21,<br />
2009<br />
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(2008).<br />
J.G. Kirk et al., Plasma Phys. C<strong>on</strong>trol. Fus., 51, 085008<br />
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[15] N.B. Narozhnyi, Sov. Phys. JETP 27, 360 (1968).<br />
[16] V.N. Baier et al., arXiv:0912.5250v1[hep-ph] (2010).
Abstract<br />
Numerical simulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QED cascades <strong>in</strong> <strong>in</strong>tense laser fields ∗<br />
Due to the dramatic progress <strong>in</strong> laser technology [1] a<br />
novel area <str<strong>on</strong>g>of</str<strong>on</strong>g> laser-matter <strong>in</strong>teracti<strong>on</strong> at ultra-high <strong>in</strong>tensity<br />
is aris<strong>in</strong>g. The Extreme Light Infrastructure project<br />
(ELI) [2] is aim<strong>in</strong>g at gett<strong>in</strong>g access to <strong>in</strong>tensity levels up to<br />
10 26 W/cm 2 . Therefore it is timely to <strong>in</strong>vestigate the structure<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the QED vacuum and the <strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charged<br />
particles with extreme fields. In particular, cascades <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
electr<strong>on</strong>s, positr<strong>on</strong>s, and phot<strong>on</strong>s may arise limit<strong>in</strong>g the <strong>in</strong>crease<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser <strong>in</strong>tensity bey<strong>on</strong>d a given threshold due<br />
to the depleti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser. In additi<strong>on</strong>, it is found that the<br />
super-<strong>in</strong>tense laser field is capable <str<strong>on</strong>g>of</str<strong>on</strong>g> restor<strong>in</strong>g the energy<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and positr<strong>on</strong>s and the dynamical quantum efficiency<br />
parameter by efficient accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and<br />
positr<strong>on</strong>s <strong>in</strong> the laser field. This novel restorati<strong>on</strong> effect<br />
may become a dom<strong>in</strong>at<strong>in</strong>g feature <str<strong>on</strong>g>of</str<strong>on</strong>g> laser-matter <strong>in</strong>teracti<strong>on</strong><br />
at ultra-high <strong>in</strong>tensities. In the present paper we report<br />
about the current status our simulati<strong>on</strong> framework based <strong>on</strong><br />
semi-classical transport equati<strong>on</strong>s for electr<strong>on</strong>s, positr<strong>on</strong>s<br />
and phot<strong>on</strong>s.<br />
INTRODUCTION<br />
The purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> the present c<strong>on</strong>tributi<strong>on</strong> is to present<br />
the current status <str<strong>on</strong>g>of</str<strong>on</strong>g> our simulati<strong>on</strong> code capable <str<strong>on</strong>g>of</str<strong>on</strong>g> model<strong>in</strong>g<br />
new physical phenomena at ultra-high laser fields<br />
(I ≥ 10 24 W/cm 2 ) as they will be available <strong>in</strong> the<br />
near future (ELI). The presence <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g electromagnetic<br />
fields reveals a range <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum electrodynamic effects,<br />
which may significantly change the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> relativistic<br />
plasma. At laser <strong>in</strong>tensities <strong>in</strong> excess <str<strong>on</strong>g>of</str<strong>on</strong>g> I ≥ 10 24 W/cm 2<br />
n<strong>on</strong>-classical effects like radiati<strong>on</strong> reacti<strong>on</strong>, vacuum polarizati<strong>on</strong>,<br />
and electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong> due to cascad<strong>in</strong>g<br />
become important and ultimately dom<strong>in</strong>ant. Due to the<br />
ubiquitous complexity <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>teracti<strong>on</strong> at ultra-high fields<br />
simulati<strong>on</strong>s will have crucial importance. A c<strong>on</strong>certed research<br />
effort will be required because c<strong>on</strong>venti<strong>on</strong>al plasma<br />
model<strong>in</strong>g techniques, while adequate for classical laserplasma<br />
<strong>in</strong>teracti<strong>on</strong> or very small numbers <str<strong>on</strong>g>of</str<strong>on</strong>g> particles, cannot<br />
access the length and time scales relevant to illumnate<br />
the real structure and dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g field QED.<br />
The paper is organized as follows. First we review and<br />
outl<strong>in</strong>e elementary quantum processes used <strong>in</strong> our code.<br />
They are s<strong>in</strong>gle phot<strong>on</strong> emissi<strong>on</strong> by electr<strong>on</strong>s and positr<strong>on</strong>s<br />
and pair creati<strong>on</strong> by hard phot<strong>on</strong>s <strong>in</strong> str<strong>on</strong>g laser fields<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> arbitrary c<strong>on</strong>figurati<strong>on</strong>. Next we proceed with stat<strong>in</strong>g<br />
appropriate k<strong>in</strong>etic equati<strong>on</strong>s, which <strong>in</strong>clude elementary<br />
quantum processes. The presence <str<strong>on</strong>g>of</str<strong>on</strong>g> high <strong>in</strong>tensity laser<br />
∗ This work was supported by DFG project RU-633/1-1 and the<br />
Cluster-<str<strong>on</strong>g>of</str<strong>on</strong>g>-Excellence ’Munich Centre for Advance Phot<strong>on</strong>ics’ (MAP)<br />
N. Elk<strong>in</strong>a and H. Ruhl<br />
LMU München<br />
fields allows a semiclassical approach for the descripti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the particle moti<strong>on</strong> <strong>in</strong> electromagnetic fields. This leads<br />
to the formulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> semi-classical transport equati<strong>on</strong>s. As<br />
a first step for the <strong>in</strong>corporati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>l<strong>in</strong>ear QED effects<br />
<strong>in</strong>to the model<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>-positr<strong>on</strong>-phot<strong>on</strong> plasma we<br />
state a set <str<strong>on</strong>g>of</str<strong>on</strong>g> k<strong>in</strong>etic equati<strong>on</strong>s to describe QED effects. The<br />
k<strong>in</strong>etic equati<strong>on</strong>s can be reformulated as a set <str<strong>on</strong>g>of</str<strong>on</strong>g> moment<br />
equati<strong>on</strong>s for quasi-elements. S<strong>in</strong>ce phase space can grow<br />
rapidly <strong>in</strong> ultra-str<strong>on</strong>g fields adaptive management <str<strong>on</strong>g>of</str<strong>on</strong>g> phase<br />
space by adaptive weights for quasi-elements has to be <strong>in</strong>troduced<br />
<strong>in</strong> order to keep the required resoluti<strong>on</strong> <strong>in</strong> phase<br />
space. Next we present results <str<strong>on</strong>g>of</str<strong>on</strong>g> prelim<strong>in</strong>ary simulati<strong>on</strong>s<br />
<strong>in</strong> order to illustrate the power <str<strong>on</strong>g>of</str<strong>on</strong>g> our numerical approach.<br />
F<strong>in</strong>ally a short c<strong>on</strong>clusi<strong>on</strong> is given.<br />
ELEMENTARY QED EFFECTS IN<br />
STRONG LASER FIELDS<br />
The process <str<strong>on</strong>g>of</str<strong>on</strong>g> cascad<strong>in</strong>g can be represented by two<br />
coupled processes <str<strong>on</strong>g>of</str<strong>on</strong>g> hard phot<strong>on</strong> emissi<strong>on</strong> and pair producti<strong>on</strong><br />
via the absorbti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> n ”s<str<strong>on</strong>g>of</str<strong>on</strong>g>t” phot<strong>on</strong>s (<str<strong>on</strong>g>of</str<strong>on</strong>g> energy<br />
¯hω ∼ 1 eV) from the laser field<br />
e ± + n¯hωl → γ + e ± , (1)<br />
γ + n¯hωl → e + + e − . (2)<br />
The created charged particles are ultra-relativistic (γ ≫<br />
1) and are exposed to electromagnetic fields <str<strong>on</strong>g>of</str<strong>on</strong>g> ultrarelativistic<br />
(a0 ≫ 1, χ ∼ 1) but still sub-critical (F ≪ ES)<br />
<strong>in</strong>tensities, where F = √ F µν √<br />
Fµν = ⃗E 2 − B⃗ 2 . The en-<br />
ergy <str<strong>on</strong>g>of</str<strong>on</strong>g> hard phot<strong>on</strong>s <strong>in</strong> the simulati<strong>on</strong>s are assumed to be <strong>in</strong><br />
the range <str<strong>on</strong>g>of</str<strong>on</strong>g> ¯hω > mc 2 . The theoretical approach used <strong>in</strong><br />
this work is based <strong>on</strong> the fact that <strong>in</strong> the rest frame <str<strong>on</strong>g>of</str<strong>on</strong>g> highly<br />
relativistic charged particles any field can be c<strong>on</strong>sidered as<br />
c<strong>on</strong>stant and crossed. This implies that E 2 − H 2 ≈ 0 and<br />
⃗E · ⃗ H ≈ 0. C<strong>on</strong>stant crossed fields can be derived from<br />
a vector potential <str<strong>on</strong>g>of</str<strong>on</strong>g> the k<strong>in</strong>d A µ = a µ k · x, where a µ is<br />
a c<strong>on</strong>stant polarizati<strong>on</strong> vector. The functi<strong>on</strong>al form for the<br />
vector potential <str<strong>on</strong>g>of</str<strong>on</strong>g> a plane wave is A µ = a µ (k · x), where<br />
a µ would now be a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> k·x. The appropriate transiti<strong>on</strong><br />
amplitudes <strong>on</strong> a tree level are calculated with the help<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Volkov soluti<strong>on</strong>s for the fermi<strong>on</strong>s [4]. However, for c<strong>on</strong>stant<br />
crossed fields <strong>on</strong>e better makes use <str<strong>on</strong>g>of</str<strong>on</strong>g> the soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the Dirac equati<strong>on</strong> for a c<strong>on</strong>stant external magnetic field,<br />
s<strong>in</strong>ce any c<strong>on</strong>stant crossed field can be transformed <strong>in</strong>to<br />
a pure magnetic field <strong>in</strong> an appropriate reference frame.<br />
Radiati<strong>on</strong> emissi<strong>on</strong> by electr<strong>on</strong>s or positr<strong>on</strong>s <strong>in</strong> a c<strong>on</strong>stant<br />
magnetic field is naturally c<strong>on</strong>troled by the dimensi<strong>on</strong>less<br />
parameter B⊥ϵ/(mEs), where B⊥ is the magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
external magnetic field normal to the particle momentum,
ϵ is the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle, m is the mass <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle,<br />
and Es = m2 /(e¯h) is the Schw<strong>in</strong>ger field. This<br />
can be understood by observ<strong>in</strong>g that the peak <str<strong>on</strong>g>of</str<strong>on</strong>g> the emitted<br />
radiati<strong>on</strong> energy <str<strong>on</strong>g>of</str<strong>on</strong>g> an energetic electr<strong>on</strong> <strong>in</strong> a c<strong>on</strong>stant<br />
magnetic field versus its energy is approximately given by<br />
(¯hω0/ϵ) (ϵ/m) 3 = B⊥ϵ/(mEs), where ω0 = eB⊥/ϵ.<br />
Further details can be found <strong>in</strong> [5]. In the lab frame this<br />
parameter becomes the quantum efficiency parameters χ<br />
χ = e¯h<br />
√<br />
− (F µν pν) 2<br />
m3 , (3)<br />
where pν is the electr<strong>on</strong> or positr<strong>on</strong> 4-momentum. In the<br />
classical radiati<strong>on</strong> realm the emitted phot<strong>on</strong> energies are<br />
much smaller than the charged particle energy. Hence,<br />
χ ≪ 1 holds. In the quantum realm χ ≫ 1 is valid. The<br />
transiti<strong>on</strong> rate for radiati<strong>on</strong> emissi<strong>on</strong> depends <strong>on</strong> a sec<strong>on</strong>d<br />
quantum parameter κ given by<br />
κ = e¯h<br />
√<br />
− (F µν kν) 2<br />
m3 , (4)<br />
where kν is the phot<strong>on</strong> 4-momentum. The angle <strong>in</strong>tegrated<br />
transiti<strong>on</strong> rate for phot<strong>on</strong> emissi<strong>on</strong> is<br />
where<br />
dWγ<br />
dω<br />
(∫ ∞<br />
m2<br />
= −α<br />
ϵ2 dz Ai(z) (5)<br />
x<br />
[<br />
2<br />
+<br />
x + κ √ ] )<br />
x ∂zAi(x) ,<br />
[<br />
κ<br />
x =<br />
χ (χ − κ)<br />
] 2<br />
3<br />
, 0 ≤ κ < χ . (6)<br />
The rate holds for radiati<strong>on</strong> emissi<strong>on</strong> by electr<strong>on</strong>s or<br />
positr<strong>on</strong>s. In the simulati<strong>on</strong>s we assume that<br />
′<br />
⃗k = ⃗p + ⃗p (7)<br />
holds for the momenta. For the energy balance we f<strong>in</strong>d<br />
q + ω = ϵ ′<br />
+ ϵ . (8)<br />
Mak<strong>in</strong>g use <str<strong>on</strong>g>of</str<strong>on</strong>g> Eqns. (7) and (8) the energy taken from the<br />
external laser field is<br />
q =<br />
√<br />
ϵ 2 + 2ωϵ ′ (1 − v ′ cos θ) − ϵ ,<br />
where θ is the angle between the emitted phot<strong>on</strong> and the<br />
outgo<strong>in</strong>g electr<strong>on</strong> or positr<strong>on</strong> ⃗p ′<br />
and v ′<br />
= |⃗p ′<br />
|/ϵ ′<br />
is the<br />
velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> or positr<strong>on</strong> after the radiati<strong>on</strong> process.<br />
For electr<strong>on</strong>s or positr<strong>on</strong>s reta<strong>in</strong><strong>in</strong>g large momenta ⃗p ′<br />
al<strong>on</strong>g ⃗k after the emissi<strong>on</strong> process v ′<br />
cos θ ≈ 1 holds. The<br />
external field has to deliver <strong>on</strong>ly little energy, q ≈ 0, <strong>in</strong> that<br />
case. Given the electr<strong>on</strong> or positr<strong>on</strong> 4-momentum and the<br />
external field c<strong>on</strong>text χ can be calculated. In a sec<strong>on</strong>d step<br />
κ is obta<strong>in</strong>ed for permissible phot<strong>on</strong> emissi<strong>on</strong> and hence x.<br />
With the help <str<strong>on</strong>g>of</str<strong>on</strong>g> the cross<strong>in</strong>g symmetry the angle <strong>in</strong>tegrated<br />
transiti<strong>on</strong> rate for e + e−-pair creati<strong>on</strong> is given by<br />
where<br />
dW e + e −<br />
dϵ−<br />
= α m2<br />
ω2 (∫ ∞<br />
y<br />
+<br />
[<br />
κ<br />
y =<br />
χ (κ − χ)<br />
[<br />
2<br />
y − κ √ ]<br />
y<br />
] 2<br />
3<br />
dz Ai(z) (9)<br />
)<br />
∂zAi(y) ,<br />
, 0 ≤ χ < κ . (10)<br />
The follow<strong>in</strong>g k<strong>in</strong>ematic relati<strong>on</strong>s are used when pairs are<br />
created<br />
⃗ k = ⃗p+ + ⃗p− , (11)<br />
q + ω = ϵ+ + ϵ− . (12)<br />
Mak<strong>in</strong>g use <str<strong>on</strong>g>of</str<strong>on</strong>g> Eqns. (11) and (12) the energy taken from<br />
the external laser field is<br />
√<br />
q = ϵ2 − + 2ωϵ+ (1 − v+ cos θ) − ϵ− ,<br />
where θ is the angle between the phot<strong>on</strong> and the emitted<br />
positr<strong>on</strong>, and v+ = |⃗p+|/ϵ+ is the velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the positr<strong>on</strong>.<br />
Given a phot<strong>on</strong> 4-vector and the field c<strong>on</strong>text κ can be obta<strong>in</strong>ed.<br />
Next it is possible to calculate χ for any permissible<br />
electr<strong>on</strong> energy and hence y.<br />
If χ and κ are large enough we f<strong>in</strong>d that phot<strong>on</strong>s generate<br />
electr<strong>on</strong>s and positr<strong>on</strong>s and the latter aga<strong>in</strong> phot<strong>on</strong>s.<br />
The cha<strong>in</strong> process leads to exp<strong>on</strong>ential growth <str<strong>on</strong>g>of</str<strong>on</strong>g> particles<br />
<strong>in</strong> the simulati<strong>on</strong>. The soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the underly<strong>in</strong>g transport<br />
equati<strong>on</strong>s for uniform circular polarized light is approximately<br />
N e + e −(t) = N e + e −(0) e Γ t , (13)<br />
where the growth rate Γ can be estimated as<br />
Γ = α µ 1<br />
√<br />
m ω c2 4 ,<br />
¯h<br />
µ = E<br />
.<br />
α Es<br />
(14)<br />
Here E is the external field.<br />
Quantum efficiency becomes very large <strong>in</strong> circular<br />
purely electric fields. Hence we c<strong>on</strong>sider two counterpropagat<strong>in</strong>g<br />
circular laser beams <strong>in</strong> the center <str<strong>on</strong>g>of</str<strong>on</strong>g> which<br />
there is a rotat<strong>in</strong>g electric field. Figure 1 shows the situati<strong>on</strong><br />
and the simulati<strong>on</strong> results. Al<strong>on</strong>g the purple l<strong>in</strong>e electr<strong>on</strong>s<br />
are <strong>in</strong>jected <strong>in</strong>to the laser focus at 600 MeV. Red and<br />
green l<strong>in</strong>es represent sec<strong>on</strong>dary electr<strong>on</strong>s and positr<strong>on</strong>s. As<br />
a ma<strong>in</strong> result we f<strong>in</strong>d that the mean energy per particles is<br />
not decreas<strong>in</strong>g <strong>in</strong> the course <str<strong>on</strong>g>of</str<strong>on</strong>g> the cascade development.<br />
The c<strong>on</strong>trary could be expected s<strong>in</strong>ce the number <str<strong>on</strong>g>of</str<strong>on</strong>g> sec<strong>on</strong>dary<br />
particles grows exp<strong>on</strong>entially. The f<strong>in</strong>d<strong>in</strong>g implies<br />
enhanced energy depositi<strong>on</strong>. In fact, the empirical observati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> almost c<strong>on</strong>stant mean energy per particle represents<br />
exp<strong>on</strong>entially grow<strong>in</strong>g energy depositi<strong>on</strong> <strong>in</strong> the evolv<strong>in</strong>g<br />
plasma.
Figure 1: Cascad<strong>in</strong>g <strong>in</strong> a laser focus at I = 3 · 10 24 W/cm 2<br />
<strong>in</strong>itiated by primary electr<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> energy ε = 600 MeV <strong>in</strong>jected<br />
<strong>in</strong>to the laser focus. The multiplicity <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles<br />
is about 2.5, which represents the <strong>on</strong>set <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-l<strong>in</strong>ear<br />
QED <strong>in</strong> the laser-focus.<br />
KINETIC SIMULATION OF THE QED<br />
PLASMA<br />
Relevant processes <strong>in</strong> ultra-<strong>in</strong>tense laser fields are the<br />
generati<strong>on</strong> and annihilati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s dur<strong>in</strong>g the <strong>in</strong>teracti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser with electr<strong>on</strong>s and positr<strong>on</strong>s and e + e−-pair creati<strong>on</strong> with the help <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser and exist<strong>in</strong>g phot<strong>on</strong>s. Appropriate<br />
transport equati<strong>on</strong>s can be formulated. They are<br />
solved with the help <str<strong>on</strong>g>of</str<strong>on</strong>g> the quasi-elements.<br />
Neglect<strong>in</strong>g higher order effects as the annihilati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
pairs and classical radiati<strong>on</strong> reacti<strong>on</strong> a system c<strong>on</strong>sist<strong>in</strong>g<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s, positr<strong>on</strong>s, and radiati<strong>on</strong> can be described by<br />
transport equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the follow<strong>in</strong>g k<strong>in</strong>d<br />
(<br />
∂t + ⃗v · ∂⃗x ± ⃗ )<br />
F · ∂⃗p f±(⃗x, ⃗p, t) (15)<br />
∫<br />
=<br />
ω>ω0 ∫<br />
−f±(⃗x, ⃗p, t)<br />
∫<br />
+<br />
d 3 k W ⃗ E, ⃗ B<br />
γ ( ⃗k, ⃗p + ⃗k) f±(⃗x, ⃗p + ⃗k, t)<br />
ω>ω0<br />
ω>ω0<br />
where ⃗ ( )<br />
F = |e| ⃗E + ⃗v × B⃗<br />
d 3 k W ⃗ E, ⃗ B<br />
γ ( ⃗k, ⃗p)<br />
d 3 k W ⃗ E, ⃗ B<br />
e + e −( ⃗ k, ⃗p) fγ(⃗x, ⃗ k, t) ,<br />
is the Lorentz force and<br />
(<br />
∂t + ∂ω<br />
∂⃗ )<br />
· ∂⃗x fγ(⃗x,<br />
k ⃗k, t) (16)<br />
∫<br />
= d 3 p W ⃗ E, ⃗ B<br />
γ ( ⃗k, ⃗p) [f+(⃗x, ⃗p, t) + f−(⃗x, ⃗p, t)]<br />
−fγ(⃗x, ⃗ ∫<br />
k, t)<br />
d 3 p W ⃗ E, ⃗ B<br />
e + e −( ⃗ k, ⃗p) ,<br />
which for ω < ω0 have to be coupled to Maxwell’s equa-<br />
ti<strong>on</strong>s with the current<br />
∫<br />
⃗j(⃗x, t) = e d 3 p ⃗v [f+(⃗x, ⃗p, t) − f−(⃗x, ⃗p, t)] . (17)<br />
It has to be made sure that radiati<strong>on</strong> stored <strong>in</strong> ⃗ E, ⃗ B and<br />
fγ does not lead to double count<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> radiati<strong>on</strong>, hence the<br />
ω > ω0 threshold.<br />
S<str<strong>on</strong>g>of</str<strong>on</strong>g>t phot<strong>on</strong>s are described classically by means <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
Maxwell equati<strong>on</strong>s for the electromagnetic fields<br />
1 ∂<br />
c<br />
⃗ E<br />
∂t = ∇ × ⃗ B − 4π ∑<br />
∫<br />
⃗p±<br />
q±<br />
c mγ f±d⃗p , (18)<br />
− 1 ∂<br />
c<br />
⃗ B<br />
∂t = ∇ × ⃗ E , (19)<br />
∇ · ⃗ E = −4π ∑ ∫<br />
q± f±d⃗p , (20)<br />
∇ · ⃗ B = 0 . (21)<br />
The complete set <str<strong>on</strong>g>of</str<strong>on</strong>g> equati<strong>on</strong>s with the discussed restricti<strong>on</strong>s<br />
can be solved for <strong>in</strong>tensities I ≥ 10 24 W/cm 2 , which<br />
are planned for ELI. For illustrati<strong>on</strong> a simulati<strong>on</strong> setup is<br />
shown <strong>in</strong> Figure 1, where the cascade is <strong>in</strong>itiated by an electr<strong>on</strong><br />
beam <str<strong>on</strong>g>of</str<strong>on</strong>g> energy ε = 600 MeV collid<strong>in</strong>g with a focused<br />
laser field <str<strong>on</strong>g>of</str<strong>on</strong>g> two circularly polarized counter-propagat<strong>in</strong>g<br />
Gaussian laser beams. The observed multiplicity is about<br />
2.5.<br />
EVENT GENERATOR<br />
To solve the transport equati<strong>on</strong>s we make use <str<strong>on</strong>g>of</str<strong>on</strong>g> an extended<br />
versi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Particle-In-Cell (PIC) method [6].<br />
The PIC method samples the phase space with a f<strong>in</strong>ite number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles and proceeds by <strong>in</strong>tegrat<strong>in</strong>g k<strong>in</strong>etic<br />
equati<strong>on</strong>s <strong>in</strong> time by advanc<strong>in</strong>g quasi-particles al<strong>on</strong>g their<br />
characteristics with<strong>in</strong> phase space.<br />
We trace the moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-electr<strong>on</strong>s and quasipositr<strong>on</strong>s<br />
<strong>in</strong> momentum and coord<strong>in</strong>ate space, whereas for<br />
hard phot<strong>on</strong>s we utilize the ray trac<strong>in</strong>g approximati<strong>on</strong>. Our<br />
numerical algorithm works as follows: At each time step<br />
tn = n ∆t we advance the positi<strong>on</strong>s and momenta <str<strong>on</strong>g>of</str<strong>on</strong>g> all<br />
the particles present <strong>in</strong> the simulati<strong>on</strong> box by solv<strong>in</strong>g their<br />
equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong>. With the help <str<strong>on</strong>g>of</str<strong>on</strong>g> the event generator<br />
we check if an electr<strong>on</strong> or a positr<strong>on</strong> is go<strong>in</strong>g to emit a phot<strong>on</strong><br />
between tn < t < tn + ∆t and if any <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong>s<br />
present is go<strong>in</strong>g to produce a pair. With the help <str<strong>on</strong>g>of</str<strong>on</strong>g> ⃗p n and<br />
⃗E n at tn the efficiency parameter χ n is evaluated. Next<br />
the total transiti<strong>on</strong> probability Wγ is computed. In order<br />
to remove the <strong>in</strong>frared divergencies we restrict the <strong>in</strong>tegrati<strong>on</strong>s<br />
over phot<strong>on</strong>s to k0 > ε <strong>in</strong> the transport equati<strong>on</strong>s.<br />
We assume that electr<strong>on</strong>s or positr<strong>on</strong>s present <strong>in</strong> the simulati<strong>on</strong><br />
box emit a phot<strong>on</strong> between tn < t < tn + ∆t if<br />
Wγ ∆t < r, where 0 ≤ r < 1 is a uniformly distributed<br />
random number. If the above <strong>in</strong>equality is fulfilled the energy<br />
εγ = ¯hωγ <str<strong>on</strong>g>of</str<strong>on</strong>g> the emitted phot<strong>on</strong> is obta<strong>in</strong>ed as a root
<str<strong>on</strong>g>of</str<strong>on</strong>g> the sampl<strong>in</strong>g equati<strong>on</strong><br />
1<br />
εγ ∫<br />
Wγ<br />
ε0<br />
dWγ(εγ)<br />
dεγ = r<br />
dεγ<br />
′ , (22)<br />
where 0 ≤ r ′ < 1 is a sec<strong>on</strong>d random number. The directi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> propagati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the newly emitted phot<strong>on</strong> is assumed<br />
to be parallel to the ⃗p n <str<strong>on</strong>g>of</str<strong>on</strong>g> the emitt<strong>in</strong>g electr<strong>on</strong> or positr<strong>on</strong>.<br />
Their momenta after emissi<strong>on</strong> are found from the c<strong>on</strong>servati<strong>on</strong><br />
laws. For pair creati<strong>on</strong> the event generator works<br />
the same way apart from the regularizati<strong>on</strong> parameter ε0,<br />
which is not needed <strong>in</strong> this case.<br />
ADAPTIVE MANAGEMENT METHOD<br />
FOR QUASIPARTICLES<br />
In ultra-<strong>in</strong>tense laser fields the number <str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-elements<br />
Npart cannot be c<strong>on</strong>sidered c<strong>on</strong>stant. The evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> particle<br />
number depends critically <strong>on</strong> the laser and seed particle<br />
parameters. Seed particles can be electr<strong>on</strong>s, positr<strong>on</strong>s,<br />
and phot<strong>on</strong>s. The c<strong>on</strong>trol parameters are the quantum efficiency<br />
parameters χ and κ. The qualitative threshold for<br />
the <strong>on</strong>set <str<strong>on</strong>g>of</str<strong>on</strong>g> cascad<strong>in</strong>g is χ ∼ κ ∼ 1. At those c<strong>on</strong>diti<strong>on</strong>s<br />
the number <str<strong>on</strong>g>of</str<strong>on</strong>g> particles grows exp<strong>on</strong>entially. In<br />
order to handle exp<strong>on</strong>ential grows with a limited number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles we implement a new techniques <str<strong>on</strong>g>of</str<strong>on</strong>g> ref<strong>in</strong>ement/coarsen<strong>in</strong>g<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> phase space represented by quasiparticles.<br />
Our approach to keep phase space adequately resolved<br />
c<strong>on</strong>sists <str<strong>on</strong>g>of</str<strong>on</strong>g> two basic algorithms for particle splitt<strong>in</strong>g<br />
and merg<strong>in</strong>g.<br />
Splitt<strong>in</strong>g<br />
Splitt<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles is straightforward. The splitted<br />
quasi-particle’s weight, momentum and energy are the<br />
sum <str<strong>on</strong>g>of</str<strong>on</strong>g> the same quantities <str<strong>on</strong>g>of</str<strong>on</strong>g> the result<strong>in</strong>g smaller quasiparticles.<br />
Particle splitt<strong>in</strong>g is used to <strong>in</strong>crease the number <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
quasi-particles per computati<strong>on</strong>al cell. Quasi-particles are<br />
split <strong>in</strong>to two equal smaller quasi-particles. The momenta<br />
and/or positi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the produced quasi-particles should be<br />
slightly different from each other <strong>in</strong> order to reach better<br />
statistical sampl<strong>in</strong>g. This approach is probably not the most<br />
accurate <strong>on</strong>e but c<strong>on</strong>serves exactly all moments <str<strong>on</strong>g>of</str<strong>on</strong>g> the distributi<strong>on</strong><br />
functi<strong>on</strong>.<br />
Merg<strong>in</strong>g<br />
The merg<strong>in</strong>g algorithm is somewhat more complicated.<br />
Roughly speak<strong>in</strong>g we have to replace the orig<strong>in</strong>al set <str<strong>on</strong>g>of</str<strong>on</strong>g> N<br />
quasi-particles by a new set <str<strong>on</strong>g>of</str<strong>on</strong>g> M particles, where N > M.<br />
Under this transformati<strong>on</strong> the basic properties <str<strong>on</strong>g>of</str<strong>on</strong>g> the distributi<strong>on</strong><br />
functi<strong>on</strong> have to be kept, i.e. f(N) ≃ f(M). This<br />
can be achieved by mak<strong>in</strong>g use <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>stra<strong>in</strong>ts def<strong>in</strong>ed<br />
by the c<strong>on</strong>servati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moments. Let us assume that the<br />
distributi<strong>on</strong> functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s, positr<strong>on</strong>s or phot<strong>on</strong>s<br />
is represented by a certa<strong>in</strong> number <str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles<br />
f(px, py, pz, t) = ∑<br />
wi δ(⃗p − ⃗pi(t)) S(⃗r − ⃗ri(t)) , (23)<br />
i<br />
where wi is the weight <str<strong>on</strong>g>of</str<strong>on</strong>g> the quasi-particle, which can<br />
change <strong>in</strong> the course <str<strong>on</strong>g>of</str<strong>on</strong>g> the simulati<strong>on</strong> and S is the shape<br />
factor <str<strong>on</strong>g>of</str<strong>on</strong>g> the quasi-particle <strong>in</strong> coord<strong>in</strong>ate space [6]. Tak<strong>in</strong>g<br />
moments <str<strong>on</strong>g>of</str<strong>on</strong>g> this distributi<strong>on</strong> functi<strong>on</strong>, <strong>on</strong>e can obta<strong>in</strong> the<br />
weight or mass, the momentum, the energy, and so <strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the plasma. Let us denote the weight <str<strong>on</strong>g>of</str<strong>on</strong>g> N quasi-elements<br />
<strong>in</strong> a cluster <str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles to be merged by W, the momentum<br />
by ⃗ P , and the energy by ϵ. Then the c<strong>on</strong>servati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> mass implies<br />
N∑<br />
wi = W = c<strong>on</strong>st .<br />
i=1<br />
Momentum c<strong>on</strong>servati<strong>on</strong> is implied by<br />
N∑<br />
wi ⃗ui = ⃗ P = c<strong>on</strong>st<br />
i=1<br />
and energy is c<strong>on</strong>served if<br />
N∑<br />
i=1<br />
wi<br />
√<br />
1 + u2 i = ε = c<strong>on</strong>st .<br />
To keep W , ⃗ P , and ε c<strong>on</strong>served the group <str<strong>on</strong>g>of</str<strong>on</strong>g> N orig<strong>in</strong>al<br />
quasi-particles has to be merged to at least two rema<strong>in</strong><strong>in</strong>g<br />
quasi-particles as is shown <strong>in</strong> Figure 2. The weight<br />
w 1/2 and momentum ⃗p 1/2 <str<strong>on</strong>g>of</str<strong>on</strong>g> the two new quasi-elements<br />
is given by<br />
w1,2 = W<br />
2 , ⃗p1,2 = ⃗ P<br />
2 ± ⃗ Q . (24)<br />
In 2D the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the vector ⃗ Q can be def<strong>in</strong>ed as follows<br />
( )<br />
−Py<br />
⃗Q = |Q| , (25)<br />
Px<br />
where the quantity | ⃗ Q| can be obta<strong>in</strong>ed from the energy<br />
c<strong>on</strong>servati<strong>on</strong> c<strong>on</strong>diti<strong>on</strong> as<br />
| ⃗ Q| = 1√<br />
ε2 − P 2 − W 2 . (26)<br />
2<br />
Thus, <strong>in</strong> 2D momentum space the momenta for newly<br />
merged particles can be found exactly. In the 3D case the<br />
perpendicular comp<strong>on</strong>ent ⃗ Q lies <strong>in</strong> the plane perpendicular<br />
to ⃗ P , i.e. the number <str<strong>on</strong>g>of</str<strong>on</strong>g> possible directi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> ⃗ Q is <strong>in</strong>f<strong>in</strong>ite.<br />
In order to assign the vector directi<strong>on</strong> we suggest to use the<br />
local anisotropy <str<strong>on</strong>g>of</str<strong>on</strong>g> the phase space represented by the cluster<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles. Mak<strong>in</strong>g use <str<strong>on</strong>g>of</str<strong>on</strong>g> the mean variati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the distributi<strong>on</strong> functi<strong>on</strong> we can choose the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
⃗Q. The test results <str<strong>on</strong>g>of</str<strong>on</strong>g> the code show that even for the case<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> exp<strong>on</strong>ential growth <str<strong>on</strong>g>of</str<strong>on</strong>g> the number <str<strong>on</strong>g>of</str<strong>on</strong>g> the real particles<br />
due to cascad<strong>in</strong>g, particle resampl<strong>in</strong>g based <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>servati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the local phase space topology can significantly reduce<br />
the number <str<strong>on</strong>g>of</str<strong>on</strong>g> computati<strong>on</strong>al particles while reta<strong>in</strong><strong>in</strong>g<br />
the features <str<strong>on</strong>g>of</str<strong>on</strong>g> the distributi<strong>on</strong> functi<strong>on</strong> and keep<strong>in</strong>g mass,<br />
momentum, and energy c<strong>on</strong>served.
Figure 2: Cluster<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles with adaptive<br />
weights (color-coded) <strong>in</strong> phase space.<br />
Figure 3: Development <str<strong>on</strong>g>of</str<strong>on</strong>g> the e ± γ cascade <strong>in</strong> a rotat<strong>in</strong>g<br />
electric field. The number <str<strong>on</strong>g>of</str<strong>on</strong>g> created pairs as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
time obta<strong>in</strong>ed from an unref<strong>in</strong>ed simulati<strong>on</strong> (green) [3] and<br />
<strong>on</strong>e with phase space ref<strong>in</strong>ement (blue).<br />
The two basic algorithms <str<strong>on</strong>g>of</str<strong>on</strong>g> particle splitt<strong>in</strong>g and merg<strong>in</strong>g<br />
are used to c<strong>on</strong>trol the number <str<strong>on</strong>g>of</str<strong>on</strong>g> computati<strong>on</strong>al quasiparticles.<br />
The results are compared <strong>in</strong> Figure 3, where the<br />
number <str<strong>on</strong>g>of</str<strong>on</strong>g> the real particles represented by a f<strong>in</strong>ite number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-particles agrees with the results <str<strong>on</strong>g>of</str<strong>on</strong>g> an unref<strong>in</strong>ed<br />
simulati<strong>on</strong> reported <strong>in</strong> [3].<br />
SUMMARY AND CONCLUSIONS<br />
We have presented the current status <str<strong>on</strong>g>of</str<strong>on</strong>g> a new simulati<strong>on</strong><br />
framework for full scale simulati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> QED cascad<strong>in</strong>g<br />
<strong>in</strong> str<strong>on</strong>g laser fields. A novel feature <str<strong>on</strong>g>of</str<strong>on</strong>g> cascades <strong>in</strong><br />
an ultra-str<strong>on</strong>g laser field is the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s,<br />
positr<strong>on</strong>s, and energetic (hundreds <str<strong>on</strong>g>of</str<strong>on</strong>g> MeV) phot<strong>on</strong>s at <strong>in</strong>tensity<br />
levels far below the Sch<strong>in</strong>ger limit. S<strong>in</strong>ce the str<strong>on</strong>g<br />
external field is capable <str<strong>on</strong>g>of</str<strong>on</strong>g> accelerat<strong>in</strong>g charged particles to<br />
ultra-relativistic energies cascades can be triggered by <strong>in</strong>i-<br />
tially slow electr<strong>on</strong>s and positr<strong>on</strong>s that are <strong>in</strong>jected <strong>in</strong>to the<br />
str<strong>on</strong>g field regi<strong>on</strong>. The appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> cascades raises pr<strong>in</strong>cipal<br />
questi<strong>on</strong> about the accessibility <str<strong>on</strong>g>of</str<strong>on</strong>g> higher laser fields<br />
s<strong>in</strong>ce the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles can deplete the<br />
laser field. This questi<strong>on</strong> requires further <strong>in</strong>vestigati<strong>on</strong> by<br />
tak<strong>in</strong>g exact energy c<strong>on</strong>servati<strong>on</strong> and the reverse processes<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> pair annihilati<strong>on</strong> <strong>in</strong>to account, which can both limit the<br />
growth <str<strong>on</strong>g>of</str<strong>on</strong>g> the number <str<strong>on</strong>g>of</str<strong>on</strong>g> particles. We also plan to extend<br />
our model further by the <strong>in</strong>corporati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the elastic collisi<strong>on</strong>s<br />
between particles and Compt<strong>on</strong> scatter<strong>in</strong>g, which<br />
may be important for the design <str<strong>on</strong>g>of</str<strong>on</strong>g> new γ-sources.<br />
REFERENCES<br />
[1] Drake, P., High-energy-density physics, <strong>Physics</strong> Today 63, 28<br />
(2010).<br />
[2] http://www.extreme-light-<strong>in</strong>frastructure.eu/<br />
[3] Elk<strong>in</strong>a, N. V., Fedotov, A. M., Kostyukov, I. Y., Legkov,<br />
M. V., Narozhny, N. B., Nerush, E. N., Ruhl, H., QED cascades<br />
<strong>in</strong>duced by circularly polarized laser fields, accepted<br />
for publicati<strong>on</strong> by Phys. Rev. ST. Accel. Beams, 2011 and<br />
http://arxiv.org/abs/1010.4528.<br />
[4] Nikishov, A. I., Ritus, V. I., Sov. Phys. JETP 25, 1135 (1964).<br />
[5] Lifshitz, E. M., Pitaevskii, L. P., and Berestetskii, V. B.,<br />
Landau-Lifshitz Course <str<strong>on</strong>g>of</str<strong>on</strong>g> Theoretical <strong>Physics</strong>, Vol. 4: Quantum<br />
Electrodynamics, Reed Educati<strong>on</strong>al and Pr<str<strong>on</strong>g>of</str<strong>on</strong>g>essi<strong>on</strong>al<br />
Publish<strong>in</strong>g, Oxford, 1982.<br />
[6] Ruhl, H., Introducti<strong>on</strong> to Computati<strong>on</strong>al Methods <strong>in</strong> Many<br />
Body <strong>Physics</strong>, eds. M. B<strong>on</strong>itz and D. Semkat, R<strong>in</strong>t<strong>on</strong> Press,<br />
(2001).
Abstract<br />
SCHWINGER LIMIT ATTAINABILITY WITH EXTREME LIGHT ∗<br />
S. V. Bulanov † , T. Zh. Esirkepov, J. K. Koga, KPSI-JAEA, Kizugawa, Kyoto, Japan<br />
S. S. Bulanov, Univ. <str<strong>on</strong>g>of</str<strong>on</strong>g> California, Berkeley, USA<br />
A. Thomas, Univ. <str<strong>on</strong>g>of</str<strong>on</strong>g> Michigan, Ann Arbor, USA<br />
High <strong>in</strong>tensity collid<strong>in</strong>g laser pulses can create abundant<br />
electr<strong>on</strong>-positr<strong>on</strong> pair plasma. This process can prevent<br />
reach<strong>in</strong>g the critical field <str<strong>on</strong>g>of</str<strong>on</strong>g> Quantum Electrodynamics<br />
at which vacuum breakdown and polarizati<strong>on</strong> occur.<br />
C<strong>on</strong>sider<strong>in</strong>g the pairs are seeded by the Schw<strong>in</strong>ger mechanizm,<br />
it is shown that the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> radiati<strong>on</strong> fricti<strong>on</strong> and<br />
the electr<strong>on</strong>-positr<strong>on</strong> avalanche development <strong>in</strong> vacuum depend<br />
<strong>on</strong> the electromagnetic wave polarizati<strong>on</strong>. For circularly<br />
polarized collid<strong>in</strong>g pulses these effects dom<strong>in</strong>ate<br />
not <strong>on</strong>ly the particle moti<strong>on</strong> but also the evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
pulses. While for l<strong>in</strong>early polarized pulses, where the electr<strong>on</strong>s<br />
(positr<strong>on</strong>s) oscillate al<strong>on</strong>g the electric field, these effects<br />
are not as str<strong>on</strong>g. There is an apparent analogy <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
these cases with circular and l<strong>in</strong>ear electr<strong>on</strong> accelerators<br />
with the corresp<strong>on</strong>d<strong>in</strong>g c<strong>on</strong>stra<strong>in</strong><strong>in</strong>g and reduced roles <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
synchrotr<strong>on</strong> radiati<strong>on</strong> losses.<br />
INTRODUCTION<br />
The lasers nowadays provide <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the most powerful<br />
sources <str<strong>on</strong>g>of</str<strong>on</strong>g> electromagnetic (EM) radiati<strong>on</strong> under laboratory<br />
c<strong>on</strong>diti<strong>on</strong>s and thus <strong>in</strong>spire the fast grow<strong>in</strong>g area<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> high field science aimed at the explorati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> novel<br />
physical processes [1]. Reach<strong>in</strong>g <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
10 23 W/cm 2 and above will br<strong>in</strong>g us to experimentally unexplored<br />
regimes. At such <strong>in</strong>tensities the laser <strong>in</strong>teracti<strong>on</strong><br />
with matter becomes str<strong>on</strong>gly dissipative, due to efficient<br />
EM energy transformati<strong>on</strong> <strong>in</strong>to high energy gamma rays<br />
[1, 2]. These gamma-phot<strong>on</strong>s <strong>in</strong> the laser field may produce<br />
electr<strong>on</strong>-positr<strong>on</strong> pairs via the Breit-Wheeler process<br />
[3]. Then the pairs accelerated by the laser generate high<br />
energy gamma quanta and so <strong>on</strong> [4], and thus the c<strong>on</strong>diti<strong>on</strong>s<br />
for the avalanche type discharge are produced at the<br />
<strong>in</strong>tensity ≈ 10 25 W/cm 2 . The occurrence <str<strong>on</strong>g>of</str<strong>on</strong>g> such ”showers”<br />
was foreseen by Heisenberg and Euler [5]. In Ref.<br />
[6] a c<strong>on</strong>clusi<strong>on</strong> is made that depleti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser energy<br />
<strong>on</strong> the electr<strong>on</strong>-positr<strong>on</strong>-gamma-ray plasma (EPGP) creati<strong>on</strong><br />
could limit atta<strong>in</strong>able EM wave <strong>in</strong>tensity and could<br />
prevent approach<strong>in</strong>g the critical quantum electrodynamics<br />
(QED) field. This field [5, 7] is also called the Schw<strong>in</strong>ger<br />
field, ES = m 2 ec 3 /e¯h corresp<strong>on</strong>d<strong>in</strong>g to the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> ≈<br />
10 29 W/cm 2 .<br />
The particle-antiparticle pair creati<strong>on</strong> by the Schw<strong>in</strong>ger<br />
field cannot be described with<strong>in</strong> the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> perturbati<strong>on</strong><br />
theory and sheds light <strong>on</strong> the n<strong>on</strong>l<strong>in</strong>ear QED prop-<br />
∗ Work supported by the M<strong>in</strong>istry <str<strong>on</strong>g>of</str<strong>on</strong>g> Educati<strong>on</strong>, Culture, Sports, Science<br />
and Technology (MEXT) <str<strong>on</strong>g>of</str<strong>on</strong>g> Japan, Grant-<strong>in</strong>-Aid for Scientific Research,<br />
Project No. 20244065.<br />
† bulanov.sergei@jaea.go.jp<br />
erties <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum [8]. Understand<strong>in</strong>g the vacuum breakdown<br />
mechanisms is challeng<strong>in</strong>g for other n<strong>on</strong>l<strong>in</strong>ear quantum<br />
field theories [9] and for astrophysics [10]. Reach<strong>in</strong>g<br />
this field limit has been c<strong>on</strong>sidered as <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the most<br />
<strong>in</strong>trigu<strong>in</strong>g scientific problems. Dem<strong>on</strong>strati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the processes<br />
associated with the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>l<strong>in</strong>ear QED, such<br />
as vacuum polarizati<strong>on</strong> and vacuum electr<strong>on</strong>-positr<strong>on</strong> pair<br />
producti<strong>on</strong>, will be <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the ma<strong>in</strong> challenges for extreme<br />
high power laser physics [1, 11].<br />
Below we discuss the atta<strong>in</strong>ability <str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger field<br />
with high power lasers (see also Ref. [12]). We compare<br />
the role <str<strong>on</strong>g>of</str<strong>on</strong>g> radiati<strong>on</strong> dissipative effects <strong>in</strong> the moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
electr<strong>on</strong>s (and positr<strong>on</strong>s) produced via the Schw<strong>in</strong>ger effect<br />
and show their dependence <strong>on</strong> the EM wave polarizati<strong>on</strong>.<br />
3D EM FIELD CONFIGURATION<br />
Pair creati<strong>on</strong> is determ<strong>in</strong>ed by the Po<strong>in</strong>care <strong>in</strong>variants<br />
[13] F = (E 2 − B 2 )/2, G = (E · B) and requires the<br />
first <strong>in</strong>variant F be positive. This c<strong>on</strong>diti<strong>on</strong> can be fulfilled<br />
<strong>in</strong> the vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g> the ant<strong>in</strong>odes <str<strong>on</strong>g>of</str<strong>on</strong>g> collid<strong>in</strong>g EM waves,<br />
or/and <strong>in</strong> the c<strong>on</strong>figurati<strong>on</strong> formed by several focused EM<br />
pulses, [14]. This EM c<strong>on</strong>figurati<strong>on</strong> locally can be approximated<br />
by an oscillat<strong>in</strong>g TM mode with poloidal electric<br />
and toroidal magnetic fields. The magnetic field <strong>in</strong> spherical<br />
coord<strong>in</strong>ates R, θ, ϕ is given by<br />
a0 s<strong>in</strong>(ω0t)<br />
B(R, θ) = eϕ<br />
(8πR) 1/2 Jn+1/2(k0R)L l n(cos θ), (1)<br />
where a0 = eE0/mecω0, k0 = ω0/c, Jν(x) and L l n(x) are<br />
the Bessel functi<strong>on</strong> and associated Legendre polnomials.<br />
The electric field is equal to E = ik0(∇ × B). In cyl<strong>in</strong>drical<br />
coord<strong>in</strong>ates r, ϕ, z the z-comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric<br />
field oscillates <strong>in</strong> vertical directi<strong>on</strong>, ∼ a0 cos(ω0t), the ϕcomp<strong>on</strong>ent<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field vanishes <strong>on</strong> the axis be<strong>in</strong>g<br />
l<strong>in</strong>early proporti<strong>on</strong>al to the radius, ∼ (a0/8)k0r s<strong>in</strong>(ω0t),<br />
and the radial comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field is relatively<br />
small, ∼ 0.1a0k 2 0rz cos(ω0t). The EM field and first<br />
Po<strong>in</strong>care <strong>in</strong>variant F(r, z) are shown <strong>in</strong> Fig. 1. We see<br />
that the EM field is localized <strong>in</strong> a regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> width less than<br />
the laser wavelength, λ0 = 2π/k0. The sec<strong>on</strong>d <strong>in</strong>variant is<br />
equal to zero, G = 0.<br />
Probability <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong><br />
Us<strong>in</strong>g expressi<strong>on</strong> for the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>-positr<strong>on</strong><br />
pair creati<strong>on</strong> [5, 7] and expand<strong>in</strong>g F(r, z) <strong>in</strong> the vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
its maximum we f<strong>in</strong>d that the pairs are created <strong>in</strong> a small<br />
4-volume near the electric field maximum with the charac-
10<br />
k 0z<br />
0<br />
-10<br />
0<br />
k 0r<br />
a) 1<br />
10<br />
0<br />
-10<br />
0<br />
F/a0 2<br />
Figure 1: a) The vector field shows r- and z-comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the poloidal electric field <strong>in</strong> the r, z plane for the TM mode.<br />
The color density shows the toroidal magnetic field distributi<strong>on</strong>,<br />
Bϕ(r, z). b) The first Po<strong>in</strong>care <strong>in</strong>variant F(r, z).<br />
teristic size<br />
1<br />
0.5<br />
πr 2 0z0t0 ≈ 53/2 λ 4 0<br />
16π 5 c<br />
k 0r<br />
( a0<br />
aS<br />
10<br />
k 0z<br />
10<br />
b)<br />
) 2<br />
. (2)<br />
Here, we <strong>in</strong>troduce aS = eES/meω0c = mec 2 /¯hω0. Integrat<strong>in</strong>g<br />
over the 4-volume the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair creati<strong>on</strong><br />
[15] we obta<strong>in</strong> the number <str<strong>on</strong>g>of</str<strong>on</strong>g> pairs produced per wave<br />
period,<br />
53/2 4π3 a4 (<br />
0 exp − πaS<br />
)<br />
, (3)<br />
a0<br />
i. e. the first pairs can be observed for an <strong>on</strong>e-micr<strong>on</strong> wavelength<br />
laser <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> 2×10 27 W/cm 2 , which<br />
corresp<strong>on</strong>ds to a0/aS ≈ 0.075, i.e. a characteristic size, r0,<br />
approximately equal to 0.04λ0.<br />
Electr<strong>on</strong> orbit near magnetic null po<strong>in</strong>t<br />
In the regi<strong>on</strong>, where the magnetic field vanishes, the<br />
electr<strong>on</strong> oscillates al<strong>on</strong>g the electric field. For an electr<strong>on</strong><br />
generated at small but f<strong>in</strong>ite radius r0 ≪ λ0 the magnetic<br />
field bends its trajectory outwards. By solv<strong>in</strong>g the electr<strong>on</strong><br />
equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> l<strong>in</strong>earized about the soluti<strong>on</strong> corresp<strong>on</strong>d<strong>in</strong>g<br />
to ultrarelativistic electr<strong>on</strong> oscillati<strong>on</strong>s <strong>in</strong> the zdirecti<strong>on</strong>,<br />
i.e. a0ω0t ≫ 1, we can f<strong>in</strong>d the electr<strong>on</strong> trajectories,<br />
which are described <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> modified Bessel<br />
functi<strong>on</strong>s, Iν(x),:<br />
pz(t) = meca0ω0t, (4)<br />
pr(t) = mec a0k0r0ω0t<br />
23/2 (<br />
ω0t<br />
I1<br />
23/2 )<br />
,<br />
r(t) =<br />
(5)<br />
a0r0<br />
(<br />
ω0t<br />
I1<br />
23/2 23/2 )<br />
+<br />
[ (<br />
a0r0ω0t ω0t<br />
I0<br />
16 23/2 ) (<br />
ω0t<br />
+ I2<br />
23/2 )]<br />
. (6)<br />
Here r0 is the <strong>in</strong>itial electr<strong>on</strong> coord<strong>in</strong>ate, which is <str<strong>on</strong>g>of</str<strong>on</strong>g>the<br />
order <str<strong>on</strong>g>of</str<strong>on</strong>g> λ0(5a0/4π 3 aS) 1/2 . The <strong>in</strong>stability growth rate is<br />
approximately equal to half the EM field frequency, ω0/2,<br />
pr ≈<br />
( a0<br />
8<br />
)<br />
k0r0(ω0t) 2 , (7)<br />
i. e. the electr<strong>on</strong> rema<strong>in</strong>s <strong>in</strong> the close vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g> the zeromagnetic<br />
field regi<strong>on</strong> leav<strong>in</strong>g it al<strong>on</strong>g the z-directi<strong>on</strong>.<br />
EM RADIATION EMISSION<br />
Frequency spectrum<br />
The electr<strong>on</strong> oscillat<strong>in</strong>g al<strong>on</strong>g the electric field emits<br />
the high frequency EM radiati<strong>on</strong> with the power ≈<br />
(2πre/3λ0)ωemec 2 γ 2 e proporti<strong>on</strong>al to the square <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong><br />
energy. In order to f<strong>in</strong>d the angular distributi<strong>on</strong> and<br />
frequency spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> <strong>in</strong> this case we should<br />
take <strong>in</strong>to account its dependence <strong>on</strong> the retarded time:<br />
t ′ = t − n · r(t)/c. Here n is the unit vector <strong>in</strong> the directi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> observati<strong>on</strong> and r(t) is the electr<strong>on</strong> coord<strong>in</strong>ate.<br />
Introduc<strong>in</strong>g the angle η between vectors n and r(t),<br />
n · r(t) = |r(t)| cos η, (8)<br />
we can f<strong>in</strong>d that <strong>in</strong> the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> oscillati<strong>on</strong>s,<br />
η = 0, the radiati<strong>on</strong> <strong>in</strong>tensity vanishes. The maxima <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
radiated power corresp<strong>on</strong>d to the angle ηm, for large γe,<br />
<strong>in</strong>versely proporti<strong>on</strong>al to the particle energy: ηm ≈ 1/2γe.<br />
Fourier comp<strong>on</strong>ents for 4-vector potential <str<strong>on</strong>g>of</str<strong>on</strong>g> the EM<br />
field accord<strong>in</strong>g to Ref. [13] are<br />
A µ (ω) = e<br />
R<br />
∫+∞<br />
−∞<br />
u µ<br />
c exp<br />
{ [<br />
iω<br />
t − 1<br />
n · r(t)<br />
c<br />
where u µ = p µ /meγe is the four-velocity and<br />
]}<br />
dt, (9)<br />
r(t) = ey(c/ω0)Arcs<strong>in</strong> [βm s<strong>in</strong>(ω0t)] , (10)<br />
with βm = a0/ √ 1 + a2 0 is the electr<strong>on</strong> coord<strong>in</strong>ate.<br />
Expand<strong>in</strong>g the phase <strong>in</strong> expressi<strong>on</strong> (9),<br />
{ ( )<br />
}<br />
cos θ<br />
Φ(t) = ω t − Arcs<strong>in</strong> [βm s<strong>in</strong>(ω0t)] , (11)<br />
ω0<br />
<strong>on</strong> small parameters, γ −1<br />
e,m and ω0t, for θ = θm ≈ 1/2γe,m,<br />
we obta<strong>in</strong><br />
Φ(t) ≈ ω<br />
[<br />
(1 − βm cos θ) t +<br />
( βm cos θ<br />
6γ 2 e,mω0<br />
)<br />
(ω0t) 3<br />
]<br />
.<br />
(12)<br />
Us<strong>in</strong>g the Airy <strong>in</strong>tegral, we can f<strong>in</strong>d the y-comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the 4-vector potential <str<strong>on</strong>g>of</str<strong>on</strong>g> EM field (9). Tak<strong>in</strong>g <strong>in</strong>to account<br />
smallness <str<strong>on</strong>g>of</str<strong>on</strong>g> the angle θ ≈ θm, θ = ψ/2γe,m ≪ 1, and<br />
present<strong>in</strong>g cos θ <strong>in</strong> the form cos θ ≈ 1 − ψ2 /2(2γe.m) 2 we<br />
can obta<strong>in</strong> the radiati<strong>on</strong> power density pL(ω, ψ).<br />
The power emitted by the electr<strong>on</strong> is given by the <strong>in</strong>tegral<br />
∫+∞<br />
pL(ω) = pL(ω, ψ)dψ. (13)<br />
−∞
To f<strong>in</strong>d it we use the <strong>in</strong>tegral calculated <strong>in</strong> Ref [17] and<br />
obta<strong>in</strong><br />
dpL 16πre<br />
=<br />
dω 33/2 mec<br />
λ0<br />
2<br />
( ) 2 ( )<br />
ω 2 ω<br />
FL<br />
, (14)<br />
ω0 3<br />
where<br />
∫<br />
FL(z) = z<br />
z<br />
ω0γ 2 e,m<br />
∞<br />
K 5/3(η) dη − zK 2/3 (z). (15)<br />
Maximum frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> emitted by oscillat<strong>in</strong>g<br />
electr<strong>on</strong> corresp<strong>on</strong>ds to ωm ≈ 0.21ω0γ 2 e,m. As we see,<br />
compared to the case <str<strong>on</strong>g>of</str<strong>on</strong>g> circular polarizati<strong>on</strong>, the l<strong>in</strong>early<br />
oscillat<strong>in</strong>g electr<strong>on</strong> emissi<strong>on</strong> is weaker with the phot<strong>on</strong> frequency<br />
<strong>in</strong> a factor γe,m smaller.<br />
Radiati<strong>on</strong> fricti<strong>on</strong> effects<br />
To take <strong>in</strong>to account the radiati<strong>on</strong> fricti<strong>on</strong> we use equati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a radiat<strong>in</strong>g electr<strong>on</strong> [13]. We can estimate<br />
the regime where the radiati<strong>on</strong> fricti<strong>on</strong> can become<br />
relatively large by compar<strong>in</strong>g the energy losses with the<br />
maximal energy ga<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong> accelerated by the<br />
electric field, E ˙(+)<br />
≈ ω0 mec2a0, i.e. ω0 mec2a0 =<br />
εradω0mec2 γ2 e , where<br />
εrad = 4πre/3λ0, (16)<br />
with re = e2 /mec2 - classical electr<strong>on</strong> radius. As is apparent,<br />
although an electr<strong>on</strong> mov<strong>in</strong>g al<strong>on</strong>g the oscillat<strong>in</strong>g<br />
electric field loses energy, radiati<strong>on</strong> fricti<strong>on</strong> effects may be-<br />
come important <strong>on</strong>ly at a0 = 2ε −1<br />
rad<br />
, i.e. at the electric field<br />
E0 = 3m 2 ec 4 /e 3 , which is <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> the critical electric<br />
field <str<strong>on</strong>g>of</str<strong>on</strong>g> classical electrodynamics (see also Ref. [15]).<br />
This is 137 times larger than the field ES.<br />
DIMENSIONLESS PARAMETERS IN QED<br />
In QED the charged particle <strong>in</strong>teracti<strong>on</strong> with EM fields is<br />
determ<strong>in</strong>ed by relativistically and gauge <strong>in</strong>variant parameters<br />
[18] χe = [(Fµνpν) 2 ] 1/2 /mecES. The parameter, χe,<br />
characterizes the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the gamma-phot<strong>on</strong> emissi<strong>on</strong><br />
by the electr<strong>on</strong> with Lorentz factor γe. It is <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
order <str<strong>on</strong>g>of</str<strong>on</strong>g> the ratio E/ES <strong>in</strong> the electr<strong>on</strong> rest frame <str<strong>on</strong>g>of</str<strong>on</strong>g> reference.<br />
Another parameter, χγ = [(Fµν¯hkν) 2 ] 1/2 /mecES,<br />
is similar to χe with the phot<strong>on</strong> 4-momentum, ¯hkµ, <strong>in</strong>stead<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> 4-momentum, pµ. It characterizes the probability<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong> due to the collisi<strong>on</strong><br />
between the high energy phot<strong>on</strong> and EM field. QED<br />
effects come <strong>in</strong>to play when the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> a phot<strong>on</strong> emitted<br />
by an electr<strong>on</strong> becomes comparable to the electr<strong>on</strong> k<strong>in</strong>etic<br />
energy, i.e., for ¯hωm = mec 2 γe. In a l<strong>in</strong>early polarized<br />
oscillat<strong>in</strong>g electric field the maximum frequency <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
emitted phot<strong>on</strong>s, ωm, is proporti<strong>on</strong>al γ 2 0, and, therefore,<br />
quantum effects should be <strong>in</strong>corporated <strong>in</strong>to the theoretical<br />
descripti<strong>on</strong> at the electr<strong>on</strong> energy corresp<strong>on</strong>d<strong>in</strong>g to the<br />
gamma-factor γ L Q = mec 2 /0.21 ¯hω0, which is above the<br />
Schw<strong>in</strong>ger limit. We see that <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> moti<strong>on</strong><br />
<strong>in</strong> a l<strong>in</strong>early polarized oscillat<strong>in</strong>g electric field neither<br />
radiati<strong>on</strong> fricti<strong>on</strong> nor quantum recoil effects are important.<br />
THRESHOLD OF AVALANCHE<br />
DISCHARGE<br />
Reach<strong>in</strong>g the threshold <str<strong>on</strong>g>of</str<strong>on</strong>g> an avalanche type discharge<br />
with EPGP generati<strong>on</strong> discussed <strong>in</strong> Refs. [4, 6] requires<br />
high enough values <str<strong>on</strong>g>of</str<strong>on</strong>g> the parameters χe and χγ def<strong>in</strong>ed<br />
above because for χγ ≪ 1 the rate <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair creati<strong>on</strong> is<br />
exp<strong>on</strong>entially small [19],<br />
( 2 m<br />
W (χγ) ≈ α<br />
ec4 ) (<br />
χγ exp − 8<br />
)<br />
. (17)<br />
3χγ<br />
¯h 2 ωγ<br />
In the limit χγ ≫ 1 the pair creati<strong>on</strong> rate is given by<br />
( 2 m<br />
W (χγ) ≈ α<br />
ec4 )<br />
(χγ) 2/3<br />
¯h 2 ωγ<br />
(18)<br />
(for details see Ref. [18]). Here ¯hωγ is the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
phot<strong>on</strong> which creates an electr<strong>on</strong>-positr<strong>on</strong> pair.<br />
S<strong>in</strong>ce for γe ≥ γQ the phot<strong>on</strong> is emitted by the electr<strong>on</strong><br />
(positr<strong>on</strong>) <strong>in</strong> a narrow angle almost parallel to the electr<strong>on</strong><br />
momentum with the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> energy,<br />
the parameters χe and χγ are approximately equal<br />
to each other. The parameter χe can be expressed via the<br />
electric and magnetic field as (see Ref. [18])<br />
χ 2 (<br />
e =<br />
E<br />
γe<br />
ES<br />
+ p × B<br />
mecES<br />
) 2<br />
( ) 2<br />
p · E<br />
−<br />
. (19)<br />
mecES<br />
In order to f<strong>in</strong>d the threshold for the avalanche development<br />
we need to estimate the QED parameter χe. The c<strong>on</strong>diti<strong>on</strong><br />
for avalanche development corresp<strong>on</strong>d<strong>in</strong>g to the parameter<br />
χe should become <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> unity with<strong>in</strong> <strong>on</strong>e<br />
tenth <str<strong>on</strong>g>of</str<strong>on</strong>g> the EM field period (e.g. see Ref. [6]). Due to the<br />
trajectory bend<strong>in</strong>g by the magnetic field the electr<strong>on</strong> transverse<br />
momentum changes as p⊥ ≈ (a0/16)k0r0(ω0t) 2 ,<br />
where k0r0 = (2.5a0/πas) 1/2 , Eq. (2). Assum<strong>in</strong>g ω0t<br />
to be equal to 0.1 π, we obta<strong>in</strong> from Eq. (19) that χe becomes<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> unity, i.e. the avalanche can start, at<br />
a0/aS ≈ 0.105, which corresp<strong>on</strong>ds to the laser <strong>in</strong>tensity<br />
4 × 10 27 W/cm 2 . The radiati<strong>on</strong> losses <strong>in</strong> this limit can be<br />
described as the synchrotr<strong>on</strong> losses <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong> with the<br />
energy ≈ mec 2 mov<strong>in</strong>g <strong>in</strong> the magnetic field a0(k0r0)/8.<br />
Us<strong>in</strong>g formulae for synchrotr<strong>on</strong> radiati<strong>on</strong> [13], it is easy<br />
to show that they do not become significant until a0 ≈<br />
5 × 10 4 . At that limit the Schw<strong>in</strong>ger mechanism provides<br />
approximately 5 × 10 5 pairs per <strong>on</strong>e-period.<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> two collid<strong>in</strong>g circularly polarized EM<br />
waves the result<strong>in</strong>g electric field rotates with frequency ω0<br />
be<strong>in</strong>g c<strong>on</strong>stant <strong>in</strong> magnitude. The power emitted by the<br />
electr<strong>on</strong> is ≈ εradω0mec 2 γ 4 e. This is a factor <str<strong>on</strong>g>of</str<strong>on</strong>g> γ 2 e larger<br />
than <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear polarizati<strong>on</strong>. The properties <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
radiati<strong>on</strong> emitted by rotat<strong>in</strong>g electr<strong>on</strong> are well known from<br />
the theory <str<strong>on</strong>g>of</str<strong>on</strong>g> synchrotr<strong>on</strong> radiati<strong>on</strong> [13, 15] and from Ref.<br />
[16]. In the limit γe ≫ 1 the emitted power is proporti<strong>on</strong>al<br />
to the fourth power <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> energy. The radiati<strong>on</strong><br />
is directed almost al<strong>on</strong>g the electr<strong>on</strong> momentum be<strong>in</strong>g localized<br />
with<strong>in</strong> the angle <strong>in</strong>versely proporti<strong>on</strong>al to the electr<strong>on</strong><br />
energy: δη ≈ 1/γe. The frequency spectrum given
y the well known expressi<strong>on</strong> [13] has a maximum frequency,<br />
ωm = 0.29ω0γ 3 e , proporti<strong>on</strong>al to the cube <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electr<strong>on</strong> energy. This is a factor <str<strong>on</strong>g>of</str<strong>on</strong>g> γe larger than <strong>in</strong> the<br />
case <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>ear polarizati<strong>on</strong>. For the electr<strong>on</strong> rotat<strong>in</strong>g <strong>in</strong> the<br />
circularly polarized collid<strong>in</strong>g EM waves the emitted power<br />
becomes equal to the maximal energy ga<strong>in</strong> at the field amplitude<br />
a0 = arad = ε −1/3<br />
rad . For the laser wavelength<br />
λ0 = 0.8 µm εrad = 2.2 × 10−8 . The normalized amplitude<br />
arad is ≈ 400 corresp<strong>on</strong>d<strong>in</strong>g to the laser <strong>in</strong>tensity<br />
Irad = 4.5 × 1023W / cm2 .<br />
We represent the electric field and the electr<strong>on</strong> momentum<br />
<strong>in</strong> the complex form:<br />
and<br />
E = Ey + iEz = E0 exp ( −iω0t) (20)<br />
p = py + ipz = P exp ( −i(ω0t − φ)) , (21)<br />
where φ is the phase equal to the angle between the electric<br />
field vector and the electr<strong>on</strong> momentum. In the stati<strong>on</strong>ary<br />
regime, when the electr<strong>on</strong> rotates with c<strong>on</strong>stant energy, the<br />
equati<strong>on</strong>s for the electr<strong>on</strong> energy, γe = [1+(P/mec) 2 ] 1/2 ,<br />
and for the phase φ have the form<br />
a 2 0 = ( γ 2 e − 1 ) ( 1 + ε 2 radγ 6) e , (22)<br />
tan φ = − 1<br />
εradγ3 . (23)<br />
e<br />
In the limit <str<strong>on</strong>g>of</str<strong>on</strong>g> weak radiati<strong>on</strong> damp<strong>in</strong>g, a0 ≪ ε −1/3<br />
rad ,<br />
the absolute value <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> momentum is proporti<strong>on</strong>al<br />
to the electric field magnitude, P = meca0, while<br />
<strong>in</strong> the regime <str<strong>on</strong>g>of</str<strong>on</strong>g> dom<strong>in</strong>ant radiati<strong>on</strong> damp<strong>in</strong>g effects, i.e. at<br />
a0 ≫ ε −1/3<br />
rad , it is given by P = mec (a0/εrad) 1/4 . For<br />
the momentum dependence given by this expressi<strong>on</strong> the<br />
power radiated by an electr<strong>on</strong> is Pγ,C = ω0mec 2 a0, i.e.<br />
the energy obta<strong>in</strong>ed from the driv<strong>in</strong>g electromagnetic wave<br />
is completely re-radiated <strong>in</strong> the form <str<strong>on</strong>g>of</str<strong>on</strong>g> high energy gamma<br />
rays. At a0 ≈ ε −1/3<br />
rad we have for the gamma phot<strong>on</strong> energy<br />
¯hωγ = 0.29¯hω0a3 (<br />
3 2<br />
rad ≈ 0.45¯hω0 mc /e ) . For example,<br />
if λ0 ≈ 0.8 µm and a0 ≈ 400 the circularly polarized laser<br />
pulse <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tensity Irad = 4.5 × 1023 W/cm2 generates a<br />
burst <str<strong>on</strong>g>of</str<strong>on</strong>g> gamma phot<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> energy about 20 MeV with the<br />
durati<strong>on</strong> determ<strong>in</strong>ed either by the laser pulse durati<strong>on</strong> or by<br />
the decay time <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse <strong>in</strong> a plasma.<br />
In Fig. 2a we show a dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> γ and φ <strong>on</strong> the<br />
EM field amplitude, a, for the dimensi<strong>on</strong>less parameter<br />
εrad = 10−8 , obta<strong>in</strong>ed by numerical soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Eqs. (22,<br />
23). Here the horiz<strong>on</strong>tal axis is normalized <strong>on</strong> ε −1/3<br />
rad and<br />
the vertical axis is normalized <strong>on</strong> (am/εrad) 1/4 .<br />
The parameter χe can be expressed via the electric field,<br />
E, as (see Ref. [18])<br />
χe = |E|<br />
(<br />
m<br />
mecES<br />
2 ec 2 + |P| 2 s<strong>in</strong> 2 ) 1/2<br />
φ , (24)<br />
where φ is an angle between the electr<strong>on</strong> momentum and<br />
the electric field. As we see from Fig. 2 the angle φ tends<br />
to zero at large electric field.<br />
ϕ<br />
Figure 2: Dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> γ and φ <strong>on</strong> the EM field amplitude,<br />
a, for the dimensi<strong>on</strong>less parameter εrad = 10−8 . The<br />
horiz<strong>on</strong>tal axis is normalized <strong>on</strong> ε −1/3<br />
rad and the vertical axis<br />
is normalized <strong>on</strong> (am/εrad) 1/4 .<br />
S<strong>in</strong>ce <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> circular polarizati<strong>on</strong> ωm is proporti<strong>on</strong>al<br />
to the cube <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> gamma-factor quantum effects<br />
should be <strong>in</strong>corporated <strong>in</strong>to the theoretical descripti<strong>on</strong><br />
at γe ≈ γ C Q = (mec 2 /0.29 ¯hω0) 1/2 ≈ 1300. For γe = a0<br />
this limit is reached at the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> ≈ 3.4 ×10 24 W/cm 2 .<br />
The electr<strong>on</strong> moti<strong>on</strong> should be described with<strong>in</strong> the framework<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> quantum mechanics. These effects change the<br />
radiative loss functi<strong>on</strong> (see Ref. [18]). In the quantum<br />
regime, it is necessary to take <strong>in</strong>to account not <strong>on</strong>ly radiative<br />
damp<strong>in</strong>g effects but also recoil momentum effects,<br />
which change the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> because<br />
the outgo<strong>in</strong>g phot<strong>on</strong> carries away the momentum<br />
¯hkm = ¯hωm/c.<br />
In the regime when the radiati<strong>on</strong> fricti<strong>on</strong> effects are important,<br />
i.e. when a0 ≫ ε −1/3<br />
rad , the angle φ between the<br />
electr<strong>on</strong> momentum and the electric field is small be<strong>in</strong>g<br />
equal to ( εrada3 ) −1/4,<br />
0 i. e. the electr<strong>on</strong> moves almost<br />
<strong>in</strong> the electric field directi<strong>on</strong>. The electr<strong>on</strong> momentum is<br />
given by P = mec (a0/εrad) 1/4 . This yields an estimati<strong>on</strong><br />
χe ≈<br />
( a0<br />
γ<br />
a 2 S εrad<br />
b)<br />
a<br />
) 1/2<br />
. (25)<br />
This becomes greater than unity for a0 > εrada2 S ≈<br />
5.5 × 103 , which corresp<strong>on</strong>ds to the laser <strong>in</strong>tensity equal<br />
to 6 × 1025W/cm2 . In Ref. [6] an avalanche threshold<br />
<strong>in</strong>tensity several times lower has been found neglect<strong>in</strong>g<br />
the effects <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> fricti<strong>on</strong> force (see also [19]).<br />
However, the radiati<strong>on</strong> fricti<strong>on</strong> time is <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
(<br />
trad = 1/ω0 εrada3 ) 1/2,<br />
0 which for a0 ≈ 5.5 × 103 is<br />
approximately <strong>on</strong>e tenth <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser period. Hence the radiati<strong>on</strong><br />
fricti<strong>on</strong> effects do not prevent the EPGP cascade development<br />
for circularly polarized collid<strong>in</strong>g waves. Such a<br />
prolific electr<strong>on</strong>-positr<strong>on</strong> pair and gamma ray creati<strong>on</strong> [4]<br />
should result <strong>in</strong> the EPGP generati<strong>on</strong>.<br />
While creat<strong>in</strong>g and then accelerat<strong>in</strong>g the electr<strong>on</strong>positr<strong>on</strong><br />
pairs the laser pulse generates an electric current<br />
and EM field. The electric field <strong>in</strong>duced <strong>in</strong>side the EPGP
cloud with a size <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser wavelength, λ0<br />
can be estimated to be<br />
Epol = 2πe(n+ + n−)λ0. (26)<br />
Here n+ ≈ n− are the electr<strong>on</strong> and positr<strong>on</strong> density, respectively.<br />
Coherent scatter<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse away from<br />
the focus regi<strong>on</strong> occurs when the polarizati<strong>on</strong> electric field<br />
becomes equal to the laser electric field. This yields for the<br />
electr<strong>on</strong> and positr<strong>on</strong> density n+ ≈ n− = E/4πeλ0. The<br />
particle number per λ 3 0 volume is about a0λ0/re. This is a<br />
factor a0 smaller than required for the laser energy depleti<strong>on</strong>.<br />
CONCLUSION<br />
In c<strong>on</strong>clusi<strong>on</strong>, the high enough laser <strong>in</strong>tensity pulse<br />
with arbitrary polarizati<strong>on</strong> plus high enough density <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
seed electr<strong>on</strong>s, e.g. generated <strong>in</strong> the laser <strong>in</strong>teracti<strong>on</strong> with<br />
solid targets can provide necessary and sufficient c<strong>on</strong>diti<strong>on</strong>s<br />
for the avalanche development, [4]. Instead, <strong>in</strong> vacuum,<br />
when the seed electr<strong>on</strong>s(positr<strong>on</strong>s) are created via<br />
the Schw<strong>in</strong>ger mechanism, we see a fundamental difference<br />
between the circularly and l<strong>in</strong>early polarized waves.<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the circularly polarized EM wave the electr<strong>on</strong><br />
radiati<strong>on</strong> is str<strong>on</strong>g and the threshold for the avalanche<br />
is low enough for avalanche start<strong>in</strong>g at the laser <strong>in</strong>tensity<br />
well below the Schw<strong>in</strong>ger limit. S<strong>in</strong>ce, as noted <strong>in</strong> Ref.<br />
[4], the electr<strong>on</strong>-positr<strong>on</strong> avalanche parameters are <strong>in</strong>sensitive<br />
to the seed electr<strong>on</strong>s (positr<strong>on</strong>s), the parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
Schw<strong>in</strong>ger created pairs become hidden and can hardly be<br />
revealed. C<strong>on</strong>trary to this, <strong>in</strong> the l<strong>in</strong>early polarized EM<br />
wave is more favorable for the realizati<strong>on</strong> and reach<strong>in</strong>g<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> ”pure” Schw<strong>in</strong>ger electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong>. An<br />
electr<strong>on</strong> mov<strong>in</strong>g al<strong>on</strong>g the electric field with velocity and<br />
accelerati<strong>on</strong> parallel to the field emits much fewer phot<strong>on</strong>s<br />
with substantially lower energy neither experienc<strong>in</strong>g<br />
the radiati<strong>on</strong> fricti<strong>on</strong> nor quantum recoil effects. We see<br />
an analogy <str<strong>on</strong>g>of</str<strong>on</strong>g> these cases with circular and l<strong>in</strong>ear electr<strong>on</strong><br />
accelerators with the corresp<strong>on</strong>d<strong>in</strong>g c<strong>on</strong>stra<strong>in</strong><strong>in</strong>g and reduced<br />
roles <str<strong>on</strong>g>of</str<strong>on</strong>g> synchrotr<strong>on</strong> radiati<strong>on</strong> losses. The electr<strong>on</strong>positr<strong>on</strong><br />
pair creati<strong>on</strong> <strong>in</strong> the Breit-Wheeler type process is<br />
also suppressed because the key parameters χe and χγ dependence<br />
<strong>on</strong> the electr<strong>on</strong> and phot<strong>on</strong> momentum, <strong>in</strong> the<br />
laser field with the same <strong>in</strong>tensity,is much weaker. The<br />
l<strong>in</strong>ear polarizati<strong>on</strong> has apparent advantages for approach<strong>in</strong>g<br />
the regimes <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong> <strong>in</strong> vacuum<br />
and n<strong>on</strong>l<strong>in</strong>ear vacuum polarizati<strong>on</strong> important for fundamental<br />
science.<br />
We thank S. G. Bochkarev, V. Yu. Bychenkov, P. Chen,<br />
G. Dunne, N. V. Elk<strong>in</strong>a, E. Esarey, A. M. Fedotov, V. F.<br />
Frolov, D. Habs, T. Henzl, M. Kando, Y. Kato, K. K<strong>on</strong>do,<br />
G. Korn, N. B. Narozhny, W. Rozmus, H. Ruhl, and A. I.<br />
Zelnikov for discussi<strong>on</strong>s.<br />
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PAIR CREATION IN QED-STRONG PULSED LASER FIELDS ∗<br />
I. V. Sokolov † , SPRL, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Michigan, Ann Arbor, MI 48109, USA<br />
N. M. Naumova ‡ , LOA, ENSTA - Ecole Polytechnique - CNRS, 91761 Palaiseau, France<br />
J. A. Nees, CUOS and FOCUS Center, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Michigan, Ann Arbor, MI 48109, USA<br />
G. A. Mourou, ILE, ENSTA - Ecole Polytechnique - CNRS, 91761 Palaiseau, France<br />
Abstract<br />
Electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong> is am<strong>on</strong>g the QEDeffects<br />
known to occur <strong>in</strong> a str<strong>on</strong>g laser pulse <strong>in</strong>teracti<strong>on</strong><br />
with a counter-propagat<strong>in</strong>g electr<strong>on</strong> beam. In this<br />
regime multiple pairs may be generated from a s<strong>in</strong>gle beam<br />
electr<strong>on</strong>, some <str<strong>on</strong>g>of</str<strong>on</strong>g> the newborn particles be<strong>in</strong>g capable <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
further pair producti<strong>on</strong>. Radiati<strong>on</strong> back-reacti<strong>on</strong> prevents<br />
avalanche development and limits pair creati<strong>on</strong>. The system<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tegro-differential k<strong>in</strong>etic equati<strong>on</strong>s for electr<strong>on</strong>s,<br />
positr<strong>on</strong>s and γ-phot<strong>on</strong>s is solved numerically.<br />
INTRODUCTION<br />
The effects <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum electrodynamics (QED) may occur<br />
<strong>in</strong> a str<strong>on</strong>g laser pulse <strong>in</strong>teracti<strong>on</strong> with a counterpropagat<strong>in</strong>g<br />
electr<strong>on</strong> beam. In the well-known experiment [1]<br />
these effects were weak and barely observable. If the laser<br />
pulse <strong>in</strong>tensity is <strong>in</strong>creased up to J ≥ 5 · 10 22 W/cm 2 the<br />
QED effects c<strong>on</strong>trol the laser-beam <strong>in</strong>teracti<strong>on</strong> and result<br />
<strong>in</strong> multiple pair producti<strong>on</strong> from a s<strong>in</strong>gle beam electr<strong>on</strong>.<br />
QED-str<strong>on</strong>g laser fields<br />
In QED an electric field, E, should be treated as str<strong>on</strong>g if<br />
it exceeds the Schw<strong>in</strong>ger limit: E ≥ ES = mec 2 /(|e| ¯ λC)<br />
(see [2]). Such field is potentially capable <str<strong>on</strong>g>of</str<strong>on</strong>g> separat<strong>in</strong>g a<br />
virtual electr<strong>on</strong>-positr<strong>on</strong> pair provid<strong>in</strong>g an energy, which<br />
exceeds the electr<strong>on</strong> rest mass energy, mec 2 , to a charge,<br />
e = −|e|, over an accelerati<strong>on</strong> length as small as the Compt<strong>on</strong><br />
wavelength, ¯ λC = ¯h/(mec) ≈ 3.9 · 10 −11 cm. Typical<br />
effects <strong>in</strong> QED str<strong>on</strong>g fields are: electr<strong>on</strong>-positr<strong>on</strong> pair creati<strong>on</strong><br />
from high-energy phot<strong>on</strong>s, high-energy phot<strong>on</strong> emissi<strong>on</strong><br />
from electr<strong>on</strong>s or positr<strong>on</strong>s and the cascade development<br />
(see [3, 4]) result<strong>in</strong>g from the first two processes.<br />
QED-str<strong>on</strong>g fields may be created <strong>in</strong> the focus <str<strong>on</strong>g>of</str<strong>on</strong>g> an<br />
ultra-bright laser. C<strong>on</strong>sider QED-effects <strong>in</strong> a relativistically<br />
str<strong>on</strong>g pulsed field [3]:<br />
|a| ≫ 1, a = eA<br />
, (1)<br />
mec2 with A be<strong>in</strong>g the vector potential <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave. In the laboratory<br />
frame <str<strong>on</strong>g>of</str<strong>on</strong>g> reference the electric field is not QEDstr<strong>on</strong>g<br />
for achieved laser <strong>in</strong>tensities, J ∼ 10 22 W/cm 2 [5],<br />
and even for the J ∼ 10 25 W/cm 2 <strong>in</strong>tensity projected [6].<br />
∗ One <str<strong>on</strong>g>of</str<strong>on</strong>g> us (I.S.) is supported by the DOE NNSA under the Predictive<br />
Science Academic Alliances Program by grant DE-FC52-08NA28616.<br />
† igorsok@umich.edu<br />
‡ natalia.naumova@ensta-paristech.fr<br />
N<strong>on</strong>etheless, a counterpropagat<strong>in</strong>g particle <strong>in</strong> a 1D wave,<br />
a(ξ), ξ = ωt−(k·x), may experience a QED-str<strong>on</strong>g field,<br />
E0 = |dA/dξ|ω(E − p∥)/c, because the laser frequency,<br />
ω = c/ ¯ λ, is Doppler upshifted <strong>in</strong> the frame <str<strong>on</strong>g>of</str<strong>on</strong>g> reference<br />
co-mov<strong>in</strong>g with the electr<strong>on</strong>. Herewith the electr<strong>on</strong> dimensi<strong>on</strong>less<br />
energy, E, and its momentum are related to mec2 ,<br />
and mec corresp<strong>on</strong>d<strong>in</strong>gly, and subscript ∥ herewith denotes<br />
the vector projecti<strong>on</strong> <strong>on</strong> the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave propagati<strong>on</strong>.<br />
The Lorentz-transformed field exceeds the Schw<strong>in</strong>ger<br />
limit, if χ ∼ E0/ES ≥ 1. Numerical values <str<strong>on</strong>g>of</str<strong>on</strong>g> the parameter,<br />
χ, may be expressed <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the local <strong>in</strong>stantaneous<br />
<strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser wave, J:<br />
χ = 3 ¯λC<br />
2 ¯λ (E − p <br />
<br />
∥) <br />
da<br />
<br />
dξ ≈ (E − p∥) 1.4 · 103 √<br />
J<br />
1023 [W/cm 2 ] .<br />
(2)<br />
This dependence shown <strong>in</strong> Fig.1 dem<strong>on</strong>strate that χ parameter<br />
<strong>on</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> tens can be achieved with <strong>KEK</strong><br />
or SLAC electr<strong>on</strong> beams us<strong>in</strong>g available or foreseen <strong>in</strong> the<br />
future ultra<strong>in</strong>tense laser pulses.<br />
Figure 1: Dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> χ <strong>on</strong> energy <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s counterpropagat<strong>in</strong>g<br />
to laser pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> various <strong>in</strong>tensities, accord<strong>in</strong>g<br />
to Eq.(2).<br />
Radiati<strong>on</strong> back-reacti<strong>on</strong><br />
The creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pairs <strong>in</strong> QED-str<strong>on</strong>g fields is a particular<br />
form <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> losses from charged particles. At high<br />
χ an <strong>in</strong>termediate stage <strong>in</strong> the pair creati<strong>on</strong> process is the<br />
emanati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a high-energy phot<strong>on</strong> by a charged particle:<br />
e → γ, e. This phot<strong>on</strong> is then absorbed <strong>in</strong> the str<strong>on</strong>g field,<br />
generat<strong>in</strong>g an electr<strong>on</strong>-positr<strong>on</strong> pair: γ → e, p.
A way to quantify the irreversible radiati<strong>on</strong> losses has<br />
been found <strong>in</strong> [7]. Specifically, <strong>in</strong> the 1D wave field the<br />
transfer <str<strong>on</strong>g>of</str<strong>on</strong>g> energy, ∆E, from the wave to a particle may be<br />
<strong>in</strong>terpreted as the absorpti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> some number <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s.<br />
Accord<strong>in</strong>gly, the momentum from the absorbed phot<strong>on</strong>s is<br />
added to the parallel momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle. So, both<br />
energy and parallel momentum are not c<strong>on</strong>served, however,<br />
their difference is: ∆(E − p∥) = 0. To get the Lorentz<strong>in</strong>variant<br />
formulati<strong>on</strong>, <strong>in</strong>troduce the four-vector <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle<br />
momentum, p = (E, p), and the wave four-vector,<br />
k = ( ω<br />
c , k) for the 1D wave field. Their four-dot-product,<br />
(k · p) = ω(E − p∥)/c, is c<strong>on</strong>served <strong>in</strong> any particle <strong>in</strong>teracti<strong>on</strong><br />
with the 1D wave field, <strong>in</strong>clud<strong>in</strong>g its moti<strong>on</strong>, phot<strong>on</strong><br />
emissi<strong>on</strong>, pair creati<strong>on</strong> etc. The sum <str<strong>on</strong>g>of</str<strong>on</strong>g> this quantity over<br />
all particles <strong>in</strong> the f<strong>in</strong>al state is equal to that for the particles<br />
<strong>in</strong> the <strong>in</strong>itial state.<br />
The radiati<strong>on</strong> losses, thereby limit the cascad<strong>in</strong>g pair<br />
creati<strong>on</strong>. Particularly, emissi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> s<str<strong>on</strong>g>of</str<strong>on</strong>g>ter γ phot<strong>on</strong>s even<br />
may be described with<strong>in</strong> the radiati<strong>on</strong> force approximati<strong>on</strong>,<br />
which is traditi<strong>on</strong>ally used to account for the radiati<strong>on</strong><br />
back-reacti<strong>on</strong> (see [8, 9, 10, 11]).<br />
The discussed processes are described by the k<strong>in</strong>etic<br />
equati<strong>on</strong>s for the <strong>in</strong>volved particles (electr<strong>on</strong>s, positr<strong>on</strong>s, γphot<strong>on</strong>s).<br />
For circularly polarized 1D wave <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>stant amplitude,<br />
the system <str<strong>on</strong>g>of</str<strong>on</strong>g> three 1D <strong>in</strong>tegro-differential k<strong>in</strong>etic<br />
equati<strong>on</strong>s is reducible to a large system <str<strong>on</strong>g>of</str<strong>on</strong>g> ODEs, which is<br />
solved here numerically.<br />
ELECTRON IN QED-STRONG FIELD<br />
The emissi<strong>on</strong> probability <strong>in</strong> the QED-str<strong>on</strong>g 1D wave<br />
field may be found <strong>in</strong> Secti<strong>on</strong>s 40,90,101 <strong>in</strong> [12]. However,<br />
to simulate highly dynamical effects <strong>in</strong> pulsed fields,<br />
<strong>on</strong>e needs a reformulated emissi<strong>on</strong> probability, related to<br />
short time <strong>in</strong>tervals (not (−∞, +∞)), which is rederived<br />
<strong>in</strong> Appendix A <strong>in</strong> [13] with careful attenti<strong>on</strong> to c<strong>on</strong>sistent<br />
problem formulati<strong>on</strong>.<br />
Aga<strong>in</strong>, the energy, ¯hω ′ , and momentum, ¯hk ′ , <str<strong>on</strong>g>of</str<strong>on</strong>g> the emitted<br />
phot<strong>on</strong> are normalized to mec 2 and mec. The fourdot-product,<br />
(k · p), is the moti<strong>on</strong>al <strong>in</strong>variant for an electr<strong>on</strong><br />
and it is also c<strong>on</strong>served <strong>in</strong> the process <str<strong>on</strong>g>of</str<strong>on</strong>g> emissi<strong>on</strong>:<br />
(k · pi) = (k · k ′ ) + (k · pf ). A subscript i, f denotes the<br />
electr<strong>on</strong> <strong>in</strong> the <strong>in</strong>itial (i) or f<strong>in</strong>al (f) state.<br />
In the 1D wave field the emissi<strong>on</strong> probability may be<br />
c<strong>on</strong>veniently related to the <strong>in</strong>terval <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave phase, dξ,<br />
which should be taken al<strong>on</strong>g the electr<strong>on</strong> trajectory. The<br />
<strong>in</strong>terval <str<strong>on</strong>g>of</str<strong>on</strong>g> time, dt, and that <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> proper time,<br />
dτe, are related to dξ as follows: dτe = dt/E = dξ/[c(k ·<br />
p)]. The phase volume element for the emitted phot<strong>on</strong> is<br />
chosen <strong>in</strong> the form d2k ′ ⊥d(k·k′ ). The emissi<strong>on</strong> probability,<br />
dWfi/(dξd(k · k ′ )), is <strong>in</strong>tegrated over d2k ′ ⊥ , therefore, it<br />
is related to the element <str<strong>on</strong>g>of</str<strong>on</strong>g> the phase volume, d(k · k ′ ) (see<br />
detail <strong>in</strong> Appendix A <strong>in</strong> [13]):<br />
dWfi<br />
d(k · k ′ )dξ = α (∫ ∞<br />
r K5/3(y)dy + κrK2/3(r) )<br />
√<br />
3πλC(k ¯ · pi) 2<br />
, (3)<br />
κ = (k · k′ )χe<br />
(k · pi) , r = (k · k′ )<br />
χe(k · pf ) , χe = 3<br />
2 (k·pi)<br />
<br />
<br />
<br />
da<br />
<br />
dξ ¯ λC.<br />
Here Kν(r) is the MacD<strong>on</strong>ald functi<strong>on</strong> and α = e 2 /(c¯h).<br />
Collisi<strong>on</strong> <strong>in</strong>tegral<br />
In str<strong>on</strong>g fields we <strong>in</strong>troduce χ-parameter not <strong>on</strong>ly for<br />
electr<strong>on</strong>s but also for γ-phot<strong>on</strong>s and relate the emissi<strong>on</strong><br />
probability to dχγ ∝ d(k · k ′ ):<br />
χγ = 3<br />
2 (k · k′ <br />
<br />
) <br />
da<br />
<br />
dξ ¯ <br />
dWfi <br />
λC, = α <br />
da<br />
<br />
dχγdξ dξ we→γ,e χe→χγ ,<br />
√ [<br />
∫<br />
(4)<br />
∞ ]<br />
3<br />
χγrK2/3(r) + K5/3(y)dy ,<br />
w e→γ,e<br />
χe→χγ =<br />
2πχ 2 e<br />
(5)<br />
Here r = χγ/[χe(χe − χγ)], χγ ≤ χe. The electr<strong>on</strong> parameter,<br />
χe, is taken for the <strong>in</strong>itial state and its value <strong>in</strong> the<br />
f<strong>in</strong>al state is χe − χγ.<br />
The distributi<strong>on</strong> functi<strong>on</strong>s for electr<strong>on</strong>s and phot<strong>on</strong>s may<br />
be also <strong>in</strong>tegrated over p⊥ and k ′ ⊥ corresp<strong>on</strong>d<strong>in</strong>gly. Thus<br />
<strong>in</strong>tegrated functi<strong>on</strong>s are distributed over (k · p), (k · k ′ ). We<br />
can parameterize locally these distributi<strong>on</strong>s via χe ∝ (k·p),<br />
χγ ∝ (k · k ′ ) and <strong>in</strong>troduce the 1D distributi<strong>on</strong> functi<strong>on</strong>s,<br />
fe(χe) and fγ(χγ).<br />
The collisi<strong>on</strong> <strong>in</strong>tegral (see [14]) describes the change<br />
<strong>in</strong> the particle distributi<strong>on</strong>s due to emissi<strong>on</strong> and accounts<br />
for the electr<strong>on</strong>s, leav<strong>in</strong>g the given phase volume, dχe,<br />
and those arriv<strong>in</strong>g <strong>in</strong>to it with<strong>in</strong> the <strong>in</strong>terval, d ˜ ξ =<br />
α|da/dξ|dξ = 2αcχedτe/(3 ¯ λC):<br />
δfe(χe)<br />
d˜ =<br />
ξ<br />
∫ ∞<br />
χe<br />
fe(χ)w e→γ,e<br />
χ→χ−χedχ−fe(χe) ∫ χe<br />
0<br />
δfγ(χγ)<br />
d˜ =<br />
ξ<br />
∫ ∞<br />
χγ<br />
r<br />
w e→γ,e<br />
χe→χ dχ,<br />
fe(χ)w e→γ,e<br />
χ→χγ dχ. (6)<br />
PHOTON IN QED-STRONG FIELD<br />
The absorpti<strong>on</strong> probability for phot<strong>on</strong>s <strong>in</strong> the 1D field is<br />
derived <strong>in</strong> Appendix B <strong>in</strong> [13]. An electr<strong>on</strong>-positr<strong>on</strong> pair<br />
(e,p) is generated <strong>in</strong> the phot<strong>on</strong> absorpti<strong>on</strong> with the c<strong>on</strong>servati<strong>on</strong><br />
law: (k · k ′ ) = (k · pe) + (k · pp).<br />
The phase volume element for the created electr<strong>on</strong>, aga<strong>in</strong><br />
is chosen <strong>in</strong> the form d2p⊥d(k · p). The absorpti<strong>on</strong> probability,<br />
dWfi/(dξd(k · pe)), is <strong>in</strong>tegrated over the transversal<br />
momenta comp<strong>on</strong>ents and related to the element <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
phase volume <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>, d(k · pe), result<strong>in</strong>g <strong>in</strong> the follow<strong>in</strong>g<br />
collisi<strong>on</strong> <strong>in</strong>tegral:<br />
δ − fe,p(χe,p)<br />
d ˜ ξ<br />
=<br />
∫ ∞<br />
χe,p<br />
fγ(χγ)w γ→e,p<br />
χγ→χe dχγ, (7)<br />
∫ χγ<br />
δ−fγ(χγ) d˜ = −fγ(χγ)<br />
ξ<br />
0<br />
Here r = χγ/[χe(χγ − χe)], χe = χγ − χp ≤ χγ and<br />
√ [<br />
∫ ∞ ]<br />
3<br />
χγrK2/3(r) − K5/3(y)dy .<br />
w γ→e,p<br />
χγ→χe =<br />
2πχ 2 γ<br />
w γ→e,p<br />
χγ→χe dχe. (8)<br />
r<br />
(9)
Figure 2: Distributi<strong>on</strong> functi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and positr<strong>on</strong>s, fe,p(χ), and a spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> emissi<strong>on</strong>, χγfγ(χ)/χ0, after the<br />
<strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 8-GeV electr<strong>on</strong>s with <strong>on</strong>e, five and ten cycles <str<strong>on</strong>g>of</str<strong>on</strong>g> a laser pulse <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tensity J ≈ 2 · 10 23 W/cm 2 (so that<br />
χ ≈ 2E[GeV ] — see Eq.(2)). Here fe − fp is the distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the beam electr<strong>on</strong>s and ∫ (fe − fp)dχ = 1.<br />
Figure 3: Pair producti<strong>on</strong> (upper panel) and energy exchange<br />
between electr<strong>on</strong>s, phot<strong>on</strong>s, positr<strong>on</strong>s (lower panel)<br />
as functi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> phase for the simulati<strong>on</strong> presented <strong>in</strong> Fig.2.<br />
SOLUTION FOR KINETIC EQUATIONS<br />
As l<strong>on</strong>g as the distributi<strong>on</strong> functi<strong>on</strong>s are <strong>in</strong>tegrated over<br />
the transversal comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> momentum and expressed <strong>in</strong><br />
terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the moti<strong>on</strong>al <strong>in</strong>tegrals, (k · pe,p), their evoluti<strong>on</strong> is<br />
c<strong>on</strong>trolled by the collisi<strong>on</strong> <strong>in</strong>tegrals:<br />
(<br />
δ + + δ − + δ (rf))<br />
fe,p,γ.<br />
∂fe,p,γ( ˜ ξ, (k · pe,p,γ))<br />
∂ ˜ =<br />
ξ<br />
(10)<br />
The derivatives, ∂/∂ ˜ ξ, are taken at c<strong>on</strong>stant (k · p). The<br />
term δ (rf) accounts for the radiati<strong>on</strong> losses for the phot<strong>on</strong>s<br />
with χγ ≤ ϵ ≪ 1 excluded from the other two terms <strong>in</strong>-<br />
tenti<strong>on</strong>ally (see details <strong>in</strong> Ref.[7]). Eqs.(10) are easy-tosolve<br />
for the 1D wave field <str<strong>on</strong>g>of</str<strong>on</strong>g> any shape, however, for<br />
circularly polarized wave <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>stant amplitude the soluti<strong>on</strong><br />
is especially simple. In this case (k · p) are different<br />
from χ by a c<strong>on</strong>stant factor, and Eqs.(10) may be solved<br />
with derivatives, ∂/∂ ˜ ξ, at c<strong>on</strong>stant χ for the functi<strong>on</strong>s,<br />
fe,p,γ( ˜ ξ, χe,p,γ).<br />
We solve Eqs.(10) numerically, by discretiz<strong>in</strong>g them at a<br />
uniform grid, χj = j∆χ, j = 1, 2, 3..., N, with the choice<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> ∆χ = 0.1. The ˜ ξ-dependent distributi<strong>on</strong> functi<strong>on</strong>s at<br />
this grid obey the system <str<strong>on</strong>g>of</str<strong>on</strong>g> 3N ODEs, which is <strong>in</strong>tegrated<br />
numerically.<br />
NUMERICAL EXAMPLES<br />
<strong>KEK</strong> parameters: χ = 30<br />
At <strong>in</strong>itializati<strong>on</strong>, electr<strong>on</strong>s with fe(χe) = δ(χe − χ0),<br />
χ0 = 30, counterpropagate <strong>in</strong> the circularly polarized wave<br />
field with |da/dξ| = 220. This choice corresp<strong>on</strong>ds to the<br />
8-GeV electr<strong>on</strong> beam and the laser <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> J ≈ 2 ·<br />
1023 W/cm 2 for λ = 0.8µm, to be achieved so<strong>on</strong>.<br />
In Fig.2 the beam-wave <strong>in</strong>teracti<strong>on</strong> is traced dur<strong>in</strong>g ξ<br />
2π =<br />
10 cycles <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>cident laser pulse (≈ 27 fs). The <strong>in</strong>itial<br />
beam electr<strong>on</strong> energy is rapidly c<strong>on</strong>verted <strong>in</strong>to γ-phot<strong>on</strong>s<br />
with high χγ, which then rapidly produce pairs, the typical<br />
rates <str<strong>on</strong>g>of</str<strong>on</strong>g> the processes be<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>verse<br />
light period. However, the larger fracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the new particles<br />
is born at χ ≤ 1, with str<strong>on</strong>gly reduced pair producti<strong>on</strong><br />
rate. Slow absorpti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s with χγ ∼ 1−2 ma<strong>in</strong>ta<strong>in</strong>s<br />
pair producti<strong>on</strong> even after tens <str<strong>on</strong>g>of</str<strong>on</strong>g> wave periods, as shown <strong>in</strong><br />
Fig.3 (upper panel). In Fig.3 the lower panel shows the evoluti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the functi<strong>on</strong>s ∑<br />
∑<br />
i<br />
χife,i/χ0,<br />
i χifp,i/χ0,<br />
∑<br />
and<br />
i χifγ,i/χ0, which represent the energy porti<strong>on</strong>s <strong>in</strong> the<br />
corresp<strong>on</strong>d<strong>in</strong>g sorts <str<strong>on</strong>g>of</str<strong>on</strong>g> particles. These plots dem<strong>on</strong>strate<br />
that for 10-cycle laser pulse ≈ 90% <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>itial electr<strong>on</strong><br />
energy is c<strong>on</strong>verted <strong>in</strong>to phot<strong>on</strong>s, the rest part is split between<br />
electr<strong>on</strong>s and positr<strong>on</strong>s.<br />
SLAC parameters: χ = 90<br />
Us<strong>in</strong>g 46-GeV electr<strong>on</strong> beam and laser pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tensity<br />
J ≈ 5 · 10 22 W/cm 2 , the value <str<strong>on</strong>g>of</str<strong>on</strong>g> χ = 90 can be
Figure 4: Pair producti<strong>on</strong> vs time for 46.6 GeV electr<strong>on</strong>s <strong>in</strong>teract<strong>in</strong>g with laser pulse <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tensity J ≈ 5 · 10 22 W/cm 2 .<br />
Plots are presented <strong>in</strong> l<strong>in</strong>ear and log − log scale. Dashed l<strong>in</strong>e is ∝ ξ 2 .<br />
achieved <strong>in</strong> their <strong>in</strong>teracti<strong>on</strong>. An evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pair producti<strong>on</strong><br />
for these parameters is shown <strong>in</strong> Fig.4 (see other plots<br />
for the same parameters <strong>in</strong> Ref.[7]). The results for such<br />
high <strong>in</strong>itial value <str<strong>on</strong>g>of</str<strong>on</strong>g> χ dem<strong>on</strong>strate almost quadratic dependence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> pair producti<strong>on</strong>. This observati<strong>on</strong> is <strong>in</strong> agreement<br />
with simple estimati<strong>on</strong>s as follows.<br />
The change <strong>in</strong> the pair producti<strong>on</strong> is proporti<strong>on</strong>al to the<br />
number <str<strong>on</strong>g>of</str<strong>on</strong>g> rigid phot<strong>on</strong>s:<br />
dN e − e +<br />
dξ<br />
∝ Nγ<br />
dWabsorpti<strong>on</strong><br />
.<br />
dξ<br />
In the same time, the change <strong>in</strong> the number <str<strong>on</strong>g>of</str<strong>on</strong>g> rigid phot<strong>on</strong>s<br />
is proporti<strong>on</strong>al to <strong>in</strong>itial number <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s:<br />
dNγ<br />
dξ<br />
∝ Ne,0<br />
dWemissi<strong>on</strong><br />
.<br />
dξ<br />
Assum<strong>in</strong>g a c<strong>on</strong>stant value for <strong>in</strong>itial number <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s,<br />
Ne,0, we obta<strong>in</strong> dependences for the number <str<strong>on</strong>g>of</str<strong>on</strong>g> rigid phot<strong>on</strong>s<br />
and the number <str<strong>on</strong>g>of</str<strong>on</strong>g> pairs:<br />
what agrees with our results.<br />
Nγ ∝ ξ, N e − e + ∝ ξ 2 ,<br />
CONCLUSION<br />
We see that the laser-beam <strong>in</strong>teracti<strong>on</strong> may be accompanied<br />
by multiple pair producti<strong>on</strong>. The <strong>in</strong>itial energy <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
beam electr<strong>on</strong> is efficiently spent for creat<strong>in</strong>g pairs with<br />
significantly lower energies as well as s<str<strong>on</strong>g>of</str<strong>on</strong>g>ter γ-phot<strong>on</strong>s.<br />
This effect may be used for produc<strong>in</strong>g a pair plasma. It<br />
could also be employed to deactivati<strong>on</strong> after-use electr<strong>on</strong><br />
beams, reduc<strong>in</strong>g radiati<strong>on</strong> hazard.<br />
The way to solve the k<strong>in</strong>etic equati<strong>on</strong>s is accurate and<br />
it does not employ the M<strong>on</strong>te-Carlo method. The soluti<strong>on</strong><br />
can be used to benchmark numerical methods designed to<br />
simulate processes <strong>in</strong> QED-str<strong>on</strong>g laser fields.<br />
REFERENCES<br />
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Oxford, 1981).
LASER ACCELERATION UP TO BLACK HOLES<br />
AND B-MESON DECAY *<br />
H. Hora, # Deptm. Theor. Phys. UNSW Sydney, Australia<br />
R. Castillo, T. Stait-Gardner, BMSci. U. Western Sydney Campbelltown, Australia<br />
D.H.H. H<str<strong>on</strong>g>of</str<strong>on</strong>g>fmann, Dept. Nucl. Phys. TU Darmstadt, Germany<br />
G.H. Miley, Dept. Nucl. Plasma & Radiolog. Eng<strong>in</strong>. Univ. Ill<strong>in</strong>ois USA<br />
P. Lalousis, IESL/FORTH, Herakli<strong>on</strong>, Greece<br />
Abstract<br />
Studies about laser produced pair producti<strong>on</strong> are followed<br />
up from early stages. The pair producti<strong>on</strong> by vacuum<br />
polarizati<strong>on</strong> was discussed with laser produced<br />
accelerati<strong>on</strong> up to the values at black holes lead<strong>in</strong>g to the<br />
discovery <str<strong>on</strong>g>of</str<strong>on</strong>g> a difference between Hawk<strong>in</strong>g and Unruh<br />
radiati<strong>on</strong>. It was clarified that producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> anti-hydrogen<br />
is at least milli<strong>on</strong> times more efficient than by present day<br />
accelerator technology. Another applicati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
ultrahigh laser fields is to focus them <strong>in</strong>to the collisi<strong>on</strong><br />
area <str<strong>on</strong>g>of</str<strong>on</strong>g> the LHC with the possibility to study the details <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the B-mes<strong>on</strong> dacay. This <str<strong>on</strong>g>of</str<strong>on</strong>g>fers an access to detect more<br />
details about CP violati<strong>on</strong> and Bs mes<strong>on</strong>s and possible<br />
signs <str<strong>on</strong>g>of</str<strong>on</strong>g> new particles <strong>on</strong> the horiz<strong>on</strong>. The available lasers<br />
with picosec<strong>on</strong>d pulses are developed to exawatts power<br />
what is <strong>in</strong>terest<strong>in</strong>g also for study<strong>in</strong>g ultra-<strong>in</strong>tense shock<br />
waves <strong>in</strong> astrophysics and result<strong>in</strong>g nuclear reacti<strong>on</strong>s.<br />
INTRODUCTION AND INITIAL RESULTS<br />
Us<strong>in</strong>g the very high <strong>in</strong>tensity laser radiati<strong>on</strong> with the<br />
electric and magnetic fields E and H far above any values<br />
applied before, led to very many new physics phenomena<br />
and last not least to the realizati<strong>on</strong> that the opened<br />
n<strong>on</strong>l<strong>in</strong>ear physics opens a new dimensi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> explorati<strong>on</strong><br />
where <strong>in</strong>deed the l<strong>in</strong>ear physics needs to be based <strong>on</strong><br />
higher accuracy data than needed before [1]. Examples<br />
appeared where results <strong>in</strong> l<strong>in</strong>ear physics were completely<br />
wr<strong>on</strong>g compared to the truth <strong>in</strong> n<strong>on</strong>l<strong>in</strong>ear physics <strong>in</strong><br />
c<strong>on</strong>trast to earlier happen<strong>in</strong>g gradual differences or<br />
approximati<strong>on</strong>s <strong>on</strong>ly. This all developed not <strong>on</strong>ly due to<br />
techniques to produce higher and higher laser <strong>in</strong>tensities,<br />
mostly realized by chirped pulse amplificati<strong>on</strong> CPA [2],<br />
but also from realiz<strong>in</strong>g to generate relativistic effects.<br />
After first c<strong>on</strong>siderati<strong>on</strong>s how to produce relativistic<br />
c<strong>on</strong>diti<strong>on</strong>s for pair producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s [3] the<br />
c<strong>on</strong>diti<strong>on</strong>s were elaborated for the laser fields produc<strong>in</strong>g<br />
quiver moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s with energies above mc 2 [4].<br />
The steps to c<strong>on</strong>clude the c<strong>on</strong>diti<strong>on</strong>s for produc<strong>in</strong>g<br />
anti-prot<strong>on</strong>s [5] were parallel to estimati<strong>on</strong>s [6] and<br />
experiments where <strong>in</strong>dicati<strong>on</strong>s for the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
very first laser produced positr<strong>on</strong>s were reported [7].<br />
With respect to the quiver moti<strong>on</strong> and drift for prot<strong>on</strong> pair<br />
producti<strong>on</strong>, the advantages <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>on</strong>g wave length laser<br />
pulses were <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>terest [8]. After estimati<strong>on</strong>s with anti-<br />
* Dedicated to Pr<str<strong>on</strong>g>of</str<strong>on</strong>g>essor Chiyoe Yamanaka, Osaka<br />
University, to his 88 th year.<br />
# h.hora@unsw.edu.au<br />
1<br />
hydrogen for space research became known, the soluti<strong>on</strong><br />
with lasers were <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>terest because the efficiency was<br />
more than <strong>on</strong>e milli<strong>on</strong> times higher with lasers due to the<br />
available much higher particle density than with<br />
accelerator techniques. It was proved that a missi<strong>on</strong> to the<br />
next fix star with<strong>in</strong> a reas<strong>on</strong>able time <str<strong>on</strong>g>of</str<strong>on</strong>g> 50 years can <strong>on</strong>ly<br />
be d<strong>on</strong>e with laser produced anti-hydrogen fuel [9].<br />
Thanks to the CPA technique, sub-picosec<strong>on</strong>d laser<br />
pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> 2 PW produced the first c<strong>on</strong>siderable number <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
positr<strong>on</strong>s [10] f<strong>in</strong>ally arriv<strong>in</strong>g [11] at record <strong>in</strong>tensities<br />
positr<strong>on</strong> beam above any other method.<br />
PAIR PRODUCTION BY VACUUM<br />
POLARIZATION<br />
Pair producti<strong>on</strong> <strong>in</strong> vacuum was from the beg<strong>in</strong>n<strong>in</strong>g<br />
c<strong>on</strong>sidered [3][4][5] where a laser <strong>in</strong>tensity above 10 28<br />
W/cm 2 was needed [12] and specified to the well known<br />
higher value later. The accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s by the<br />
electric field <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser was close to the values <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
Hawk<strong>in</strong>g radiati<strong>on</strong> and the Unruh radiati<strong>on</strong> at the black<br />
holes. This was studied <strong>in</strong> c<strong>on</strong>necti<strong>on</strong> with the black body<br />
radiati<strong>on</strong> which fields are <str<strong>on</strong>g>of</str<strong>on</strong>g> the same order and where the<br />
electr<strong>on</strong>s at thermal equilibrium were not l<strong>on</strong>ger<br />
follow<strong>in</strong>g the Fermi-Dirac statistics [13]. Further studies<br />
clarified that there was a difference between the Hawk<strong>in</strong>g<br />
and the Unruh radiati<strong>on</strong> [14] with a relati<strong>on</strong> to the<br />
Casimir effect [15][16]. These results were based <strong>on</strong> the<br />
theory <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> accelerati<strong>on</strong> <strong>in</strong> vacuum [17] as a<br />
basically n<strong>on</strong>l<strong>in</strong>ear effect [1]. The essential aspects <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
these studies are as follows.<br />
The Unruh effect is a phenomen<strong>on</strong> whereby an<br />
accelerated observer travell<strong>in</strong>g through a true vacuum<br />
state—that is the ground state |0> which will be referred<br />
to here as the M<strong>in</strong>kowski vacuum—will experience<br />
themselves to be immersed <strong>in</strong> a thermal blackbody<br />
distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> particles [16]; Before compar<strong>in</strong>g the<br />
thermal radiati<strong>on</strong> experienced by an accelerated observer<br />
to the Hawk<strong>in</strong>g radiati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a black hole a brief digressi<strong>on</strong><br />
<strong>in</strong>to the physical nature <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum is appropriate.<br />
The M<strong>in</strong>kowski vacuum is a physical vacuum with<br />
pairs <str<strong>on</strong>g>of</str<strong>on</strong>g> virtual particles manifest<strong>in</strong>g for short durati<strong>on</strong>s<br />
c<strong>on</strong>t<strong>in</strong>uously and, unlike the pre-quantum field theory<br />
vacuum, has observable effects <strong>on</strong> physical systems (e.g.<br />
the f<strong>in</strong>e structure <str<strong>on</strong>g>of</str<strong>on</strong>g> the atomic hydrogen spectrum and<br />
the Casimir effect). Tak<strong>in</strong>g the Casimir effect as an<br />
example, two parallel mirrors placed <strong>in</strong> a vacuum will<br />
experience an attractive force <strong>in</strong>versely proporti<strong>on</strong>al to<br />
the forth power <str<strong>on</strong>g>of</str<strong>on</strong>g> the distance separat<strong>in</strong>g them as a result
<str<strong>on</strong>g>of</str<strong>on</strong>g> the quantum fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum. Essentially<br />
l<strong>on</strong>g wavelength virtual particles cannot manifest between<br />
the c<strong>on</strong>duct<strong>in</strong>g mirrors result<strong>in</strong>g <strong>in</strong> a decreased energy<br />
density between the mirrors compared with the vacuum<br />
surround<strong>in</strong>g them where there is no such restricti<strong>on</strong>. The<br />
Casimir effect is symbolic <str<strong>on</strong>g>of</str<strong>on</strong>g> the physical nature <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
quantum vacuum.<br />
The quantum field is best decomposed for an<br />
accelerated observer us<strong>in</strong>g a different basis than the<br />
standard momentum basis used <strong>in</strong> quantum field theory;<br />
this basis be<strong>in</strong>g related to the standard basis by the<br />
Bogoliubov transformati<strong>on</strong>s. These transformati<strong>on</strong>s play<br />
an <strong>in</strong>tegral part <strong>in</strong> analyses <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh effect. The<br />
particle number operator differs too and does not give<br />
zero when applied to the M<strong>in</strong>kowski vacuum state (which<br />
is not identical to the R<strong>in</strong>dler vacuum state). The result is,<br />
as stated above, that an accelerated observer <strong>in</strong> a pure<br />
vacuum will experience themselves <strong>in</strong> a heat bath with a<br />
blackbody distributi<strong>on</strong>.<br />
Thus a state without particles to an <strong>in</strong>ertial observer<br />
will be seen to c<strong>on</strong>ta<strong>in</strong> particles by an accelerated<br />
observer. The dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> temperature up<strong>on</strong><br />
accelerati<strong>on</strong> is, T = 2πckBa/ħ, where c is the velocity <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
light, and a is the accelerati<strong>on</strong>. If a is <strong>in</strong>terpreted as the<br />
accelerati<strong>on</strong> at the event horiz<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a black hole then the<br />
same equati<strong>on</strong> describes the temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> the thermal<br />
radiati<strong>on</strong> emitted from a black hole via the process <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Hawk<strong>in</strong>g radiati<strong>on</strong>. The similarity <str<strong>on</strong>g>of</str<strong>on</strong>g> the equati<strong>on</strong>s and<br />
the equivalence pr<strong>in</strong>ciple <str<strong>on</strong>g>of</str<strong>on</strong>g> general relativity h<strong>in</strong>t that the<br />
mechanisms for the radiati<strong>on</strong> may be the same but this is<br />
not the case. C<strong>on</strong>sider the follow<strong>in</strong>g.<br />
Hawk<strong>in</strong>g radiati<strong>on</strong> is sometimes described as result<strong>in</strong>g<br />
from pair producti<strong>on</strong> near the horiz<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a black hole with<br />
<strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the virtual particles escap<strong>in</strong>g and becom<strong>in</strong>g real<br />
and the other disappear<strong>in</strong>g <strong>in</strong>to the black hole [15]. All<br />
observers experience Hawk<strong>in</strong>g radiati<strong>on</strong> while <strong>on</strong>ly<br />
accelerated observers experience the Unruh effect.<br />
Furthermore, an observer <strong>on</strong> earth is effectively <strong>in</strong> an<br />
accelerated coord<strong>in</strong>ate system via the equivalence<br />
pr<strong>in</strong>ciple and hence should observe the surround<strong>in</strong>g<br />
vacuum to have a temperature due to the Unruh effect but<br />
the earth does not emit Hawk<strong>in</strong>g radiati<strong>on</strong> and neither do<br />
other gravitati<strong>on</strong>al bodies without event horiz<strong>on</strong>s. The<br />
Unruh effect results from a different mechanism to that <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Hawk<strong>in</strong>g radiati<strong>on</strong>; it is local, be<strong>in</strong>g experienced <strong>on</strong>ly by<br />
accelerated observers [14].<br />
PETAWATT LASER PULSES FOR<br />
B-MESON DIAGNOSTICS<br />
The present day available PW laser pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> subpicosec<strong>on</strong>d<br />
durati<strong>on</strong> and the next higher powers can be<br />
used for important studies <str<strong>on</strong>g>of</str<strong>on</strong>g> the details <str<strong>on</strong>g>of</str<strong>on</strong>g> B-mes<strong>on</strong><br />
diagnostics because their lifetimes are <strong>on</strong> the same time<br />
scale. This diagnostics at collider beam <strong>in</strong>teracti<strong>on</strong>s with<br />
lasers was studied before [18] for the c<strong>on</strong>diti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
Large Electr<strong>on</strong> Positr<strong>on</strong> (LEP) collider and can now be<br />
extended for the c<strong>on</strong>diti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> B-mes<strong>on</strong>s, e.g. at the Large<br />
Hadr<strong>on</strong> Collider LHC or similar B-mes<strong>on</strong> factories [19].<br />
2<br />
A prototype <str<strong>on</strong>g>of</str<strong>on</strong>g> this technique was given by the<br />
<strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 16 W/cm 2 laser <strong>in</strong>tensities <strong>in</strong> low density<br />
helium [20]. It was expected from theory that a radial<br />
emissi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s from the focus should c<strong>on</strong>vert half<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the quiver energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s <strong>in</strong>to energy <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
translative moti<strong>on</strong>. The measured radially emitted keV<br />
electr<strong>on</strong>s corresp<strong>on</strong>ded exactly to the expected theory.<br />
The c<strong>on</strong>servati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong>s leads<br />
to a slightly forward directi<strong>on</strong> parallel to the laser axis.<br />
This was measured <strong>in</strong> [21] <strong>in</strong> agreement with the earlier<br />
predicti<strong>on</strong> [22]. In the same way, the charged particles<br />
generated <strong>in</strong> the focus <str<strong>on</strong>g>of</str<strong>on</strong>g> the collider when be<strong>in</strong>g <strong>in</strong> the<br />
focus <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser beam, will get an upshift <str<strong>on</strong>g>of</str<strong>on</strong>g> energy and<br />
a change <str<strong>on</strong>g>of</str<strong>on</strong>g> directi<strong>on</strong>. The PW laser pulses and even the<br />
better exawatt (EW) laser pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> few fs durati<strong>on</strong> [23]<br />
can then follow up the tim<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> generat<strong>in</strong>g or<br />
annihilat<strong>in</strong>g process <str<strong>on</strong>g>of</str<strong>on</strong>g> the B-mes<strong>on</strong> generati<strong>on</strong> and the<br />
decay processes. The importance is evident for further<br />
analyz<strong>in</strong>g the different types <str<strong>on</strong>g>of</str<strong>on</strong>g> B-mes<strong>on</strong>s where <strong>in</strong>sights<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the CP violati<strong>on</strong> strange bos<strong>on</strong>s Bs will be <strong>in</strong>terest<strong>in</strong>g<br />
<strong>on</strong> the way <str<strong>on</strong>g>of</str<strong>on</strong>g> a “possible new particles <strong>on</strong> the horiz<strong>on</strong>”<br />
[24].<br />
The theory is based <strong>on</strong> electro-dynamic <strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the laser radiati<strong>on</strong> with the particles as known from<br />
plasma <strong>in</strong>teracti<strong>on</strong> as the n<strong>on</strong>l<strong>in</strong>ear force given by [25]<br />
fNL = ∇•[EE + HH − 0.5(E 2 + H 2 )1<br />
+ (1+(∂/∂t)/ω)(n 2 −1)EE]/(4π)<br />
− (∂/∂t) E × H/(4πc) (1)<br />
(see Eq. 8.88 <str<strong>on</strong>g>of</str<strong>on</strong>g> Ref. 1991 [25]) where 1 is the unity<br />
tensor, c the vacuum speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light. The value n is the<br />
(complex) refractive <strong>in</strong>dex determ<strong>in</strong>ed by the laser<br />
frequency ω and the electr<strong>on</strong>-i<strong>on</strong> collisi<strong>on</strong> frequency ν <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
a plasma<br />
n = 1 – (ne/nec)/(1 + iν/ω), (2)<br />
where ne is the electr<strong>on</strong> density, nec is the critical electr<strong>on</strong><br />
density where the plasma frequency ωp is equal to the<br />
laser frequency ω. The dielectric properties <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum<br />
polarizati<strong>on</strong> are to be <strong>in</strong>cluded appropriately for the pair<br />
producti<strong>on</strong> <strong>in</strong> vacuum. The derivati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this force with<br />
<strong>in</strong>clusi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the dielectric plasma properties for the n<strong>on</strong>transient<br />
case s<strong>in</strong>ce 1969 [25] was based <strong>on</strong> momentum<br />
c<strong>on</strong>servati<strong>on</strong>. The f<strong>in</strong>al complete transient case, Eq. (1), is<br />
known s<strong>in</strong>ce 1985 based <strong>on</strong> symmetry where it was<br />
proved later that this and <strong>on</strong>ly this is the Lorentz and<br />
gauge <strong>in</strong>variant descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the n<strong>on</strong>l<strong>in</strong>ear force.<br />
For simplified <strong>on</strong>e-dimensi<strong>on</strong>al geometry and<br />
perpendicular laser irradiati<strong>on</strong>, the force (1) can be<br />
reduced to the time averaged value<br />
fNL = − (∂/∂x)(E 2 +H 2 )/(8π)<br />
= − (ωp/ω) 2 (∂/∂x)(Ev 2 /n)/(16π), (3)<br />
where Ev is the amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field <strong>in</strong> vacuum.<br />
The last expressi<strong>on</strong> is rem<strong>in</strong>d<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the formulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
p<strong>on</strong>deromotive force <strong>in</strong> electrostatics and is sometimes<br />
called “radiati<strong>on</strong> pressure accelerati<strong>on</strong>”.<br />
The relativistic limits for the emissi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the charged<br />
particles from the collider area with a laser focus are<br />
given for laser <strong>in</strong>tensities <str<strong>on</strong>g>of</str<strong>on</strong>g> neodymium glass lasers [19]
(1) charged B-mes<strong>on</strong>s<br />
Irel = 1.2×10 25 W/cm 2 ∆ε = 2.41 keV<br />
(2) prot<strong>on</strong>s or antiprot<strong>on</strong>s from the B-decay<br />
Irel = 3.9×10 26 W/cm 2 ∆ε = 424 eV<br />
(3) charged π-mes<strong>on</strong>s from B-mes<strong>on</strong>s decay:<br />
Irel = 2.73×10 23 W/cm 2 ∆ε = 31.5 keV<br />
The size <str<strong>on</strong>g>of</str<strong>on</strong>g> the lasers for PW-fs pulses are comparably<br />
compact such that the diagnostics with an additi<strong>on</strong>al laser<br />
focus may not be a too difficult problem. The signals<br />
from the detectors for comparable cases with and without<br />
the laser will then be d<strong>on</strong>e by functi<strong>on</strong>al analytical<br />
fold<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>formati<strong>on</strong> about the time dependence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
creati<strong>on</strong>, decay and annihilati<strong>on</strong> processes <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
numerous types <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles,<br />
Figure 1. Genu<strong>in</strong>e two fluid hydrodynamic computati<strong>on</strong>s<br />
[32][33] <str<strong>on</strong>g>of</str<strong>on</strong>g> the i<strong>on</strong> density <strong>in</strong> solid DT after irradiati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
a laser pulse <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 20 W/cm 2 <str<strong>on</strong>g>of</str<strong>on</strong>g> ps durati<strong>on</strong> at the times 22<br />
ps (dashed) and 225ps after the <strong>in</strong>itiati<strong>on</strong>.<br />
EXAWATT LASER PULSES FOR SHOCK<br />
WAVES AND NUCLEAR REACTIONS<br />
Studies with the advanced PW to EW laser pulses are<br />
important also for exotic c<strong>on</strong>diti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> shock waves <strong>in</strong><br />
astrophysics [26], ultrahigh accelerati<strong>on</strong>s and for related<br />
<strong>in</strong>teracti<strong>on</strong>s <strong>in</strong>clud<strong>in</strong>g nuclear mechanisms. The essential<br />
difference to the usual thermal pressure generati<strong>on</strong><br />
processes <strong>in</strong> plasmas is the direct c<strong>on</strong>versi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> laser<br />
energy <strong>in</strong>to particle moti<strong>on</strong>. This can be seen from the<br />
n<strong>on</strong>l<strong>in</strong>ear forces <strong>in</strong>clud<strong>in</strong>g the optical resp<strong>on</strong>se s<strong>in</strong>ce 1969<br />
[25] expressed <strong>in</strong> Eq. (1). The then predicted ultrahigh<br />
accelerati<strong>on</strong>s were first measured by Sauerbrey by the<br />
Doppler effect at target <strong>in</strong>teracti<strong>on</strong> with above TW-ps<br />
laser pulses. The n<strong>on</strong>l<strong>in</strong>ear force driven accelerati<strong>on</strong>s<br />
were 10 20 cm/s 2 [27] <strong>in</strong> c<strong>on</strong>trast to comparable<br />
accelerati<strong>on</strong>s with thermal-pressures <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 15 cm/s 2 [28].<br />
The high accelerati<strong>on</strong> was <strong>in</strong> full agreement with the<br />
theory [29] and could then be used to ignite solid state<br />
density fusi<strong>on</strong> fuel deuterium tritium DT [30]. This is a<br />
3<br />
rather simplified scheme <str<strong>on</strong>g>of</str<strong>on</strong>g> ignit<strong>in</strong>g hydrogen-bor<strong>on</strong>11<br />
with produc<strong>in</strong>g less radioactive radiati<strong>on</strong> per generated<br />
energy than burn<strong>in</strong>g coal [31].<br />
In order to show the velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the react<strong>in</strong>g fusi<strong>on</strong><br />
flame – similar to cases <strong>in</strong> astrophysics – Fig. 1 shows the<br />
computati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the i<strong>on</strong> density <strong>in</strong> frozen DT at ps laser<br />
irradiati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 20 W/cm 2 . The reacti<strong>on</strong> fr<strong>on</strong>t at the<br />
<strong>in</strong>teracti<strong>on</strong> follow<strong>in</strong>g the ps igniti<strong>on</strong> at later times when<br />
propagat<strong>in</strong>g through the solid density DT can be seen<br />
where compressi<strong>on</strong>s up to four times the solid state are<br />
generated with<strong>in</strong> the mov<strong>in</strong>g short depth shock wave.<br />
This numerical result automatically agrees with the factor<br />
four <str<strong>on</strong>g>of</str<strong>on</strong>g> the Rank<strong>in</strong>e-Hug<strong>on</strong>iot shock wave theory. The<br />
shock velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> 1550 km/s is <strong>in</strong> the range known for this<br />
type <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>teracti<strong>on</strong>. For later times the fusi<strong>on</strong> flame shows<br />
more and more a deviati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the density pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile differ<strong>in</strong>g<br />
from the simplified shock wave theory. This is evident<br />
from the output <str<strong>on</strong>g>of</str<strong>on</strong>g> the fast velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the generated alpha<br />
particles when mov<strong>in</strong>g <strong>in</strong>to the untouched solid DT by<br />
gradually chang<strong>in</strong>g there the c<strong>on</strong>diti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> densities and<br />
temperatures. However the velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the entire flame is<br />
remarkably unchanged. More properties are given <strong>in</strong> the<br />
references, however, the genu<strong>in</strong>e two-fluid computati<strong>on</strong>s<br />
arrive at many more details than known from the <strong>on</strong>efluid<br />
computati<strong>on</strong> [30][31]. It is important to note that<br />
these studies are aimed to apply ps laser pulses <strong>in</strong> the<br />
range <str<strong>on</strong>g>of</str<strong>on</strong>g> 30 PW up to nearly EW. Generaliz<strong>in</strong>g the<br />
preced<strong>in</strong>g computati<strong>on</strong>s [30][31], the genu<strong>in</strong>e two-fluid<br />
hydrodynamics [32][33] is used <strong>in</strong> order to follow up the<br />
details <str<strong>on</strong>g>of</str<strong>on</strong>g> the generated very high electric fields <strong>in</strong> the<br />
shock fr<strong>on</strong>ts and to c<strong>on</strong>firm most <str<strong>on</strong>g>of</str<strong>on</strong>g> the other results<br />
calculated before with the usual <strong>on</strong>e fluid hydrodynamics.<br />
The results are <strong>in</strong>terest<strong>in</strong>g for astrophysical cases and for<br />
shock igniti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> fusi<strong>on</strong> [34] where <strong>in</strong> c<strong>on</strong>trast to the<br />
thermal pressure process, the new research now was<br />
generalized to n<strong>on</strong>-thermal n<strong>on</strong>l<strong>in</strong>ear force direct<br />
c<strong>on</strong>versi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> laser energy <strong>in</strong>to plasma moti<strong>on</strong> to reach<br />
the ultra-high accelerati<strong>on</strong>s.<br />
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Present Status <str<strong>on</strong>g>of</str<strong>on</strong>g> Ultra-<strong>in</strong>tense Lasers and High-Field <strong>Physics</strong><br />
<strong>in</strong> the World<br />
INTRODUCTION<br />
It is the 50 th year anniversary <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>venti<strong>on</strong> by<br />
Theodor Maiman, who dem<strong>on</strong>strated laser generati<strong>on</strong> <strong>in</strong><br />
1960 with a ruby rod as shown <strong>in</strong> Fig. 1. At that time,<br />
most <str<strong>on</strong>g>of</str<strong>on</strong>g> scientists believed that the gas is the best for<br />
las<strong>in</strong>g media, while Maiman took a different way aga<strong>in</strong>st<br />
all and succeeded to be the first runner <strong>in</strong> this big race.<br />
Fig. 1 Maiman and his world first laser made <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Ruby rod. It was June <strong>in</strong> 1960.<br />
In the last fifty years, the scale, power, and focused<br />
<strong>in</strong>tensity dramatically <strong>in</strong>creased from 1kW to 10 PW, the<br />
12 order <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude <strong>in</strong>crease. The laser is now widely<br />
used <strong>in</strong> fundamental and applied science. In additi<strong>on</strong>, the<br />
laser is also used for many devices <strong>in</strong> commercial goods.<br />
In the present paper, we focus <strong>on</strong> the highest <strong>in</strong>tensity<br />
laser at the present time and to be c<strong>on</strong>structed <strong>in</strong> the near<br />
future and discuss about what k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> extreme experiment<br />
we can do. The vacuum breakdown and resultant<br />
highly-relativistic electr<strong>on</strong>-positr<strong>on</strong> pair plasma<br />
producti<strong>on</strong> are ma<strong>in</strong> topics <str<strong>on</strong>g>of</str<strong>on</strong>g> this paper. We have reached<br />
the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> 2.2×10 22 W/cm 2 <str<strong>on</strong>g>of</str<strong>on</strong>g> focused laser at Univ.<br />
Michigan [1]. The electric field <str<strong>on</strong>g>of</str<strong>on</strong>g> this <strong>in</strong>tensity is 1.4×<br />
10 12 V/cm. The energy <str<strong>on</strong>g>of</str<strong>on</strong>g> an oscillat<strong>in</strong>g electr<strong>on</strong> <strong>in</strong> this<br />
electric field can be calculated to be 10 2 mc 2 . It is natural<br />
that with the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> such str<strong>on</strong>g lasers wishes to<br />
carry out a variety <str<strong>on</strong>g>of</str<strong>on</strong>g> experiments related to N<strong>on</strong>-l<strong>in</strong>er<br />
QED and QCD.<br />
H. Takabe, ILE, Osaka University, Japan<br />
In the present paper, we review the present status <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
exist<strong>in</strong>g and planned ultra-<strong>in</strong>tense laser facilities <strong>in</strong> the<br />
world at first, and describe the physics scenario <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> pair plasma. It is menti<strong>on</strong>ed that<br />
when the laser <strong>in</strong>tensity becomes 100 times the present<br />
record and it becomes above 10 24 W/cm 2 , vacuum<br />
breakdown becomes essential to creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> highly<br />
relativistic electr<strong>on</strong> pair plasma. The avalanche effect<br />
enhances the pair creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum.<br />
M U -PW L LASERS T I IN THE WORLD<br />
A great progress <strong>in</strong> c<strong>on</strong>struct<strong>in</strong>g 10 PW laser systems<br />
<strong>in</strong> Europe is go<strong>in</strong>g by the leadership <str<strong>on</strong>g>of</str<strong>on</strong>g> G. Mourou, the<br />
founder <str<strong>on</strong>g>of</str<strong>on</strong>g> the pulse compressi<strong>on</strong> technique with CPA [2].<br />
It is amaz<strong>in</strong>g that EU decided to fund three 10 PW laser<br />
facilities <strong>in</strong> the EU member countries <str<strong>on</strong>g>of</str<strong>on</strong>g> the East Europe,<br />
Czech, Hungary, and Romania. Total budget for<br />
c<strong>on</strong>structi<strong>on</strong> is about 800 M Euro. Laser property <str<strong>on</strong>g>of</str<strong>on</strong>g> each<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> them is almost same but with different ma<strong>in</strong> subjects<br />
[3]. Czech’s aims at high-brightness sources <str<strong>on</strong>g>of</str<strong>on</strong>g> x-rays and<br />
particles[4], Hungary’s atto-sec<strong>on</strong>d XUV/X-ray source<br />
and its applicati<strong>on</strong> [5], and Romania’s laser-<strong>in</strong>duced<br />
nuclear physics[6]. They are three ELI modules and the<br />
site selecti<strong>on</strong> was d<strong>on</strong>e <strong>on</strong> October 1, 2009. They will also<br />
play important role <strong>in</strong> technology development for<br />
c<strong>on</strong>struct<strong>in</strong>g 200PW laser system ELI (Extreme Light<br />
Infrastructure) [7]. It is announced that the technology<br />
and site <str<strong>on</strong>g>of</str<strong>on</strong>g> ELI are to be determ<strong>in</strong>ed after 2012.<br />
On the other hand, UK will c<strong>on</strong>struct 10 PW system<br />
by modify<strong>in</strong>g and expand the present Vulcan laser system<br />
and it will be completed due by 2014-15 [8]. France plans<br />
to c<strong>on</strong>struct a s<strong>in</strong>gle beam 10 PW laser APOLLON as<br />
<strong>in</strong>ternati<strong>on</strong>al collaborati<strong>on</strong> system with ma<strong>in</strong> c<strong>on</strong>tributi<strong>on</strong><br />
by three <strong>in</strong>stituti<strong>on</strong>s, LULI, LOA, and Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Optics<br />
and it will be completed due by 2012-13. Vulcan is <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
300J, 30fs, 1shot/15 m<strong>in</strong>, and APPLLON is <str<strong>on</strong>g>of</str<strong>on</strong>g> 150J, 15 fs,<br />
1shot/m<strong>in</strong>.<br />
In Asia, APRI <strong>in</strong> Korea has 1PW laser system with 47<br />
J/30fs at 0.1 Hz [9]. In Japan, 1 PW laser is <strong>in</strong><br />
Kansai-J A E [10] A and 10 PW laser LFEX with 10 kJ/ 1ps<br />
s<strong>in</strong>gle shot base [1 1] is under c<strong>on</strong>structi<strong>on</strong> <strong>in</strong> Osaka<br />
University.
In USA, LLNL has a user’s facility, Titan laser which<br />
is 1PW system with 400fs--10ps, up to 530 J and<br />
2shots/hour [12]. At LLE, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Rochester, an<br />
ultra-<strong>in</strong>tense laser system OMEGA-EP is now <strong>in</strong><br />
operati<strong>on</strong>. It is <str<strong>on</strong>g>of</str<strong>on</strong>g> 1 P multi-kJ, W , 1ps and <strong>in</strong>tensity<br />
higher than 10 20 W/cm 2 We dem<strong>on</strong>strated that the Bethe-Heitler process is ma<strong>in</strong><br />
process to produce the pairs, although it is the two step<br />
process <strong>in</strong> the gold foil. We predicted the positr<strong>on</strong><br />
spectrum and compared with Cowan’s experimental data.<br />
What we found is that the experimental data looks like<br />
[13]. In additi<strong>on</strong>, the world- shifted by about 10 MeV to the higher energy side as you<br />
biggest laser NIF will have four beams to be multi-P It W . can see <strong>in</strong> Fig. 2 [18]. The M<strong>on</strong>te-Carlo calculati<strong>on</strong><br />
is called NIF-ARC [14]. NIF-ARC is orig<strong>in</strong>ally motivated<br />
for diagnostic purpose for NIF igniti<strong>on</strong> campaign (NIC),<br />
but it is also open to users <strong>in</strong> the world <strong>on</strong>ly for the<br />
fundamental science.<br />
PAIR PLASMAS<br />
E<strong>in</strong>ste<strong>in</strong> said “Imag<strong>in</strong>ati<strong>on</strong> is more important than<br />
knowledge”. We can spread the world <str<strong>on</strong>g>of</str<strong>on</strong>g> imag<strong>in</strong>ati<strong>on</strong> with<br />
the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> such ultra-<strong>in</strong>tense lasers. The first<br />
experiment with PW laser was d<strong>on</strong>e by T. Cowan et al.<br />
[15] and he reported photo-nuclear fissi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Uranium<br />
and positr<strong>on</strong> producti<strong>on</strong> <strong>in</strong> a gold foil. In order to analyze<br />
his experimental data <strong>on</strong> positr<strong>on</strong> producti<strong>on</strong> [16], we<br />
pursued jo<strong>in</strong>t research start<strong>in</strong>g from model<strong>in</strong>g the Trident<br />
and Bethe-Heitler processes <strong>in</strong> Fokker-Planck equati<strong>on</strong> to<br />
relativistic electr<strong>on</strong>s as ma<strong>in</strong> source for pair creati<strong>on</strong> [17].<br />
Figure 2: (a) The l<strong>on</strong>gitud<strong>in</strong>al momentum<br />
distributi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> positr<strong>on</strong> (red) and electr<strong>on</strong>s (black) at<br />
back <str<strong>on</strong>g>of</str<strong>on</strong>g> the target. The target rear is at x=0. The solid<br />
l<strong>in</strong>e <strong>in</strong>dicates the sheath field normalized by 10 12 V/m.<br />
(b) Positr<strong>on</strong> spectrum calculated by PIC. The blue<br />
dotted l<strong>in</strong>e is the electr<strong>on</strong> spectrum used <strong>in</strong> PIC<br />
calculati<strong>on</strong>.<br />
usually used <strong>in</strong> HEP field [17] predicted almost the same<br />
as Fokker-Planck case.<br />
In order to see the physics caus<strong>in</strong>g the difference, we<br />
thought that s<strong>in</strong>ce the laser-produced relativistic electr<strong>on</strong><br />
density is much higher than the case <str<strong>on</strong>g>of</str<strong>on</strong>g> accelerator beams,<br />
a str<strong>on</strong>g electric field formati<strong>on</strong> by charge separati<strong>on</strong><br />
effect, namely; plasma effect, becomes important to<br />
accelerate the created positr<strong>on</strong> at the rear side <str<strong>on</strong>g>of</str<strong>on</strong>g> the target<br />
foil [18]. We po<strong>in</strong>ted out this is critically different po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
laser case compared to the accelerator case. We have<br />
carried out two-dimensi<strong>on</strong>al PIC (Particle-<strong>in</strong>-Cell)<br />
simulati<strong>on</strong> [20] with the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong> calculated with<br />
the Fokker-Planck equati<strong>on</strong> and successes to reproduce<br />
the experimental positr<strong>on</strong> spectrum as shown <strong>in</strong> l<strong>in</strong>e with<br />
PIC. PIC simulati<strong>on</strong> calculated Maxwell equati<strong>on</strong>s c<strong>on</strong>-<br />
sistently with charged particles and the electric field the<br />
potential <str<strong>on</strong>g>of</str<strong>on</strong>g> about 20 MV is found to be produced. It<br />
should be noted that s<strong>in</strong>ce the electric field is the vertical<br />
directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the target rear surface and the positr<strong>on</strong>s are<br />
accelerated predom<strong>in</strong>antly to this directi<strong>on</strong>, we obta<strong>in</strong> a<br />
jet like positr<strong>on</strong>s with Lorentz factor about 10-20 the<br />
number <str<strong>on</strong>g>of</str<strong>on</strong>g> which is the same as those <str<strong>on</strong>g>of</str<strong>on</strong>g> AGN-jets<br />
(Cosmological jets) [21]. Recently, H. Chen et al. pub-<br />
lished several papers and they also found the importance<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field and jet-like emissi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> positr<strong>on</strong>s [22].<br />
We are now collaborat<strong>in</strong>g to dem<strong>on</strong>strate the charge<br />
Fig. 3 Positr<strong>on</strong> number scal<strong>in</strong>g from 100’s Joule<br />
regi<strong>on</strong> to Several KJ laser case.
neutral pair plasma producti<strong>on</strong> as shown <strong>in</strong> Fig. 3 with 10<br />
kJ class PW laser NIF-ARC to be completed <strong>in</strong> 2013 [23].<br />
VACUUM BRAKDOWN<br />
With more <strong>in</strong>crease <str<strong>on</strong>g>of</str<strong>on</strong>g> laser <strong>in</strong>tensity higher than<br />
10 22 W/cm 2 , we have a possibility to use all laser energy<br />
to create the pair plasma fireball. This physics is based <strong>on</strong><br />
an excit<strong>in</strong>g physics <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum breakdown with laser<br />
fields. In this case, we use the vacuum as target to<br />
produce the pair plasma. The pair plasma creati<strong>on</strong> itself is<br />
a t<strong>in</strong>y topic, but the dem<strong>on</strong>strati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> “Vacuum<br />
Breakdown” is more dramatic target for the research.<br />
The vacuum breakdown started to be <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the argu<strong>in</strong>g<br />
topics just after the paper <str<strong>on</strong>g>of</str<strong>on</strong>g> relativistic quantum<br />
mechanics with Dirac equati<strong>on</strong> [24]. N . B o W h . r ,<br />
Heisenberg, and so <strong>on</strong> predicted the vacuum breakdown<br />
as “Gedanken experiment”. It is now go<strong>in</strong>g to be a real<br />
experiment thanks to the progress <str<strong>on</strong>g>of</str<strong>on</strong>g> laser technology.<br />
When the laser <strong>in</strong>tensity exceeds 10 24 W/cm 2 , we<br />
can expect the follow<strong>in</strong>g three different physics scenario<br />
with <strong>in</strong>crease <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser <strong>in</strong>tensity.<br />
(1) Vacuum breakdown due to the pair avalanche<br />
triggered by the <strong>in</strong>duced pair producti<strong>on</strong><br />
(2) Vacuum breakdown due to the pair avalanche<br />
by sp<strong>on</strong>taneous pair producti<strong>on</strong><br />
(3) Vacuum breakdown without the avalanche near<br />
and over the Schw<strong>in</strong>ger limit<br />
When the c<strong>on</strong>cept <str<strong>on</strong>g>of</str<strong>on</strong>g> Dirac sea is proposed, dist<strong>in</strong>guished<br />
physicists predicted that if a str<strong>on</strong>g electric field is<br />
imposed <strong>in</strong> the vacuum, electr<strong>on</strong>-positr<strong>on</strong> pairs appears<br />
because <str<strong>on</strong>g>of</str<strong>on</strong>g> the tunnel<strong>in</strong>g effect <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum particles.<br />
Heisenberg’s uncerta<strong>in</strong> pr<strong>in</strong>ciple requires the existence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
quantum noise energy even <strong>in</strong> a complete vacuum. This<br />
means any particle with a f<strong>in</strong>ite energy shows its face <strong>in</strong> a<br />
very short time <strong>in</strong> the vacuum because the uncerta<strong>in</strong><br />
pr<strong>in</strong>ciple requires ∆ε∆t ≥ 1/2ℏ. If we assume ∆ε = mc 2 ,<br />
then we obta<strong>in</strong> ∆t ~ 10 -21 s. If dur<strong>in</strong>g this extremely short<br />
time, we can give the energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> the rest mass<br />
and separate the pair over a substantial distance to avoid<br />
sp<strong>on</strong>taneous annihilati<strong>on</strong>, real pair is created.<br />
The critical value <str<strong>on</strong>g>of</str<strong>on</strong>g> such extremely str<strong>on</strong>g electric<br />
field E can easily obta<strong>in</strong>ed with the relati<strong>on</strong>
field trigger<strong>in</strong>g such phenomena is calculated with<br />
particle and M<strong>on</strong>te-Carlo hybrid code [27] and it is<br />
c<strong>on</strong>cluded to be about 10 24 W/cm 2 , the value <str<strong>on</strong>g>of</str<strong>on</strong>g> which is<br />
very lower than the Schw<strong>in</strong>ger limit <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (4). It should<br />
be noted that prolific pair creati<strong>on</strong> occurs thanks to the<br />
avalanche effect expla<strong>in</strong>ed later.<br />
In order to know the radiati<strong>on</strong> emissi<strong>on</strong> by an<br />
electr<strong>on</strong> <strong>in</strong> a str<strong>on</strong>g electromagnetic field, we have to<br />
solve the soluti<strong>on</strong> for the Lagrangian
small for the electric field weaker than Schw<strong>in</strong>ger limit <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Eq. (3).<br />
It is very important to know that for the plane wave<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> laser field <strong>in</strong> the vacuum, the two scalars S and P<br />
disappear and no tunnel<strong>in</strong>g effect can be expected even<br />
with the laser <strong>in</strong>tensity higher than the Schw<strong>in</strong>ger limit <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Eq. (4). So, we are required to deform the laser field from<br />
the plane wave relati<strong>on</strong> to maximize the Lorentz<br />
<strong>in</strong>variants S and P, especially the scalar P <strong>in</strong> Eqs. (8) and<br />
(9). In order to optimize with realistic optics, not <strong>on</strong>ly the<br />
collid<strong>in</strong>g laser scheme but also str<strong>on</strong>gly tightly focused<br />
laser field is proposed [32]. In additi<strong>on</strong>, multiple collid<strong>in</strong>g<br />
laser pulses method is also proposed and evaluated<br />
quantitatively how less laser energy is enough to produce<br />
<strong>on</strong>e pair [33]. We have proposed another tightly focused<br />
laser with use <str<strong>on</strong>g>of</str<strong>on</strong>g> a radially polarized laser [34].<br />
In the above three schemes, it is c<strong>on</strong>cluded that the<br />
tunnel<strong>in</strong>g effect starts with the laser <strong>in</strong>tensity more than<br />
10 26 -10 27 W/cm 2 . Although the tunnel<strong>in</strong>g effect is not<br />
substantial like Schw<strong>in</strong>ger limit, the seed pairs are pro-<br />
duced <strong>in</strong> the pure vacuum and the prolific pair producti<strong>on</strong><br />
and the resultant vacuum breakdown might be expected<br />
by the avalanche mechanism. Of course, very precise cal-<br />
culati<strong>on</strong> is necessary with the computati<strong>on</strong>al model<strong>in</strong>g<br />
briefly menti<strong>on</strong>ed previously. In this case, solv<strong>in</strong>g the<br />
Maxwell equati<strong>on</strong>s self-c<strong>on</strong>sistently is very important to<br />
see if the laser field is scattered out without be<strong>in</strong>g focused<br />
like the case <str<strong>on</strong>g>of</str<strong>on</strong>g> pure vacuum. This is because the<br />
appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> the pairs means average refractive <strong>in</strong>dex<br />
changes as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> time at the focused po<strong>in</strong>t.<br />
When the laser <strong>in</strong>tensity reaches the Schw<strong>in</strong>ger limit<br />
Eq. (4), the pair creati<strong>on</strong> by the tunnel<strong>in</strong>g effect becomes<br />
dom<strong>in</strong>ant and sp<strong>on</strong>taneous creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the pairs occur at<br />
all the place where the value <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (7) is large enough.<br />
In this case, the <strong>in</strong>terest<strong>in</strong>g po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> research is how the<br />
avalanche effect is still important for the pair creati<strong>on</strong><br />
process. In additi<strong>on</strong>, we also expect the creati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> low<br />
energy hadr<strong>on</strong> and hadr<strong>on</strong> pairs such as π and π ± .<br />
What k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> lept<strong>on</strong> pairs and hadr<strong>on</strong> mixture or quark-<br />
glu<strong>on</strong> mixture plasma is produced is a very <strong>in</strong>terest<strong>in</strong>g<br />
subject.<br />
LABORATORY COSMOLOGY AND<br />
MODELING QUARK-GLUON PLASMA<br />
We can expect to do a variety <str<strong>on</strong>g>of</str<strong>on</strong>g> model experiments<br />
for open questi<strong>on</strong>s <strong>in</strong> high-energy astrophysics. The<br />
<strong>in</strong>vestigati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma properties <str<strong>on</strong>g>of</str<strong>on</strong>g> highly rela-<br />
tivistic electr<strong>on</strong> positr<strong>on</strong> plasmas and jets is essential for<br />
example, AGN jets and the jets related to the gamma-ray<br />
bursts [35]. In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> “quiet” pair producti<strong>on</strong> <strong>in</strong> gold<br />
foil as described relat<strong>in</strong>g to Fig. 2, the am-bipolar field<br />
accelerates the positr<strong>on</strong> to produce relativistic pair plasma<br />
<strong>in</strong> the jet-like structure. This is good to model the<br />
propagati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> pair plasma jets <strong>in</strong> the Universe.<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> “violent” pair creati<strong>on</strong> via the vacuum<br />
breakdown, it is not clear what k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> energy distributi<strong>on</strong><br />
and angular distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair plasma is produced<br />
and how we can c<strong>on</strong>trol them. In additi<strong>on</strong> the distributi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> gamma-ray is also open questi<strong>on</strong>. It may not be <strong>in</strong> the<br />
thermodynamic equilibrium state, while the physical<br />
dynamics <strong>in</strong> the highly relativistic “pair fireball” is very<br />
<strong>in</strong>terest<strong>in</strong>g matter to study l<strong>in</strong>er and n<strong>on</strong>l<strong>in</strong>ear collective<br />
phenomena as plasma physics. Whether a collisi<strong>on</strong>less<br />
shock formati<strong>on</strong> may observed after the n<strong>on</strong>l<strong>in</strong>ear stage is<br />
very hot topics <strong>in</strong> the GRB physics [36]. For example, this<br />
plasma would be a good test bed to verify and validate the<br />
computati<strong>on</strong>al model<strong>in</strong>g. It is very challeng<strong>in</strong>g, while the<br />
improvement <str<strong>on</strong>g>of</str<strong>on</strong>g> such simulati<strong>on</strong> code compared to the<br />
real experiment is very much beneficial also to the<br />
model<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> Quark-Glu<strong>on</strong> plasma (QGP), where the color<br />
force is dom<strong>in</strong>ant than the electric force. It is said that<br />
n<strong>on</strong>-Abelian system like QGP would become thermo-<br />
dynamic equilibrium <strong>in</strong> an extremely short time [37]. In<br />
additi<strong>on</strong>, we may be able to identify if the avalanche<br />
effect plays an important role <strong>in</strong> the formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP <strong>in</strong><br />
extremely t<strong>in</strong>y space and time.<br />
As schematically shown <strong>in</strong> Fig. 6, the QGP created<br />
through vacuum breakdown by color field is <str<strong>on</strong>g>of</str<strong>on</strong>g> the order<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> 10 -12 cm (=10fm) and the life time <str<strong>on</strong>g>of</str<strong>on</strong>g> QGP is about<br />
10fm/c = 3×10 -21 s. In the RHIC [38] and LHC [39]<br />
experiment for QGP, therefore, it is difficult to measure<br />
the plasma state and, for example, the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> state <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Fig. 6 Laser produced electr<strong>on</strong> pair fireball will be a<br />
model experiment <str<strong>on</strong>g>of</str<strong>on</strong>g> quark-glu<strong>on</strong> plasma (QGP)<br />
dynamics.
QGP is studied by us<strong>in</strong>g the particle energy and angle<br />
distributi<strong>on</strong> determ<strong>in</strong>ed just after the phase transiti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
liquid phase <str<strong>on</strong>g>of</str<strong>on</strong>g> QCD to free particle state which is called<br />
freeze-out [40]. In case <str<strong>on</strong>g>of</str<strong>on</strong>g> laser driven vacuum breakdown,<br />
the lept<strong>on</strong> pair plasma is generated and its spatial<br />
scale is about 10 -4 cm (=1µm) and life time is about 10 -15 s<br />
(1fs) as schematically shown <strong>in</strong> Fig. 6. S<strong>in</strong>ce the plasma<br />
size and lifetime are milli<strong>on</strong> times bigger and l<strong>on</strong>ger than<br />
t h we can e develop the Q diagnostics G to P directly ,<br />
[1] V Yan<str<strong>on</strong>g>of</str<strong>on</strong>g>sky . et al., Opt. Express16, 2109n (2008)<br />
[2] G.A.Mourou, Phys. Today 51,No.1,22(1998).<br />
[3] http://www.extreme-light-<strong>in</strong>frastructure.eu/<br />
[4] Czech: http://www.eli-beams.eu/<br />
[5] Hungary:<br />
http://www.eli-beams.eu/news-from-hungary/<br />
[6] Romania: http://eli.ifa-mg.ro/<br />
[7] ELI: http://www.eli-beams.eu/evropsky-projekt/<br />
eli-v-kostce/<br />
[8] http://www.clf.rl.ac.uk/<br />
[9] http://apri.gist.ac.kr/eng/<strong>in</strong>dex.php<br />
[10] http://wwwapr.kansai.jaea.go.jp/outl<strong>in</strong>e.html<br />
[ 1 1 ] http://www.ile.osaka-u.ac.jp/<br />
[12] https://jlf.llnl.gov/html/facilities/titan/titan.html<br />
[13] http://www.lle.rochester.edu/<br />
[14] https://e-reports-ext.llnl.gov/pdf/349575.pdf<br />
[15] T.E. Cowan et al., Phys. Rev. Lett. 84, 903 (2000).<br />
[16] T.E. Cowan et al., Laser Part. Beams 17, 773<br />
(1999).<br />
[17] K. Nakashima and H. Takabe, Phys. Plasmas 9, 1505<br />
(2002).<br />
measure the properties <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair plasma. For example,<br />
we can measure the turbulent spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the highly<br />
n<strong>on</strong>l<strong>in</strong>ear stage <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair plasma with a bright x-ray<br />
source <strong>in</strong> the range <str<strong>on</strong>g>of</str<strong>on</strong>g> atto-sec<strong>on</strong>d pulse technology to be<br />
developed <strong>in</strong> Hungary [5]. Deep understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> such<br />
plasma and verificati<strong>on</strong> and validati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the computati<strong>on</strong>al<br />
model<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum breakdown, avalanche<br />
effect, phot<strong>on</strong>-lept<strong>on</strong> <strong>in</strong>teracti<strong>on</strong>, and a variety <str<strong>on</strong>g>of</str<strong>on</strong>g> physics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> highly relativistic pair plasma are sure to be very much<br />
beneficial for understand<strong>in</strong>g the QGP physics. We can say<br />
the vacuum breakdown physics is a model experiment <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
QGP <strong>in</strong> relatively small scale laser facility.<br />
CONCLUSION<br />
With the progress <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tense laser technology, the<br />
physics to be solved <strong>in</strong> laser-matter <strong>in</strong>teracti<strong>on</strong> has<br />
already become relativistic regime for electr<strong>on</strong>s. In<br />
several years, we will come to the regime where laservacuum<br />
<strong>in</strong>teracti<strong>on</strong> becomes important and the 80-years<br />
stand<strong>in</strong>g theory <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum breakdown will be really<br />
experimentally studied. We c<strong>on</strong>cluded that the electr<strong>on</strong><br />
pair creati<strong>on</strong> can be expected from 10 24 W/cm 2 with use<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> seed electr<strong>on</strong>s and thanks to the resultant avalanche<br />
effect. When the laser <strong>in</strong>tensity exceeds more than 10 26<br />
W/cm 2 [18] T. Cowan et al, 2002: H. Takabe, <strong>in</strong> “Hadr<strong>on</strong> and<br />
Nuclear <strong>Physics</strong> 09”, Edt. A. Hosaka et al., p.<br />
388-403 (World Scientifics, 2010)<br />
[19] Y. Sentoku, private communicati<strong>on</strong>.<br />
[20] For example; L. L. Stephan, Computer <strong>Physics</strong><br />
Communicati<strong>on</strong>s 72, Issues 2-3, November 1992,<br />
Pages 144-148<br />
[21]<br />
[22]<br />
[23]<br />
[24]<br />
A. Mizuta, S. Yamada, and H. Takabe,<br />
Astrophysical J. 606 804 (2004).<br />
H. Chen et al., Phys. Plasmas 16, 122702 (2009).<br />
References there <strong>in</strong>.<br />
H. Chen, H. Takabe et al, NIF proposal (2010) ,<br />
not published, approved by NIF committee.<br />
P. Dirac, The Pr<strong>in</strong>ciple <str<strong>on</strong>g>of</str<strong>on</strong>g> Quantum Mechanics.<br />
[25]<br />
(Oxford Univ. Press, 1930).<br />
A. R. Bell and J. G. Kirk, Phys. Rev. Lett., 101,<br />
200403 (2008)<br />
[26] Landau & Lifshitz, “Theory <str<strong>on</strong>g>of</str<strong>on</strong>g> Classical Field”,<br />
Chap. 9, S.75.<br />
[27] J. G. Kirk, A. R. Bell, and I. Arka, Plasam Phys.<br />
, we can really break down the pure vacuum with<br />
[28]<br />
[29]<br />
[30]<br />
[31]<br />
[32]<br />
C<strong>on</strong>trol Fusi<strong>on</strong> 51 085008 (2009).<br />
H. Takabe, “Relativistic Plasma <strong>Physics</strong>”, J.<br />
Plasma Fusi<strong>on</strong> Res. 78 427-438 (2002), <strong>in</strong><br />
Japanese<br />
N. Elk<strong>in</strong>a and H. Ruhl, <strong>in</strong> this proceed<strong>in</strong>g<br />
N. B. Narozhny et al., JETP Letters<br />
R. Ruff<strong>in</strong>i et al., <strong>Physics</strong> Reports 487, 1-140<br />
(2010)<br />
A. M. Fedotov, Laser <strong>Physics</strong> 19, 214 (2009)<br />
help <str<strong>on</strong>g>of</str<strong>on</strong>g> pairs appear<strong>in</strong>g by quantum tunnel<strong>in</strong>g effect. W e [33] S. S. Bulanov et al., Phys. Rev. Lett. 104, 220404<br />
also po<strong>in</strong>ted out that this lept<strong>on</strong> fireball physics will be (2010)<br />
beneficial to model<strong>in</strong>g quark-glu<strong>on</strong> plasmas.<br />
[34] G. Miyaji, PhD Thesis, January, 2004<br />
[35] For example; F. D. Seward and P. A. Charles,<br />
REFERENCES<br />
“Explor<strong>in</strong>g the X-ray Universe” 2 nd editi<strong>on</strong><br />
(Cambridge, 2010)<br />
[36] T. N. Kato, Astrophysical J. 668 974-979, (2007);<br />
P. Chang, A. Spitkovsky, and J. Ar<strong>on</strong>s 674 378,<br />
(2008).<br />
[37] M. Asakawa et al., Phys. Rev. Lett. 96, 252301<br />
(2006)<br />
[38] RHIC: http://www.bnl.gov/rhic/<br />
[39] LHC: http://lhc.web.cern.ch/lhc/<br />
[40] S. A. Bass and A. Dumitru, Phys. Rev. C 61,<br />
064909 (2000)
REACHING THE SCHWINGER LIMIT WITH X-RAYS*<br />
Charles K. Rhodes, John Boguta, Alex B. Borisov, Shahab F. Khan, James W. L<strong>on</strong>gworth, John C.<br />
McCork<strong>in</strong>dale, Sankar Poopalas<strong>in</strong>gam, Erv<strong>in</strong> Racz # and Ji Zhao<br />
Laboratory for X-ray Microimag<strong>in</strong>g and Bio<strong>in</strong>formatics, Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, University <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Ill<strong>in</strong>ois at Chicago, Chicago, IL 60607-7059, USA<br />
Abstract<br />
The derivati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an elementary figure <str<strong>on</strong>g>of</str<strong>on</strong>g> merit shows<br />
that atta<strong>in</strong>ment <str<strong>on</strong>g>of</str<strong>on</strong>g> an <strong>in</strong>tensity corresp<strong>on</strong>d<strong>in</strong>g to the<br />
Schw<strong>in</strong>ger/Heisenberg Limit (~ 4.6 x 10 29 W/cm 2 ) is<br />
significantly facilitated by the use <str<strong>on</strong>g>of</str<strong>on</strong>g> coherent x-ray<br />
sources <strong>in</strong> the kiloelectr<strong>on</strong>volt regime. For the Xe(L)<br />
system at ~ 4.5 keV, a m<strong>in</strong>imum pulse energy <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 1.5 J<br />
and corresp<strong>on</strong>d<strong>in</strong>g peak power P0 ~ 300 PW are estimated.<br />
INTRODUCTION<br />
The history <str<strong>on</strong>g>of</str<strong>on</strong>g> high-<strong>in</strong>tensity n<strong>on</strong>l<strong>in</strong>ear <strong>in</strong>teracti<strong>on</strong>s, that<br />
commenced <strong>in</strong> 1961 with the observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> sec<strong>on</strong>d<br />
harm<strong>on</strong>ic radiati<strong>on</strong> [1] at 347.2 nm <strong>in</strong> crystall<strong>in</strong>e quartz,<br />
spans a range <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 10 18 <strong>in</strong> experimental <strong>in</strong>tensity and<br />
rema<strong>in</strong>s a stable, robust prov<strong>in</strong>ce <str<strong>on</strong>g>of</str<strong>on</strong>g> laser-based research<br />
after a half century. The generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> focal <strong>in</strong>tensities <strong>in</strong><br />
the 10 20 -10 21 W/cm 2 range is presently a rout<strong>in</strong>e<br />
achievement. Over a period <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 25 years, a path <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
research was cut through this field <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>l<strong>in</strong>ear<br />
phenomena that led to the development <str<strong>on</strong>g>of</str<strong>on</strong>g> a multikilovolt<br />
(~ 4.5 keV) x-ray amplifier <str<strong>on</strong>g>of</str<strong>on</strong>g> excepti<strong>on</strong>al peak brightness<br />
[2-5] that is excited by femtosec<strong>on</strong>d KrF* (248 nm)<br />
pulses and whose experimentally based power scal<strong>in</strong>g<br />
limit for a compact laboratory <strong>in</strong>strument falls <strong>in</strong> the<br />
multipetawatt realm [6]. The existence <str<strong>on</strong>g>of</str<strong>on</strong>g> advanced highenergy<br />
KrF* technology [7], that has been <strong>in</strong>dependently<br />
developed for fusi<strong>on</strong> applicati<strong>on</strong>s, could be readily<br />
adapted to extend the coherent x-ray power level <strong>in</strong>to the<br />
200 - 500 PW regime.<br />
* Sp<strong>on</strong>sored by Defense Advanced Research Projects<br />
Agency, Microsystems Technology Office (MTO),<br />
Program: Ultrabeam - The Scal<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> Coherent<br />
Amplificati<strong>on</strong> to the Gamma Ray Regi<strong>on</strong>, issued by<br />
DARPA/CMO under C<strong>on</strong>tract No. HR0011-10-C-0105.<br />
"The views and c<strong>on</strong>clusi<strong>on</strong>s c<strong>on</strong>ta<strong>in</strong>ed <strong>in</strong> this document<br />
are those <str<strong>on</strong>g>of</str<strong>on</strong>g> the authors and should not be <strong>in</strong>terpreted as<br />
represent<strong>in</strong>g the <str<strong>on</strong>g>of</str<strong>on</strong>g>ficial policies, either expressly or<br />
implied, <str<strong>on</strong>g>of</str<strong>on</strong>g> the Defense Advanced Research Projects<br />
Agency or the U.S. Government."<br />
rhodes@uic.edu<br />
# KFKI Research Institute for Particle and Nuclear<br />
<strong>Physics</strong>, EURATOM Associati<strong>on</strong>, P.O. Box 49,1525,<br />
Budapest, Hungary<br />
DISCUSSION<br />
The use <str<strong>on</strong>g>of</str<strong>on</strong>g> coherent x-rays for the atta<strong>in</strong>ment <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<strong>in</strong>tensities approach<strong>in</strong>g the Schw<strong>in</strong>ger Limit [8] <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 4.6 x<br />
10 29 W/cm 2 , a c<strong>on</strong>diti<strong>on</strong> <strong>in</strong>itially discussed by Sauter [9]<br />
and Heisenberg and Euler [10], is supported by very<br />
powerful scal<strong>in</strong>g relati<strong>on</strong>ships. A summary <str<strong>on</strong>g>of</str<strong>on</strong>g> the key<br />
physical parameters is presented <strong>in</strong> Fig. (1); the<br />
assessment gives the comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an <strong>in</strong>frared source<br />
(ħω ≅ 1eV) with a corresp<strong>on</strong>d<strong>in</strong>g source deliver<strong>in</strong>g (ħω<br />
≅ 4.5 keV) x-rays. The chief outcome is a figure <str<strong>on</strong>g>of</str<strong>on</strong>g> merit<br />
that measures the propensity to generate a high <strong>in</strong>tensity,<br />
and the x-ray illum<strong>in</strong>ator is favored over the <strong>in</strong>frared<br />
system by a factor that exceeds 10 21 , a result predicated<br />
<strong>on</strong> a relati<strong>on</strong>ship proporti<strong>on</strong>al to ω 6 . Furthermore, it is<br />
known that the overall physical situati<strong>on</strong> govern<strong>in</strong>g the<br />
atta<strong>in</strong>ment <str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger c<strong>on</strong>diti<strong>on</strong> is generally<br />
assisted by the existence <str<strong>on</strong>g>of</str<strong>on</strong>g> particularly propitious<br />
geometries <str<strong>on</strong>g>of</str<strong>on</strong>g> irradiati<strong>on</strong> [11,12].<br />
An additi<strong>on</strong>al advantage <str<strong>on</strong>g>of</str<strong>on</strong>g> an x-ray wavelength is the<br />
very large elevati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tensity characteristic <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
relativistic regime that flows from a fundamental scal<strong>in</strong>g<br />
proporti<strong>on</strong>al to ω 2 . Basically, the electric field E<br />
corresp<strong>on</strong>d<strong>in</strong>g to the relativistic regime is given by the<br />
c<strong>on</strong>diti<strong>on</strong><br />
eE<br />
mc ω<br />
= 1 (1)<br />
which, for Xe(L) radiati<strong>on</strong> with ħω ≅ 4.5 keV, yields an<br />
<strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 10 25 W/cm 2 . This value is regarded both as<br />
(a) sufficiently high to produce observable c<strong>on</strong>sequences<br />
from the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum [13] and (b) free <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
cascade limit for <strong>in</strong>tensities up to the Schw<strong>in</strong>ger/<br />
Heisenberg value, <strong>in</strong> c<strong>on</strong>trast to the <strong>in</strong>frared case [14].<br />
The ability to produce stable self-channeled propagati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tense pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> radiati<strong>on</strong> <strong>in</strong> an underdense<br />
plasma is a well established phenomen<strong>on</strong> at quantum<br />
energies below ~5 eV [15-18]. An immediate c<strong>on</strong>sequence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the availability <str<strong>on</strong>g>of</str<strong>on</strong>g> a high power source <str<strong>on</strong>g>of</str<strong>on</strong>g> coherent<br />
x-rays is the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> extend<strong>in</strong>g the generati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> these highly c<strong>on</strong>f<strong>in</strong>ed stable [16] modes <str<strong>on</strong>g>of</str<strong>on</strong>g> deeply<br />
penetrat<strong>in</strong>g propagati<strong>on</strong> to x-ray wavelengths <strong>in</strong> materials<br />
at solid density. If such channels can be pro-duced with<br />
multi-kilovolt x-rays <strong>in</strong> high-Z solids, the producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
power densities <strong>on</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 10 30 W/cm 3 are<br />
projected [19,20].
Fig. (1): Presentati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the physical parameters<br />
associated with the development <str<strong>on</strong>g>of</str<strong>on</strong>g> a figure <str<strong>on</strong>g>of</str<strong>on</strong>g> merit for<br />
the propensity to produce a high focal <strong>in</strong>tensity. The<br />
comb<strong>in</strong>ed c<strong>on</strong>siderati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the quantum energy (ħω), the<br />
radiative rate, and the characteristic area, produces a<br />
str<strong>on</strong>g scal<strong>in</strong>g relati<strong>on</strong>ship favor<strong>in</strong>g short wavelengths<br />
that is proporti<strong>on</strong>al to ω 6 . Accord<strong>in</strong>gly, <strong>in</strong> the comparis<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>frared (ħω ≅ eV) to the x-ray range (ħω ≅ 4.5<br />
keV) associated with the Xe(L) system, the figure <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
merit <strong>in</strong>creases by a factor greater than 10 21 .<br />
PHOTON STAGING CONCEPT<br />
The key c<strong>on</strong>cept for the producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> ultrahigh<br />
<strong>in</strong>tensities with x-rays is “Phot<strong>on</strong> Stag<strong>in</strong>g”. Simply stated,<br />
this is the channel<strong>in</strong>g process outl<strong>in</strong>ed <strong>in</strong> Fig. (2) elevated<br />
<strong>in</strong> both frequency ω and electr<strong>on</strong> density ne. The<br />
govern<strong>in</strong>g scal<strong>in</strong>gs are illustrated <strong>in</strong> Fig. (2); basically, it<br />
is the channel<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> x-rays <strong>in</strong> solids, a phenomen<strong>on</strong> that<br />
raises the critical power Pcr to values <strong>in</strong> the range <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 0.1-<br />
1 PW. S<strong>in</strong>ce the plasma density is ne ~ 4 – 5 x 10 24 cm -3<br />
<strong>in</strong> high-Z solids, the corresp<strong>on</strong>d<strong>in</strong>g channel diameters are<br />
compressed to ~ 100 Ǻ <strong>in</strong> materials like Fe, Au, and U.<br />
At the wavelength λx ~ 2.9 Å, for which the critical<br />
electr<strong>on</strong> density is ncr = 1.33 × 10 28 cm −3 , all fully i<strong>on</strong>ized<br />
c<strong>on</strong>densed matter is underdense, <strong>in</strong>clud<strong>in</strong>g uranium. The<br />
key requirement [15,16] for the producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a channel <strong>in</strong><br />
the underdense regime is the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a peak power<br />
Po exceed<strong>in</strong>g the critical power Pcr necessary for the<br />
development <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>f<strong>in</strong>ed propagati<strong>on</strong> with the<br />
relativistic/charge-displacement mechanism.<br />
The stability <str<strong>on</strong>g>of</str<strong>on</strong>g> the propagati<strong>on</strong> is assured by the<br />
existence <str<strong>on</strong>g>of</str<strong>on</strong>g> a robust eigenmode [16]. For the case <str<strong>on</strong>g>of</str<strong>on</strong>g> λx<br />
≅ 2.9 Å and fully i<strong>on</strong>ized uranium, Pcr ≅ 49 TW. With a<br />
pulse length τx ~ 50 as, a value that is well with<strong>in</strong> the<br />
projected performance [5,6] <str<strong>on</strong>g>of</str<strong>on</strong>g> the Xe(L) system, the<br />
critical power corresp<strong>on</strong>d<strong>in</strong>g to uranium can be achieved<br />
with a pulse energy as small as Ex ~ 3.0 mJ.<br />
Fig. (2): “Phot<strong>on</strong> Stag<strong>in</strong>g”, channel formati<strong>on</strong> <strong>in</strong> solids<br />
with x-rays. Channel formati<strong>on</strong> becomes possible with<br />
pulse energies <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 10 – 100 mJ for corresp<strong>on</strong>d<strong>in</strong>g x-ray<br />
pulse lengths <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 10 – 100 as. Enormously enhanced<br />
power compressi<strong>on</strong> is the lead<strong>in</strong>g outcome.<br />
FINDINGS<br />
The power and <strong>in</strong>tensity scal<strong>in</strong>g properties <str<strong>on</strong>g>of</str<strong>on</strong>g> channels<br />
produced <strong>in</strong> uranium by Xe(M) and Xe(L) radiati<strong>on</strong>,<br />
respectively at ħω ≅ 1 keV and ħω ≅ 4.5 keV, are<br />
summarized <strong>in</strong> Figs. (3) and (4). The calculati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
propagati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Xe(L) radiati<strong>on</strong> <strong>in</strong> uranium channels show<br />
that a power <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 300 PW is sufficient to reach the<br />
Schw<strong>in</strong>ger value <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 4.6 x 10 29 W/cm 2 , the<br />
corresp<strong>on</strong>d<strong>in</strong>g power for Xe(M) is ~ 750 PW. The overall<br />
f<strong>in</strong>d<strong>in</strong>gs for the peak power <strong>in</strong> Be and U for Xe(M) and<br />
Xe(L) needed to reach the Schw<strong>in</strong>ger <strong>in</strong>tensity are<br />
presented <strong>in</strong> Fig. (5). The practicality <str<strong>on</strong>g>of</str<strong>on</strong>g> this achievement<br />
is documented <strong>in</strong> Table I. The m<strong>in</strong>imum total energy<br />
required to reach the Schw<strong>in</strong>ger level for the 248 nm<br />
driver technology is estimated to be 10 - 15 J, a value that<br />
can certa<strong>in</strong>ly be atta<strong>in</strong>ed with the utilizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> KrF*<br />
technology at its present level <str<strong>on</strong>g>of</str<strong>on</strong>g> development [3-7].<br />
Although mechanisms <str<strong>on</strong>g>of</str<strong>on</strong>g> radiative loss, such as Compt<strong>on</strong><br />
scatter<strong>in</strong>g, Bremsstrahlung, pair producti<strong>on</strong>, and the<br />
excitati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> nuclear reacti<strong>on</strong>s, have not been <strong>in</strong>cluded <strong>in</strong><br />
this elementary analysis, these losses can be significantly<br />
mitigated by the use <str<strong>on</strong>g>of</str<strong>on</strong>g> composite low-Z/high-Z targets.
Fig. (3): Channel producti<strong>on</strong> <strong>in</strong> solid U with a Xe(L)<br />
pulse hav<strong>in</strong>g an energy <str<strong>on</strong>g>of</str<strong>on</strong>g> 1.5 J and an <strong>in</strong>cident power P0<br />
= 300 PW. The peak <strong>in</strong>tensity produced is ~ 5.6 x 10 29<br />
W/cm 2 , a value <strong>in</strong>dicat<strong>in</strong>g that the Schw<strong>in</strong>ger Limit is<br />
atta<strong>in</strong>able.<br />
Fig. (4): Channel producti<strong>on</strong> <strong>in</strong> solid U with a Xe(M)<br />
pulse hav<strong>in</strong>g an energy <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 7.5 J and an <strong>in</strong>cident power<br />
P0 = 750 PW. The peak <strong>in</strong>tensity produced is ~ 5.5 x 10 29<br />
W/cm 2 , a value <strong>in</strong>dicat<strong>in</strong>g that the Schw<strong>in</strong>ger limit is<br />
atta<strong>in</strong>able.<br />
Fig. (5): Power scal<strong>in</strong>g for Xe(M) and Xe(L) <strong>in</strong> solid Be<br />
and U associated with the atta<strong>in</strong>ment <str<strong>on</strong>g>of</str<strong>on</strong>g> an <strong>in</strong>tensity at the<br />
Schw<strong>in</strong>ger limit. S<strong>in</strong>ce the area <str<strong>on</strong>g>of</str<strong>on</strong>g> the channel scales<br />
approximately as the <strong>in</strong>verse <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma density [16],<br />
the case <str<strong>on</strong>g>of</str<strong>on</strong>g> Be requires ~ 10-fold higher <strong>in</strong>cident power.<br />
The optimal case, illustrated <strong>in</strong> Fig. (3), corresp<strong>on</strong>ds to<br />
Xe(L) <strong>in</strong> U with an <strong>in</strong>cident peak power P0 ≅ 300 PW.<br />
Table I: Tabulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> practical c<strong>on</strong>siderati<strong>on</strong>s for the<br />
c<strong>on</strong>structi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Xe(M) and Xe(L) x-ray systems capable<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> reach<strong>in</strong>g the Schw<strong>in</strong>ger Limit. The range <str<strong>on</strong>g>of</str<strong>on</strong>g> system<br />
parameters shown corresp<strong>on</strong>ds to the <strong>in</strong>tensity range <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
10 26 - 5 x 10 29 W/cm 2 . *The m<strong>in</strong>imum pulse energy<br />
shown is not corrected for the <strong>in</strong>fluence <str<strong>on</strong>g>of</str<strong>on</strong>g> radiative losses.<br />
SYSTEM PARAMETER Xe(M) Xe(L)<br />
Wavelength (eV) 900 4500<br />
Pulse Length (as) 10 5<br />
M<strong>in</strong>imum Pulse Energy<br />
(J)*<br />
~0.01–5 ~0.01–1.5<br />
Intensity Limit (W/cm 2 ) 10 26 – 5 x 10 29<br />
E-M Cascade Limit No<br />
Coherent Beam Additi<strong>on</strong><br />
Required<br />
No<br />
Pulse Rate (Hz) ~ 1<br />
Optical Beam<br />
Compressi<strong>on</strong>/Focus<strong>in</strong>g<br />
Required<br />
(Grat<strong>in</strong>gs/Optics)<br />
Prepulse C<strong>on</strong>trast C<strong>on</strong>trol<br />
Needed<br />
Aperture Scale<br />
(Wavelength)<br />
Basic New <strong>Physics</strong> <strong>in</strong><br />
Source and Interacti<strong>on</strong>s<br />
CONCLUSIONS<br />
No<br />
No<br />
~ 30 cm<br />
(248 nm)<br />
Attosec<strong>on</strong>d Excitati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Progagati<strong>on</strong>/ Cluster<br />
Excitati<strong>on</strong>/Dicke<br />
Effect/Channel Focus<strong>in</strong>g<br />
We c<strong>on</strong>clude that an <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
Schw<strong>in</strong>ger Limit can be atta<strong>in</strong>ed and that the basic<br />
physical scal<strong>in</strong>g greatly favors the use <str<strong>on</strong>g>of</str<strong>on</strong>g> coherent x-rays.<br />
S<strong>in</strong>ce the cosmological c<strong>on</strong>stant represent<strong>in</strong>g the “dark<br />
energy” is now experimentally established [21] to be ΩΛ<br />
≅ 0.73, the ability to probe directly the nature <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
complex entity that we traditi<strong>on</strong>ally c<strong>on</strong>sider as the<br />
“vacuum” can <strong>on</strong>ly be expected to yield pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ound <strong>in</strong>sights<br />
<strong>in</strong>to the basic c<strong>on</strong>cept <str<strong>on</strong>g>of</str<strong>on</strong>g> space.<br />
ACKNOWLEDMENT<br />
Two authors, A.B. Borisov and C.K. Rhodes acknowledge<br />
<strong>in</strong>formative c<strong>on</strong>versati<strong>on</strong>s with S.V. Bulanov.
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Abstract<br />
N<strong>on</strong>-l<strong>in</strong>ear QED effects by str<strong>on</strong>g magnetic field <strong>in</strong> astrophysics<br />
Kazunori Kohri<br />
Cosmophysics group, Theory Center, IPNS, <strong>KEK</strong>, Tsukuba 305-0801, Japan<br />
Department <str<strong>on</strong>g>of</str<strong>on</strong>g> Particle and Nuclear <strong>Physics</strong>, GUAS, Tsukuba, 305-0801, Japan<br />
In this talk we have reviewed possible n<strong>on</strong>-l<strong>in</strong>ear QED<br />
effects <strong>in</strong> supercritical magnetic fields with B ≫ 3 × 10 13<br />
G which sometimes appear <strong>in</strong> astrophysics. Under that circumstance,<br />
the Landau levels and the propagator <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong><br />
are significantly affected, which can <strong>in</strong>duce some n<strong>on</strong>trivial<br />
modificati<strong>on</strong>s, e.g., <strong>in</strong> equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> state <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>,<br />
or <strong>in</strong> refractive <strong>in</strong>dices <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> <strong>in</strong> the magnetized plasma.<br />
In particular, it should be quite attractive that the so-called<br />
“phot<strong>on</strong> splitt<strong>in</strong>g” would occur <strong>in</strong> that situati<strong>on</strong>, which<br />
could have dom<strong>in</strong>ated energy-loss process <str<strong>on</strong>g>of</str<strong>on</strong>g> high-energy<br />
phot<strong>on</strong>s <strong>on</strong>ly <strong>in</strong> such an extremely str<strong>on</strong>g magnetic field.<br />
INTRODUCTION<br />
Astrophysical objects which have supercritical magnetic<br />
fields B > Bc with Bc ≡ m 2 e/e ≃ 3 × 10 13 have been<br />
reported. Here me denotes electr<strong>on</strong> mass and e means<br />
electr<strong>on</strong> charge. They are known as S<str<strong>on</strong>g>of</str<strong>on</strong>g>t Gamma-ray Repeaters<br />
(SGRs) and Anomalous X-ray pulsars (AXPs) with<br />
hav<strong>in</strong>g Period <str<strong>on</strong>g>of</str<strong>on</strong>g> O(1) – O(10) sec and Period Derivative<br />
O(10 −12 )–O(10 −10 ). Nowadays they are collectively<br />
called “magnetars” [1].<br />
It has been known that pulsars have str<strong>on</strong>g magnetic<br />
fields with the order <str<strong>on</strong>g>of</str<strong>on</strong>g> B ∼ 10 12 G and periods with the<br />
order <str<strong>on</strong>g>of</str<strong>on</strong>g> O(1) sec. Therefore we may th<strong>in</strong>k that magnetars<br />
could bel<strong>on</strong>g to an another special class <str<strong>on</strong>g>of</str<strong>on</strong>g> pulsars or<br />
neutr<strong>on</strong> stars.<br />
In terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the n<strong>on</strong>-l<strong>in</strong>ear quantum electrodynamics<br />
(QED), such a str<strong>on</strong>g magnetic field is exclusively attractive.<br />
That is because we cannot make use <str<strong>on</strong>g>of</str<strong>on</strong>g> the standard<br />
perturbati<strong>on</strong> theory, i.e., the perturbative approach to calculate<br />
the scatter<strong>in</strong>g amplitudes, the self-energies and so<br />
<strong>on</strong>. Then usual results obta<strong>in</strong>ed <strong>in</strong> a weak-magnetic field<br />
limit can be completely different from those <strong>in</strong> the str<strong>on</strong>gmagnetic<br />
field limit.<br />
In the language <str<strong>on</strong>g>of</str<strong>on</strong>g> quantum field theory, the order parameter<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the perturbati<strong>on</strong> B/Bc becomes no l<strong>on</strong>ger sufficiently<br />
small <strong>in</strong> a supercritical magnetic field. In a weakmagnetic<br />
field limit, we can expand physical quantities as<br />
a power series with respect to the order parameter. For example,<br />
when we c<strong>on</strong>sider the full orders <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong><br />
propagator (which might be called “dressed propagator”)<br />
<strong>in</strong> a weak magnetic field, external magnetic field l<strong>in</strong>es can<br />
couple to the undressed propagator <strong>in</strong> the higher order calculati<strong>on</strong>s<br />
(see Fig. 1). Then the next-order term is obta<strong>in</strong>ed<br />
by multiply<strong>in</strong>g a numerical number <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> B/Bc<br />
to the undressed term. 1<br />
1 Readers can refer APPENDIX A for a <strong>in</strong>tuitive understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
reas<strong>on</strong> why B/Bc becomes the order parameter <str<strong>on</strong>g>of</str<strong>on</strong>g> the power-law expansi<strong>on</strong><br />
<strong>in</strong> the weak-field limit.<br />
Figure 1: Electr<strong>on</strong> propagator <strong>in</strong> a weak-magnetic field<br />
limit. The magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the next-order term is obta<strong>in</strong>ed<br />
by multiply<strong>in</strong>g the order parameter B/Bc. See a simple<br />
pro<str<strong>on</strong>g>of</str<strong>on</strong>g> shown <strong>in</strong> APPENDIX A.<br />
If the strength <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field is larger than the critical<br />
value, we cannot make use <str<strong>on</strong>g>of</str<strong>on</strong>g> the known techniques <strong>in</strong><br />
the perturbati<strong>on</strong> theory. Then we have to perform a fullorder<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the calculati<strong>on</strong> n<strong>on</strong>perturbatively, which means<br />
that we have to sum up all <str<strong>on</strong>g>of</str<strong>on</strong>g> the diagrams. It is notable<br />
that about this k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>perturbative calculati<strong>on</strong>s <strong>in</strong>clud<strong>in</strong>g<br />
the full-order <str<strong>on</strong>g>of</str<strong>on</strong>g> the coupl<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the external magnetic<br />
fields, the lead<strong>in</strong>g term comes <strong>on</strong>ly from the tree level, not<br />
from the higher-loop levels (See Fig. 1).<br />
Under these circumstances, some significant modificati<strong>on</strong>s<br />
from the normal l<strong>in</strong>ear QED are possible <strong>in</strong><br />
• Energy gaps <str<strong>on</strong>g>of</str<strong>on</strong>g> Landau levels larger than me, which<br />
<strong>in</strong>duce anisotropic electr<strong>on</strong> pressure<br />
• N<strong>on</strong>-zero vacuum polarizati<strong>on</strong>, which <strong>in</strong>duces anomalous<br />
refractive <strong>in</strong>dices<br />
• Possible “phot<strong>on</strong> splitt<strong>in</strong>g” as a new class <str<strong>on</strong>g>of</str<strong>on</strong>g> cool<strong>in</strong>g<br />
process for high-energy phot<strong>on</strong>s<br />
Next we will discuss the details <str<strong>on</strong>g>of</str<strong>on</strong>g> those modificati<strong>on</strong>s and<br />
their effects <strong>on</strong> phenomena <strong>in</strong> astrophysics.<br />
LANDAU LEVELS IN STRONG<br />
MAGNETIC FIELD AND ANISOTROPIC<br />
ELECTRON PRESSURE<br />
First <str<strong>on</strong>g>of</str<strong>on</strong>g> all, it is notable that the energy gap <strong>in</strong> the Landau<br />
levels <strong>in</strong> the supercritical str<strong>on</strong>g magnetic field can be<br />
larger than electr<strong>on</strong> rest mass. The electr<strong>on</strong> energy <strong>in</strong> the<br />
magnetized plasma with a uniform magnetic field al<strong>on</strong>g the<br />
z-axis B = Bˆz is given by<br />
E = √ m 2 e + p 2 z + eB(2n + 1 − α), (1)<br />
where pz means the z-comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the momentum, n denotes<br />
the <strong>in</strong>dex <str<strong>on</strong>g>of</str<strong>on</strong>g> a Landau level, and α is the sp<strong>in</strong> <strong>in</strong>dex
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>. When the magnetic field is larger than the critical<br />
value, i.e., eB ≫ m 2 e, the x- and y- comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
electr<strong>on</strong> momentums are no l<strong>on</strong>ger distributed uniformly<br />
and isotropicly because they are quantized <strong>in</strong> the x-y plane<br />
which is perpendicular to B.<br />
This anisotropic momentum <strong>in</strong>duces the corresp<strong>on</strong>d<strong>in</strong>g<br />
anisotropic dynamic pressure ˜ Pi = (n + 1/2 − α/2)eB/E<br />
with i = x and y. By averag<strong>in</strong>g the dynamic pressure <strong>in</strong> the<br />
grand can<strong>on</strong>ical ensemble, we get the pressure, Pi which<br />
corresp<strong>on</strong>ds to a diag<strong>on</strong>al comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the stress-energy<br />
tensor and is still anisotropic. This should be the pressure<br />
which can be used <strong>in</strong> the fluid dynamics [3].<br />
It was reported that this anisotropic pressure can <strong>in</strong>duce<br />
a n<strong>on</strong>standard result <strong>in</strong> supernovae. In the neutr<strong>in</strong>o-driven<br />
w<strong>in</strong>d which appears after a supernova, especially an additi<strong>on</strong>al<br />
entropy <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma can be produced by the<br />
anisotropic pressure <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> [3]. For a successful rprocess<br />
nucleosynthesis which is believed to occur after<br />
supernovae, we need the additi<strong>on</strong>al entropy per bary<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ∆S/kB ∼ 200. As is shown <strong>in</strong> Fig. 2, the<br />
producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the entropy per bary<strong>on</strong> with be<strong>in</strong>g the order<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> O(200) can be possible <strong>in</strong> case <str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>g magnetic<br />
fields such as B ∼ 10 16 G. See Ref. [3] for the further<br />
details. We expect that future developments <strong>in</strong> numerical<br />
simulati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the supernovae explosi<strong>on</strong> will reveal the details<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the current scenario with its dynamics coupled to<br />
the supercritical magnetic field.<br />
Figure 2: Produced entropy per bary<strong>on</strong> <strong>in</strong> neutr<strong>in</strong>o-driven<br />
w<strong>in</strong>d <strong>in</strong> supernovae. We need ∆S/kB ∼ 200 for a successful<br />
r-process nucleosynthesis. Further details are shown <strong>in</strong><br />
Ref. [3].<br />
Figure 3: Diagrams <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum polarizati<strong>on</strong> tensors. We<br />
could have expanded it as a series <str<strong>on</strong>g>of</str<strong>on</strong>g> the order parameter<br />
B/Bc <strong>in</strong> the perturbati<strong>on</strong> theory <strong>in</strong> a weak magnetic field<br />
limit. However, if the magnetic field is larger than the critical<br />
<strong>on</strong>e, we have to add all <str<strong>on</strong>g>of</str<strong>on</strong>g> the diagram n<strong>on</strong>perturbatively.<br />
The double l<strong>in</strong>e denotes the fully-dressed propagator<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>.<br />
NON-ZERO VACUUM POLARIZATION<br />
AND ANOMALOUS REFRACTIVE<br />
INDICES<br />
In the supercritical magnetic field, it is expected that the<br />
regularized polarizati<strong>on</strong> tensor Π µν can have a n<strong>on</strong>-zero<br />
value. By add<strong>in</strong>g the all <str<strong>on</strong>g>of</str<strong>on</strong>g> the external field l<strong>in</strong>es, we<br />
can calculate the fully <strong>on</strong>e-loop vacuum polarizati<strong>on</strong> tensor<br />
(Fig. 3). By us<strong>in</strong>g Schw<strong>in</strong>ger’s proper time method<br />
(See Appendix B for a brief review), the sum <str<strong>on</strong>g>of</str<strong>on</strong>g> the external<br />
fields can be replaced by the <strong>in</strong>tegral <str<strong>on</strong>g>of</str<strong>on</strong>g> the proper<br />
time. In the full calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum polarizati<strong>on</strong> tensor,<br />
we should have to perform 2-dim <strong>in</strong>tegral by proper<br />
time.<br />
If there were n<strong>on</strong>-zero vacuum polarizati<strong>on</strong> tensor, we<br />
expected that the dispersi<strong>on</strong> relati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> can be modified.<br />
The refractive <strong>in</strong>dices are def<strong>in</strong>ed by<br />
µ 2 = |k|2<br />
, (2)<br />
ω2 where k denotes the spatial 3-vector <str<strong>on</strong>g>of</str<strong>on</strong>g> the 4-dim phot<strong>on</strong><br />
momentum k µ with µ = 0, 1, 2, 3, and ω = k 0 = Eγ<br />
means the phot<strong>on</strong> energy.<br />
Then the wave equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> can be expressed by<br />
[ k 2 g µν k µ k ν + regΠ µν (k) ] Aµ(k) = 0. (3)<br />
In this case, the polarizati<strong>on</strong> tensor can be written by 2-dim<br />
<strong>in</strong>tegral to be<br />
Π αβ (x, x ′ )<br />
∝ Tr [ γ α G(x, x ′ )γ β G(x ′ , x) ]<br />
∝<br />
∫ ∞<br />
0<br />
ds<br />
s<br />
∫ ∞<br />
0<br />
(4)<br />
ds ′<br />
s ′ [· · ·] , (5)<br />
with the two spatial po<strong>in</strong>ts, x, and x ′ .<br />
To have n<strong>on</strong>trivial soluti<strong>on</strong>s for Aµ(k), the determ<strong>in</strong>ant<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the operator [· · ·] <strong>in</strong> the left-hand side should vanishes [5,<br />
6]. Then we obta<strong>in</strong> the soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> µ’s.<br />
In Fig. 4, we plotted refractive <strong>in</strong>dices µ1 which corresp<strong>on</strong>ds<br />
to the eigen vector <str<strong>on</strong>g>of</str<strong>on</strong>g> the polarizati<strong>on</strong> (0, 1, 0), and<br />
µ2 which corresp<strong>on</strong>ds to the other mode. The angle between<br />
B and k is taken to be cos 2 θ = 1/2.
Figure 4: Refractive <strong>in</strong>dices <str<strong>on</strong>g>of</str<strong>on</strong>g> two polarizati<strong>on</strong> modes as<br />
a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> B/Bc. µ1 corresp<strong>on</strong>ds to the eigen vector <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the polarizati<strong>on</strong> (0, 1, 0), and µ2 corresp<strong>on</strong>ds to the other<br />
mode. This is shown <strong>in</strong> Ref. [4].<br />
PHOTON SPLITTING<br />
It has been said that the most impressive phenomen<strong>on</strong> <strong>in</strong><br />
the supercritical magnetic field should be phot<strong>on</strong> splitt<strong>in</strong>g.<br />
In the weak magnetic field limit, a phot<strong>on</strong> can not split <strong>in</strong>to<br />
two phot<strong>on</strong>s. On the other hand, <strong>in</strong> the supercritical magnetic<br />
field, a phot<strong>on</strong> can scatter <str<strong>on</strong>g>of</str<strong>on</strong>g>f the external magnetic<br />
field and produce two phot<strong>on</strong>s by <strong>on</strong>e loop effects <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
dressed <strong>in</strong>ternal electr<strong>on</strong>.<br />
More precisely, γ → γ + γ might be allowed <strong>on</strong>ly k<strong>in</strong>ematically<br />
through an <strong>on</strong>e-loop diagram with the normal undressed<br />
propagator. However, Furry’s theorem [7] does<br />
not permit odd number <str<strong>on</strong>g>of</str<strong>on</strong>g> vertices. Thus, the actual diagram<br />
should start from the four-po<strong>in</strong>t box diagram which<br />
is shown <strong>in</strong> Fig. 5. Therefore the phot<strong>on</strong> splitt<strong>in</strong>g is purely<br />
n<strong>on</strong>-l<strong>in</strong>ear QED effect through the dressed electr<strong>on</strong> propagator.<br />
In this study, thus it is essential to calculate the<br />
full-orders <str<strong>on</strong>g>of</str<strong>on</strong>g> the dressed electr<strong>on</strong> propagator.<br />
Figure 5: Diagrams <str<strong>on</strong>g>of</str<strong>on</strong>g> triangle tensor for phot<strong>on</strong> splitt<strong>in</strong>g.<br />
Note that Furry’s theorem does not permit odd number <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
vertices.<br />
We have known that it is possible to write down the fullexpressi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the decay rate or the triangle tensor <str<strong>on</strong>g>of</str<strong>on</strong>g> photo<br />
splitt<strong>in</strong>g by us<strong>in</strong>g Schw<strong>in</strong>ger’s proper time method as well<br />
as the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the polarizati<strong>on</strong> tensor:<br />
Π αβδ (x, x ′ , x ′′ )<br />
∝ Tr [ γ α G(x, x ′ )γ β G(x ′ , x ′′ )γ δ G(x ′′ , x) ]<br />
∝<br />
∫ ∞<br />
0<br />
ds<br />
s<br />
∫ ∞<br />
0<br />
ds ′<br />
s ′<br />
∫ ∞<br />
0<br />
(6)<br />
ds ′′<br />
s ′′ [· · ·] , (7)<br />
with the three spatial po<strong>in</strong>ts, x, x ′ , x ′′ . In this case, the dimensi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tegral is <strong>in</strong>evitably three, which makes the<br />
calculati<strong>on</strong> to obta<strong>in</strong> a c<strong>on</strong>crete numerical number much<br />
more difficult. So far no groups have succeeded to get c<strong>on</strong>crete<br />
values <str<strong>on</strong>g>of</str<strong>on</strong>g> the rate for arbitrary physical variables, such<br />
as for B or ω by perform<strong>in</strong>g the <strong>in</strong>tegrati<strong>on</strong>s <strong>in</strong> the 3-dim<br />
proper time [8, 9].<br />
On the other hand, <strong>on</strong>ly soluti<strong>on</strong>s with both low-energy<br />
ω ≪ me and weak-field limits B ≪ Bc have been<br />
known [8]. Accord<strong>in</strong>g to those soluti<strong>on</strong>s, for example<br />
the <strong>in</strong>verse <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> attenuati<strong>on</strong>-length due to phot<strong>on</strong><br />
splitt<strong>in</strong>g can be approximately proporti<strong>on</strong>al to ∝ ω 5 B 6 .<br />
Under this circumstance, Bar<strong>in</strong>g and Hard<strong>in</strong>g<br />
(2001) [10] extrapolated those approximate soluti<strong>on</strong>s<br />
to larger values <str<strong>on</strong>g>of</str<strong>on</strong>g> B and ω and applied them to energyloss<br />
processes <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> and phot<strong>on</strong> which exist around<br />
magnetars. It is astrophysically-impressive that s<strong>in</strong>ce<br />
the magnetic field l<strong>in</strong>e and the magnetosphere have<br />
n<strong>on</strong>trivial structures around pulsars, the rate <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
e + e − -pair creati<strong>on</strong> by phot<strong>on</strong> scatter<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g>f the magnetic<br />
field (γ → e + + e − ) could be smaller than the phot<strong>on</strong><br />
splitt<strong>in</strong>g rate (γ → γ + γ). That is because the paircreati<strong>on</strong><br />
rate depends <strong>on</strong> the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the beam phot<strong>on</strong><br />
Γ ∝ B s<strong>in</strong> θkB/Bc with θkB be<strong>in</strong>g an angle between the<br />
magnetic-field l<strong>in</strong>e and the phot<strong>on</strong> momentum k although<br />
the phot<strong>on</strong> splitt<strong>in</strong>g rate does not depend <strong>on</strong> the directi<strong>on</strong><br />
under the uniform magnetic field. It might be <strong>in</strong>tuitive,<br />
for example, the former rate can become large <strong>in</strong> head-<strong>on</strong><br />
collisi<strong>on</strong>, or <strong>on</strong> the other hand it is negligible with a<br />
smaller energy than the threshold <str<strong>on</strong>g>of</str<strong>on</strong>g> the pair creati<strong>on</strong><br />
Eγ < me/ s<strong>in</strong> θkB depend<strong>in</strong>g <strong>on</strong> a value <str<strong>on</strong>g>of</str<strong>on</strong>g> θkB. It should<br />
be the ast<strong>on</strong>ish<strong>in</strong>g po<strong>in</strong>t that the authors <str<strong>on</strong>g>of</str<strong>on</strong>g> [10] proposed<br />
that the reas<strong>on</strong> why the synchrotr<strong>on</strong> radio emissi<strong>on</strong> has<br />
not been observed from magnetars is that high-energy<br />
phot<strong>on</strong>s emitted by curvature radiati<strong>on</strong> can loose their<br />
energy ma<strong>in</strong>ly through phot<strong>on</strong> splitt<strong>in</strong>g, not through the<br />
e + e − -pair creati<strong>on</strong> <strong>in</strong> the supercritical magnetic field. To<br />
check this quite an attractive scenario, we would have to<br />
succeed the full calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong>-splitt<strong>in</strong>g rate<br />
to obta<strong>in</strong> the c<strong>on</strong>crete numerical numbers as functi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
magnetic field and phot<strong>on</strong> energies.<br />
CONCLUSION<br />
In this talk, we have reviewed various astrophysical phenomena<br />
<strong>in</strong>duced by the n<strong>on</strong>-l<strong>in</strong>ear QED effect such as large<br />
Landau levels, n<strong>on</strong>-zero vacuum polarizati<strong>on</strong> tensor <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s<br />
and possible phot<strong>on</strong> splitt<strong>in</strong>g. About the rate <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>
splitt<strong>in</strong>g, no <strong>on</strong>e has succeeded to fully calculate it and obta<strong>in</strong><br />
the c<strong>on</strong>crete numerical values as functi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> arbitrary<br />
values <str<strong>on</strong>g>of</str<strong>on</strong>g> B and Eγ by perform<strong>in</strong>g the 3-dim <strong>in</strong>tegrati<strong>on</strong>.<br />
Recently hard power-low emissi<strong>on</strong>s with the phot<strong>on</strong> <strong>in</strong>dex<br />
Γ ∼ 1 from magnetars has been reported by X-ray<br />
observati<strong>on</strong> by Suzaku [11, 12]. So far we have not known<br />
the orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> such a hard comp<strong>on</strong>ent from normal pulsars.<br />
They might relate with new emissi<strong>on</strong> mechanisms <strong>in</strong>duced<br />
by the n<strong>on</strong>-l<strong>in</strong>ear QED effects <strong>in</strong> the supercritical magnetic<br />
field (See also Refs. [13, 14]).<br />
ACKNOWLEDGEMENT<br />
The author would like to thank Shoichi Yamada for l<strong>on</strong>gterm<br />
collaborati<strong>on</strong>s and c<strong>on</strong>t<strong>in</strong>uous discussi<strong>on</strong>s. He also<br />
thanks Kazuo Makishima and Teruaki Enoto for fruitful<br />
discussi<strong>on</strong>s.<br />
APPENDIX<br />
Appendix A: Analytical understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the order<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> radiative correcti<strong>on</strong><br />
In this secti<strong>on</strong>, we show that the first-order radiative correcti<strong>on</strong><br />
to the undressed electr<strong>on</strong> propagator <strong>in</strong> an external<br />
magnetic field is the undressed term multiplied by a factor<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> O(eB/m 2 e). This simply means that the order<br />
parameter <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the perturbati<strong>on</strong> due to the weak<br />
external-magnetic field could be eB/m 2 e.<br />
In Fig. 6 we showed the diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> the coupl<strong>in</strong>g between<br />
the free electr<strong>on</strong> field (solid l<strong>in</strong>e) and an external static<br />
magnetic field Aµ (wavy l<strong>in</strong>e) with a coupl<strong>in</strong>g g = −|e|.<br />
The space-time po<strong>in</strong>ts, X, Y, Z are be<strong>in</strong>g c<strong>on</strong>sidered <strong>in</strong> the<br />
X3 ∼ Y3 ∼ Z3 ∼ 0 plane.<br />
Figure 6: diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> the coupl<strong>in</strong>g between the free electr<strong>on</strong><br />
field (solid l<strong>in</strong>e) and an external static magnetic field Aµ<br />
(wavy l<strong>in</strong>e) with a coupl<strong>in</strong>g g = −|e|. The space-time<br />
po<strong>in</strong>ts, X, Y, Z are be<strong>in</strong>g c<strong>on</strong>sidered <strong>in</strong> the X3 ∼ Y3 ∼<br />
Z3 ∼ 0 plane.<br />
We c<strong>on</strong>sider the first-order propagator <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong><br />
which propagates from X to Y <strong>in</strong> real space.<br />
∫<br />
˜G(X, Y ) = dZ igγ µ ∫<br />
i<br />
Aµ(Z) dp e<br />
̸ p − me<br />
−ip(X−Z)<br />
∫<br />
i −iq(Z−Y )<br />
× dq e<br />
̸ q − me<br />
= +igZ3Bγ 2<br />
∫ ∫<br />
1<br />
dZ dp e<br />
̸ p − me<br />
−ip(X−Z)<br />
∫<br />
1<br />
× dq<br />
̸ q − me<br />
e −iq(Z−Y ) , (8)<br />
where we took Aµ(Z) = (0, 0, −Z3B, 0) with a c<strong>on</strong>stant<br />
magnetic field B, and we used γ µ Aµ(Z) = γ 2 (−Z3B) <strong>in</strong><br />
the last l<strong>in</strong>e.<br />
By <strong>in</strong>tegrati<strong>on</strong> by parts, the last <strong>in</strong>tegral is estimated to<br />
be ∫<br />
1 −iq(Z−Y )<br />
dq e<br />
̸ q − me<br />
∫<br />
1 ∂<br />
= dq<br />
̸ q − me ∂q3<br />
∫<br />
1<br />
= [· · ·] +<br />
dq<br />
i(Z3 − Y3)<br />
−iq(Z−Y ) 1<br />
e<br />
−i(Z3 − Y3)<br />
γ 3<br />
(̸ q − me) 2 e−iq(Z−Y ) ,(9)<br />
where we can omit the first term because <str<strong>on</strong>g>of</str<strong>on</strong>g> a natural<br />
boundary c<strong>on</strong>diti<strong>on</strong>. Then we get<br />
˜G(X, Y ) =<br />
Z3<br />
gBγ<br />
Z3 − Y3<br />
2 γ 3<br />
∫ ∫<br />
dZ dp e−ipX<br />
̸<br />
×<br />
∫<br />
p − me<br />
∼<br />
∫<br />
dq e−iZ(q−p)<br />
e+iqY<br />
(̸ q − me) 2<br />
i<br />
dp<br />
̸ p − me<br />
e −ip(X−Y ) eBγ2 γ 3<br />
,<br />
(̸ p − me) 2<br />
(10)<br />
where we have assumed that Z3/(Z3 − Y3) ∼ O(1), and<br />
made use <str<strong>on</strong>g>of</str<strong>on</strong>g> a delta functi<strong>on</strong>, δ(q − p). Because the momentum<br />
̸ p <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>ternal l<strong>in</strong>e is <str<strong>on</strong>g>of</str<strong>on</strong>g>f-shell, its 4-dim norm<br />
is smaller than the rest mass, √ p2 ≤ m. By us<strong>in</strong>g this<br />
approximati<strong>on</strong>, we obta<strong>in</strong> the f<strong>in</strong>al expressi<strong>on</strong>,<br />
∫<br />
˜G(X,<br />
i<br />
Y ) ∼ dp<br />
̸ p − me<br />
e −ip(X−Y ) × O( eB<br />
m2 ). (11)<br />
e<br />
From this <strong>in</strong>tegrand, we f<strong>in</strong>d that the 1st-order correcti<strong>on</strong><br />
is given by the 0-th order term by multiply<strong>in</strong>g the factor<br />
O(eB/m 2 e) <strong>in</strong> a perturbati<strong>on</strong> theory. This means that the<br />
order parameter for the perturbati<strong>on</strong> should be (eB/m 2 e).<br />
Therefore a series <str<strong>on</strong>g>of</str<strong>on</strong>g> the above calculati<strong>on</strong> could have given<br />
us a brief pro<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> what the order parameter <strong>in</strong> the perturbati<strong>on</strong><br />
theory is.<br />
Appendix B: Schw<strong>in</strong>ger’s proper-time method<br />
Here we briefly <strong>in</strong>troduce an outl<strong>in</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the Schw<strong>in</strong>ger’s<br />
proper-time method. In positi<strong>on</strong> representati<strong>on</strong>, we can<br />
write the Dirac’s delta functi<strong>on</strong> to be<br />
δ 4 (x − x ′ ) = ⟨x |1| x ′ ⟩ = ⟨x|x ′ ⟩, (12)<br />
with the propagator <strong>in</strong> positi<strong>on</strong> representati<strong>on</strong>,<br />
G(x, x ′ ) = ⟨ x ˆ G x ′⟩ . (13)<br />
Orig<strong>in</strong>ally the operator <str<strong>on</strong>g>of</str<strong>on</strong>g> the green functi<strong>on</strong> can be expressed<br />
by def<strong>in</strong>iti<strong>on</strong> as<br />
[γ µ Πµ − m] ˆ G = 1, (14)
with the can<strong>on</strong>ical momentum Πµ ≡ i∂µ + eAµ. Here the<br />
operator <str<strong>on</strong>g>of</str<strong>on</strong>g> the green functi<strong>on</strong> is rewritten to be<br />
ˆG =<br />
=<br />
=<br />
1<br />
γ µ Πµ − m<br />
γ µ Πµ + m<br />
(γ µ Πµ) 2 − m2 ∫ ∞<br />
ds<br />
0 i e−im2 s −iHs µ<br />
e (γ Πµ + m) ,<br />
(15)<br />
with H ≡ − (γ µ Πµ) 2 . In the transformati<strong>on</strong> to the<br />
last l<strong>in</strong>e, the <strong>in</strong>tegral representati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the operator was<br />
adopted.<br />
Eq. (15) means that the operator can time-evolve through<br />
the unitary matrix U(s) = e −iHs as if that would be governed<br />
by an effective time s and an effective Hamilt<strong>on</strong>ian<br />
H <strong>in</strong> quantum mechanics. Then it is notable that their<br />
mass dimensi<strong>on</strong>s should be two and m<strong>in</strong>us two for H and<br />
s, respectively. Therefor H and s do not denote the usual<br />
Hamilt<strong>on</strong>ian and time. In this situati<strong>on</strong>, this effective time s<br />
is called the “Schw<strong>in</strong>ger’s proper time” [2]. After we have<br />
formulated quantum mechanics <strong>in</strong> this system, we will be<br />
able to use this unitary matrix to discuss the time evoluti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> operators.<br />
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volume.
Abstract<br />
WIDE-BAND X-RAY OBSERVATIONS OF MAGNETARS ∗<br />
K. Makishima, Dept. <strong>Physics</strong> † and RESCEU, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo<br />
and the Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Physical and Chemical Research (RIKEN)<br />
After a brief review <str<strong>on</strong>g>of</str<strong>on</strong>g> physics <str<strong>on</strong>g>of</str<strong>on</strong>g> neutr<strong>on</strong> stars, this article<br />
focuses <strong>on</strong> so called magnetars, a special subset <str<strong>on</strong>g>of</str<strong>on</strong>g> neutr<strong>on</strong><br />
stars, which are though to have magnetic fields rach<strong>in</strong>g<br />
10 14−15 G. Recent observati<strong>on</strong>al results <str<strong>on</strong>g>of</str<strong>on</strong>g> these objects,<br />
made with the Japanese X-ray satellite Suzaku, suggest<br />
that their X-ray to gamma-ray emissi<strong>on</strong> results from<br />
phot<strong>on</strong> splitt<strong>in</strong>g process <strong>in</strong> the str<strong>on</strong>g magnetic field.<br />
INTRODUCTION<br />
As a brief <strong>in</strong>troducti<strong>on</strong> to neutr<strong>on</strong> stars, a paragraph may<br />
be spent <strong>on</strong> stellar evoluti<strong>on</strong>. As illustrated <strong>in</strong> Fig. 1, stars,<br />
first born as protostars, take different evoluti<strong>on</strong>ary paths<br />
depend<strong>in</strong>g <strong>on</strong> their <strong>in</strong>itial mass. The lightest <strong>on</strong>es end up<br />
with planets, where Coulombic repulsi<strong>on</strong> am<strong>on</strong>g i<strong>on</strong>s supports<br />
the gravity. Somewhat more massive <strong>on</strong>es become<br />
brown dwarfs, which employ electr<strong>on</strong> degenerate pressure<br />
<strong>in</strong>stead. Objects with > 0.08 times the solar mass become<br />
normal stars, where classical gas pressure generated by nuclear<br />
fusi<strong>on</strong> balances the gravity. After 10 6−10 years depend<strong>in</strong>g<br />
<strong>on</strong> their <strong>in</strong>itial mass, these stars come to their evoluti<strong>on</strong>ary<br />
endpo<strong>in</strong>ts. Lightest normal stars leave at their<br />
centers white dwarfs, which are similar to brown dwarfs<br />
except their lack <str<strong>on</strong>g>of</str<strong>on</strong>g> hydrogen. More massive <strong>on</strong>es experience<br />
gravitati<strong>on</strong>al-collapse supernova explosi<strong>on</strong>s, and their<br />
cores collapse <strong>in</strong>to neutr<strong>on</strong> stars which are supported by degenerate<br />
neutr<strong>on</strong> pressure. If the <strong>in</strong>itial mass is even higher,<br />
the f<strong>in</strong>al gravity is too str<strong>on</strong>g for a neutr<strong>on</strong> star to support,<br />
and the object becomes a black hole.<br />
This article uses the MKSA units, except that magnetic<br />
field B is expressed <strong>in</strong> Gauss (G); 1 G = 10 −4 T.<br />
DEGENERATE STELLAR OBJECTS<br />
Stellar Equilibrium<br />
The balance between the gravity and pressure p <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
spherical star can be expressed at each radius r as<br />
dp/dr = GρMr(r)/r , (1)<br />
where G, ρ and Mr(r) = ∫ r<br />
0 4πr2 ρ(r)dr are the gravitati<strong>on</strong>al<br />
c<strong>on</strong>stant, mass density, and “mass coord<strong>in</strong>ate”, respectiveky.<br />
Instead <str<strong>on</strong>g>of</str<strong>on</strong>g> exactly solv<strong>in</strong>g eq.(1), it is useful to<br />
multiply its both sides by 4πr 3 and <strong>in</strong>tegrate from r = 0 to<br />
the stellar surface r = R, to obta<strong>in</strong> a Viriar relati<strong>on</strong> as<br />
3 < p > V = −Φ ≡<br />
∫ M<br />
0<br />
GMr<br />
r dMr<br />
2 5GM<br />
∼ . (2)<br />
3R<br />
∗ Work supported by T. Enoto, Y.E. Nakagawa, and M. Nakajima<br />
† maxima@phys.s.u-tokyo.ac.jp<br />
Figure 1: A very schematic illustrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> stellar evoluti<strong>on</strong>.<br />
Abscissa at the top <strong>in</strong>dicates <strong>in</strong>itial stellar mass, while that<br />
at the bottom f<strong>in</strong>al mass. Time goes from top to bottom.<br />
Here, V is the stellar volume, means volume average,<br />
M ≡ Mr(R) is the stellar mass, Φ is self-gravitati<strong>on</strong>al<br />
energy, and the last approximati<strong>on</strong> assumes a c<strong>on</strong>stant ρ.<br />
Degenerate Stars<br />
C<strong>on</strong>sider a star which is supported by degenerate pressure<br />
pd <str<strong>on</strong>g>of</str<strong>on</strong>g> some Fermi<strong>on</strong>s, <str<strong>on</strong>g>of</str<strong>on</strong>g> which the mass is mf and<br />
density is nf. We may substite p <strong>in</strong> eq.(2) with the n<strong>on</strong>relativistic<br />
degenerate pressure <str<strong>on</strong>g>of</str<strong>on</strong>g> an ideal Fermi gas, i.e.,<br />
pd = A¯hn 5/3<br />
f /mf, where ¯h is the Dirac c<strong>on</strong>stant and A is a<br />
numerical c<strong>on</strong>stant <str<strong>on</strong>g>of</str<strong>on</strong>g> order unity. Perform<strong>in</strong>g elementary<br />
calculati<strong>on</strong>s, and us<strong>in</strong>g another numerical c<strong>on</strong>stant A ′ , we<br />
obta<strong>in</strong> a mass-radius relati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this degenerate star as<br />
R/λf = A ′ α −1<br />
g η −5/3 (M/mn) −1/3 . (3)<br />
Here, mn is the nucle<strong>on</strong> mass, η is the nucle<strong>on</strong>-to-Fermi<strong>on</strong><br />
number ratio, λf ≡ 2π¯h/mfc is the Fermi<strong>on</strong>’s Compt<strong>on</strong><br />
wavelength, and αg ≡ Gm 2 n/¯hc = 5.9 × 10 −39<br />
is a dimensi<strong>on</strong>less c<strong>on</strong>stant represent<strong>in</strong>g gravitati<strong>on</strong>al <strong>in</strong>teracti<strong>on</strong>.<br />
Thus, the stellar radius is proporti<strong>on</strong>al to the<br />
Fermi<strong>on</strong>’s Compt<strong>on</strong> wavelength. If we use the extreme<br />
relativistic expressi<strong>on</strong> for pd, an upper limit mass (Chandrasekhar<br />
mass) is derived.<br />
Brown and White Dwarfs<br />
If the Fermi<strong>on</strong>s are electr<strong>on</strong>s with λe = 2.4 × 10 −12<br />
m, the star becomes a brown dwarf (η = 1.2) or a white<br />
dwarf (η = 2.0). Normaliz<strong>in</strong>g M to the solar mass M⊙ ≡<br />
2.0 × 10 30 kg, and faithfully calculat<strong>in</strong>g A ′ , we obta<strong>in</strong><br />
R = 1.0 × 10 7 (M/M⊙) −1/3 (2/η) −5/3 m (4)<br />
which implies an object <str<strong>on</strong>g>of</str<strong>on</strong>g> the Earth’s size.
Basic C<strong>on</strong>cepts<br />
NEUTRON STARS<br />
When the stellar <strong>in</strong>terior is mostly “neutr<strong>on</strong>ized” and the<br />
neutr<strong>on</strong>s’ degenerate pressure supports the gravity, the object<br />
becomes a neutr<strong>on</strong> star (NS). From eq.(3), we f<strong>in</strong>d<br />
that a NS with M ∼ M⊙ is by 3 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude<br />
(∼ λe/λn = mn/me) smaller <strong>in</strong> radius than a white dwarf<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the same mass; approximately ∼ 10 km. However, more<br />
accurate estimates <str<strong>on</strong>g>of</str<strong>on</strong>g> the mass-radius relati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> NSs are<br />
a subject <str<strong>on</strong>g>of</str<strong>on</strong>g> yet unsolved studies, because it is affected by<br />
general relativity and the nuclear equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> state.<br />
Classificati<strong>on</strong><br />
NSs may be classified <strong>in</strong> several aspects. One is their<br />
surface magnetic fields (MFs) B, which range from < 10 9<br />
G to ∼ 10 15 . Another is whether the object is isolated or <strong>in</strong><br />
a b<strong>in</strong>ary system with another star. The classificati<strong>on</strong> from<br />
these two dimensi<strong>on</strong>s is presented <strong>in</strong> Table 1.<br />
Shown with [ ] <strong>in</strong> Table 1 is yet another classificati<strong>on</strong><br />
axis, the source energy <str<strong>on</strong>g>of</str<strong>on</strong>g> their radiati<strong>on</strong>. Isolated NSs with<br />
l<strong>on</strong>g rotati<strong>on</strong> periods (e.g.,> 10 s) can utilize <strong>on</strong>ly their<br />
<strong>in</strong>ternal energies [i], to emit blackbody radiati<strong>on</strong>. Fastrotat<strong>in</strong>g<br />
magnetized <strong>on</strong>es can spend their huge rotati<strong>on</strong>al<br />
energies [r], while those <strong>in</strong> b<strong>in</strong>aries can alternatively utilize<br />
gravitati<strong>on</strong>al energies [g] <str<strong>on</strong>g>of</str<strong>on</strong>g> accret<strong>in</strong>g materials. A<br />
subset <str<strong>on</strong>g>of</str<strong>on</strong>g> these accret<strong>in</strong>g NSs can also use nuclear energies<br />
[n], when the accreted material <strong>in</strong>termittently make<br />
nuclear fusi<strong>on</strong> <strong>on</strong> the NS surfaces (a phenomen<strong>on</strong> called<br />
X-ray bursts). F<strong>in</strong>ally, those with the str<strong>on</strong>gest B, magnetars,<br />
are thought to be magnetically powered [m].<br />
Table 1: Classificati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> neutr<strong>on</strong> stars. ∗,#<br />
B (G) isolated b<strong>in</strong>ary<br />
< 10 10 isolated NS [i](R) X-ray bursters [g,n](R)<br />
10 11−13 radio pulsars [r](B) X-ray pulsars [g](M, B)<br />
10 14−15 magnetars [m](B) —<br />
* : [ ] <strong>in</strong>dicate radiati<strong>on</strong> energy sources: [i]=<strong>in</strong>ternal,<br />
[r]=rotati<strong>on</strong>al, [n]=nuclear, [g]=gravitati<strong>on</strong>al, [m]=magnetic<br />
# : M, R and B <strong>in</strong>dicate that the mass, radius, and surface<br />
magnetic fields are measurable, respectively.<br />
Mass and Radius<br />
NSs have three important parameters; mass M, radius<br />
R, and MF B. In Table 1, we <strong>in</strong>diacate, with ( ), which <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
them can be measured <strong>in</strong> each class <str<strong>on</strong>g>of</str<strong>on</strong>g> objects.<br />
Like <strong>in</strong> other astrophysical c<strong>on</strong>texts, the b<strong>in</strong>ary envir<strong>on</strong>ment<br />
provides opportunities to measure the NS mass. C<strong>on</strong>sider<br />
a NS <strong>in</strong> a b<strong>in</strong>ary system with five unknowns; the<br />
NS mass Mns, the compani<strong>on</strong> mass Mc, orbital separati<strong>on</strong><br />
a, orbital angular frequency Ω, and orbital <strong>in</strong>cl<strong>in</strong>ati<strong>on</strong><br />
i. We can observe Ω, as well as the orbital velocity<br />
Kns = aΩ s<strong>in</strong> i/(1 + q) (with q ≡ Mns/Mc) <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
NS via pulse arrival delays, and that <str<strong>on</strong>g>of</str<strong>on</strong>g> the compani<strong>on</strong> star<br />
Kc = aqΩ s<strong>in</strong> i/(1 + q) via optical Doppler spectroscopy.<br />
Furthermore, we can utilize the Kepler’s law,<br />
G(Mns + Mc) = a 3 Ω 2 . (5)<br />
Then, if i is somehow estimated, the five variables can be<br />
all uniquely determ<strong>in</strong>ed. The results show a sharp c<strong>on</strong>centrati<strong>on</strong><br />
at Mns = 1.4M⊙ [24].<br />
The radius <str<strong>on</strong>g>of</str<strong>on</strong>g> an NS can be measured if it emits blackbody<br />
radiati<strong>on</strong> uniformly from its surface, like dur<strong>in</strong>g the<br />
X-ray bursts. Then, by observ<strong>in</strong>g the bolometric (=frequency<br />
<strong>in</strong>tegrated) flux f and the surface temperature T ,<br />
we can estimate R from the Stefan-Boltzmann law,<br />
f = 4πR 2 σT 4 /4πD 2 , (6)<br />
where D is the source distance and σ is the Stefan-<br />
Boltzmann c<strong>on</strong>stant. Measurements give R ∼ 10 km, but<br />
are not yet accurate enough to c<strong>on</strong>stra<strong>in</strong> nuclear equati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> state [19]. This is ma<strong>in</strong>ly because few subsets <strong>in</strong> Table 1<br />
provide simultaneous measurements <str<strong>on</strong>g>of</str<strong>on</strong>g> M and R.<br />
Radio Pulsar Number<br />
250<br />
Radio Pulsars<br />
8<br />
200 Accr. Pulsars<br />
Magnetar<br />
Magnetars<br />
6&<br />
150<br />
4<br />
100<br />
Accret<strong>in</strong>g<br />
50<br />
2<br />
Pulsars<br />
0<br />
8<br />
9 10 11 12 13 14 15<br />
Log Magnetic Field (Gauss)<br />
Figure 2: Distributi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> surface magnetic fields <str<strong>on</strong>g>of</str<strong>on</strong>g> neutr<strong>on</strong><br />
stars <strong>in</strong> our Galaxy. Blue represents rotati<strong>on</strong>-powered<br />
pulsars, and red magnetars, with their field strengths both<br />
estimated us<strong>in</strong>g eq.(7). Green (with the number <strong>on</strong> the right<br />
hand side) shows accreti<strong>on</strong>-powered pulsars <str<strong>on</strong>g>of</str<strong>on</strong>g> which the<br />
field <strong>in</strong>tensity is measured with eq.(11).<br />
Surface Magnetic Fields<br />
MF strengths <str<strong>on</strong>g>of</str<strong>on</strong>g> rotati<strong>on</strong>-powered NSs (radio pulsars <strong>in</strong><br />
Table 1) are estimated assum<strong>in</strong>g that they lose their rotati<strong>on</strong>al<br />
energies by emitt<strong>in</strong>g magnetic dipole radiati<strong>on</strong> as<br />
they rotates [9]. Then, the dipolar MF strength is expressed,<br />
<strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the period P and its derivative ˙ P , as<br />
B = 1.0 × 10 12<br />
√<br />
(P/0.1s)( ˙ P /1 × 10−14ss−1 ) G , (7)<br />
assum<strong>in</strong>g a can<strong>on</strong>ical moment <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>ertia. Similar results are<br />
obta<strong>in</strong>ed if we assume that the rotati<strong>on</strong>al energies are spent<br />
<strong>in</strong> particle accelerati<strong>on</strong> [7].<br />
The MF distributi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> rotati<strong>on</strong>-powered NSs (i.e., objects<br />
with [r]), thus estimated, are given <strong>in</strong> Fig. 2 <strong>in</strong> blue<br />
and red. The major peak at B ∼ 10 12 G encompasses ord<strong>in</strong>ary<br />
radio pulsars, a m<strong>in</strong>or peak at 10 8−9 G millisec<strong>on</strong>d<br />
pulsars, while red are magnetars. Green data po<strong>in</strong>ts are expla<strong>in</strong>ed<br />
<strong>in</strong> the next secti<strong>on</strong>.<br />
0
Radiati<strong>on</strong> from Rotati<strong>on</strong>-Powered Pulsars<br />
The Crab pulsar (borne <strong>in</strong> AD1054), a prototypical<br />
rotati<strong>on</strong>-powered NS, has B ∼ 3 × 10 12 G, and rotates<br />
with an angular frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> ω = 200 Hz. As a result, the<br />
<strong>in</strong>duced Lorentz electric field amounts to<br />
E ∼ RωB ∼ 10 14 Vm −1 , (8)<br />
and the electric potential to ER ∼ 6 × 10 18 V. This ultrahigh<br />
electric field will accelerate electr<strong>on</strong>s (plus probably<br />
positr<strong>on</strong>s), which then emit n<strong>on</strong>-thermal radiati<strong>on</strong> via synchrotr<strong>on</strong><br />
radiati<strong>on</strong>, <strong>in</strong>verse Compt<strong>on</strong> scatter<strong>in</strong>g, and curvature<br />
radiati<strong>on</strong> [18]. Emergent spectra <strong>in</strong>evitably span many<br />
orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude <strong>in</strong> frequency. To observe these objects,<br />
without hampered by background stellar signals, we<br />
may use either radio frequencies, or gamma-ray range as<br />
dem<strong>on</strong>strated by the Fermi Gamma-ray Space Telescope<br />
launched <strong>in</strong> 2008 June [1].<br />
ACCRETING X-RAY PULSARS<br />
X-ray Emissi<strong>on</strong><br />
C<strong>on</strong>trary to rotati<strong>on</strong>-powered NSs, accreti<strong>on</strong>-powered<br />
NSs (those with [g] <strong>in</strong> Table 1) radiate predom<strong>in</strong>antly <strong>in</strong><br />
the X-ray frequency, for the follow<strong>in</strong>g reas<strong>on</strong>. C<strong>on</strong>sider an<br />
accret<strong>in</strong>g NS with weak MF, radiat<strong>in</strong>g spherically at a lum<strong>in</strong>osity<br />
L. Then, the phot<strong>on</strong> momentum flux L/4πr 2 c exerts<br />
an outward force Fout ≡ LσTne/4πr 2 c per unit volume <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the accret<strong>in</strong>g matter, with σT be<strong>in</strong>g the Thoms<strong>on</strong> cross secti<strong>on</strong>.<br />
The maximum lum<strong>in</strong>osity LE, called Edd<strong>in</strong>gt<strong>on</strong> limit,<br />
is realized when Fout balances the <strong>in</strong>ward gravitati<strong>on</strong>al pull<br />
[right hand side <str<strong>on</strong>g>of</str<strong>on</strong>g> eq.(1)] exerted <strong>on</strong> the same volume. Assum<strong>in</strong>g<br />
η = 1.2, we then have<br />
LE = 4πGηmpcM/σT = 2.2 × 10 31 (M/1.4M⊙) W.<br />
(9)<br />
Equat<strong>in</strong>g this with the blackbody lum<strong>in</strong>osity 4πR 2 σT 4 , the<br />
blackbody temperature is obta<strong>in</strong>ed as<br />
T = 2.3 × 10 7 (L/LE) K. (10)<br />
This falls right <strong>on</strong> the X-ray energy band.<br />
X-ray phot<strong>on</strong>s have high penetrat<strong>in</strong>g power, as evidenced<br />
by medical diagnostics. However, they cannot penetrate<br />
the atmosphere, which is ∼ 50 times thicker (<strong>in</strong> Oxygen<br />
column density) than human body. As a result, a number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> X-ray astrophysics satellites have been launched. In<br />
Japan, Hakucho (launched <strong>in</strong> 1979), Tenma (1983), G<strong>in</strong>ga<br />
(1987), ASCA (1993)[21], and Suzaku (2005)[13] have<br />
been c<strong>on</strong>tribut<strong>in</strong>g. We are now prepar<strong>in</strong>g for the next missi<strong>on</strong>,<br />
ASTRO-H, to be launched <strong>in</strong> 2014.<br />
Cyclotr<strong>on</strong> Res<strong>on</strong>ances<br />
While eq.(10) applies to spherical accreti<strong>on</strong> <strong>on</strong>to weak-<br />
FM NSs, the c<strong>on</strong>diti<strong>on</strong> somewhat differs when the NS has<br />
a str<strong>on</strong>g MF (X-ray pulsars <strong>in</strong> Table 1), because the accreti<strong>on</strong><br />
flow is channeled <strong>on</strong>to the two magnetic poles, and the<br />
Thoms<strong>on</strong> cross secti<strong>on</strong> is magnetically modified [10]. As<br />
a result, the X-ray emissi<strong>on</strong> from X-ray pulsars atta<strong>in</strong>s a<br />
higher temperature, ∼ 10 8 K, or ∼ 10 keV.<br />
An outstand<strong>in</strong>g property <str<strong>on</strong>g>of</str<strong>on</strong>g> X-ray pulsars is a clear spectral<br />
feature due to electr<strong>on</strong> cyclotr<strong>on</strong> res<strong>on</strong>ance, or Cyclotr<strong>on</strong><br />
Res<strong>on</strong>ance Scatter<strong>in</strong>g Feature (CRSF), which <str<strong>on</strong>g>of</str<strong>on</strong>g>ten<br />
appears <strong>in</strong> their spectra at an energy <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Ea = ¯heB/me = 11.6(B/10 12 G) keV. (11)<br />
So far, CRSFs have been observed, sometimes <strong>in</strong> multiple<br />
harm<strong>on</strong>ics, all <strong>in</strong> absorpti<strong>on</strong>, from ∼ 15 objects [12], about<br />
half the known X-ray pulsars. Figure 3 gives <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
most prom<strong>in</strong>ent examples <str<strong>on</strong>g>of</str<strong>on</strong>g> CRSFs [17].<br />
Equati<strong>on</strong> (11) have been used to directly measure the<br />
surface MF strengths <str<strong>on</strong>g>of</str<strong>on</strong>g> X-ray pulsars. Obviously, this<br />
method gives a much higher accuracy than eq.(7) <strong>in</strong> measur<strong>in</strong>g<br />
B. The green data po<strong>in</strong>ts <strong>in</strong> Fig. 2 refer to these measurements<br />
[12]. The results are c<strong>on</strong>centrated over a rather<br />
narrow range, B = (1−4)×10 12 G, although the regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
B > 5 × 10 12 G is yet to be explored with high-sensitivity<br />
hard X-ray <strong>in</strong>struments, <strong>in</strong>clud<strong>in</strong>g the Hard X-ray Detector<br />
[20] <strong>on</strong>board Suzaku. This result argues aga<strong>in</strong>st a view,<br />
which was popular <strong>in</strong> the 1990’s, that the MF <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>gly<br />
magnetized NSs decay wit time [23].<br />
In Fig. 2, the green data po<strong>in</strong>ts appear to be rathe dist<strong>in</strong>ct<br />
from the more weakly magnetized objects. Then, the<br />
MF strengths may exhibit bimodal behavior. Based <strong>on</strong> this,<br />
we further speculate that the NS magnetism is a manifestati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> ferromagnetism <strong>in</strong> nuclear matter [12], rather than<br />
the more popular idea <str<strong>on</strong>g>of</str<strong>on</strong>g> permanent current. Those with<br />
B = 10 8−9 G can be <strong>in</strong>terpreted as entirely paramagnetic<br />
objects. The str<strong>on</strong>ger-field objects with B ∼ 10 12 G can be<br />
expla<strong>in</strong>ed if <strong>on</strong>ly ∼ 10 −4 <str<strong>on</strong>g>of</str<strong>on</strong>g> the total NS volume becomes<br />
ferromagnetic. Furthermore, we expect B ∼ 10 16 G if the<br />
entire volume <str<strong>on</strong>g>of</str<strong>on</strong>g> a NS becomes ferromagnetic. Magnetars,<br />
to be described below, may be such objects.<br />
Figure 3: Examples <str<strong>on</strong>g>of</str<strong>on</strong>g> CRSFs, measured from the transient<br />
pulsar X0331+53. The fundamental and 2nd harm<strong>on</strong>ic features<br />
are <strong>in</strong>dicated by arrows. Dashed l<strong>in</strong>es <strong>in</strong>dicate the<br />
underly<strong>in</strong>g thermal spectrum.
General behavior<br />
MAGNETARS<br />
About 15 NSs shown <strong>in</strong> Fig. 2 <strong>in</strong> red are called magnetars,<br />
and possess the follow<strong>in</strong>g comm<strong>on</strong> characteristics:<br />
1. Rotati<strong>on</strong> periods <str<strong>on</strong>g>of</str<strong>on</strong>g> P = 2 − 11 sec, and high sp<strong>in</strong>down<br />
rates as ˙<br />
P 10 −11 s s −1 , yield<strong>in</strong>g via via eq.(7)<br />
B = 10 14−15 G. (12)<br />
2. Young characteristic ages, τ ≡ P/2 ˙<br />
P = 10 2−4 yr,<br />
which is also supported by their occasi<strong>on</strong>al associati<strong>on</strong><br />
with supernova remnants.<br />
3. Sporadically recurr<strong>in</strong>g active periods, when a large<br />
number <str<strong>on</strong>g>of</str<strong>on</strong>g> short (< 1 s) bursts are emitted as <strong>in</strong> Fig. 4.<br />
4. X-ray emissi<strong>on</strong> with a lum<strong>in</strong>osity L ∼ 10 27 W, which<br />
much exceeds the sp<strong>in</strong>-down energy release.<br />
5. No evidence <str<strong>on</strong>g>of</str<strong>on</strong>g> b<strong>in</strong>ary compani<strong>on</strong>s.<br />
6. No radio emissi<strong>on</strong> unlike typical rotati<strong>on</strong> driven NSs.<br />
Figure 4: An example <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetar short bursts. These<br />
were observed with Suzaku, for <strong>on</strong>e day, from the magnetar<br />
1E1547-54 dur<strong>in</strong>g its 2009 January activity.<br />
From 4 and 5, magnetars can neither be rotati<strong>on</strong> powered<br />
nor accreti<strong>on</strong> powered. Furthermore, eq.(12) implies<br />
that the magnetic energy <str<strong>on</strong>g>of</str<strong>on</strong>g> these objects exceed their rotati<strong>on</strong>al<br />
energies. Thus, magnetars are c<strong>on</strong>sidered to be magnetically<br />
powered objects (marked with [m] <strong>in</strong> Table 1),<br />
produc<strong>in</strong>g persistent and burst X-ray radiati<strong>on</strong> by somehow<br />
releas<strong>in</strong>g their magnetic energy [22]. This <strong>in</strong>terpretati<strong>on</strong> is<br />
supported by Fig. 5, where B <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars measured with<br />
eq.(7) decreases with τ. S<strong>in</strong>ce an NS with B = 10 15 G has<br />
a magnetic energy <str<strong>on</strong>g>of</str<strong>on</strong>g> ∼ 3×10 40 J, its decay over ∼ 100 kyr<br />
(Fig. 5) would afford L ∼ 1×10 28 W, which is sufficient to<br />
expla<strong>in</strong> item 4. Magnetars may have even str<strong>on</strong>ger toroidal<br />
MF, which is mostly c<strong>on</strong>f<strong>in</strong>ed with the stellar <strong>in</strong>terior.<br />
Although we believe magnetars to be a special subset<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> NSs, their masses and radii rema<strong>in</strong> totally unknown.<br />
Therefore, we cannot tell at present whether magnetars are<br />
different from other NSs <strong>in</strong> any aspect other than the MF.<br />
Likewise, noth<strong>in</strong>g is known, either, what k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> supernova<br />
explosi<strong>on</strong>s lead to the formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars.<br />
One fasc<strong>in</strong>at<strong>in</strong>g aspect <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars is that their <strong>in</strong>ferred<br />
MF strengths [eq.(12)] exceeds the critical value,<br />
Bcr = (mec) 2 /¯he = 4.4 × 10 13 G (13)<br />
at which the CRSF energy [eq.(11)], or equivalently the<br />
Landau level separati<strong>on</strong>, reaches mec 2 . Therefore, we expect<br />
various “str<strong>on</strong>g field” effects to take place.<br />
Figure 5: The estimated surface MF strengths <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars<br />
(the same as <strong>in</strong> Fig. 2), shown as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> τ.<br />
Emissi<strong>on</strong> from Magnetars<br />
In the 20th Century, magnetars were known to emit (except<br />
short bursts) pulsat<strong>in</strong>g persistent s<str<strong>on</strong>g>of</str<strong>on</strong>g>t X-ray emissi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> which the spectrum is approximated by a ∼ 0.5 keV<br />
blackbody. Then, the European Gamma-ray observatory<br />
INTEGRAL discovered [11, 8], from several magnetars,<br />
a str<strong>on</strong>gly puls<strong>in</strong>g hard X-ray comp<strong>on</strong>ent, extend<strong>in</strong>g to<br />
∼ 100 keV with a very flat phot<strong>on</strong> <strong>in</strong>dex <str<strong>on</strong>g>of</str<strong>on</strong>g> Γ ∼ 1 (phot<strong>on</strong><br />
number flux scal<strong>in</strong>g with energy E as ∝ E −Γ ). Such<br />
a hard-slope emissi<strong>on</strong> has rarely been observed from other<br />
types <str<strong>on</strong>g>of</str<strong>on</strong>g> cosmic X-ray sources, and is difficult to expla<strong>in</strong> <strong>in</strong><br />
terms <str<strong>on</strong>g>of</str<strong>on</strong>g> know radiati<strong>on</strong> processes (e.g., synchrotr<strong>on</strong> emissi<strong>on</strong>).<br />
Therefore, this comp<strong>on</strong>ent is c<strong>on</strong>sidered to provide<br />
an important clue to the nature <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars.<br />
We have c<strong>on</strong>ducted extensive X-ray studies <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars,<br />
us<strong>in</strong>g Suzaku [2, 3, 5, 4, 6, 16] which has a superior<br />
wide-band capability realized by the HXD [20], and the<br />
gamma-ray burst satellite HETE-2 [14, 15]. Our results,<br />
cover<strong>in</strong>g both persistent and burst emissi<strong>on</strong>s, are summarized<br />
as follows.<br />
1. As exemplified by Fig. 6, about 8 magnetars we observed<br />
all exhibit persistent spectra that are composed<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a s<str<strong>on</strong>g>of</str<strong>on</strong>g>t and a hard comp<strong>on</strong>ent [2, 16, 5, 4, 6]. This<br />
re<strong>in</strong>forces the INTEGRAL discovery, and improves it.<br />
2. The s<str<strong>on</strong>g>of</str<strong>on</strong>g>t comp<strong>on</strong>ent is approximated by two blackbodies,<br />
with temperatures TL and TH which scales as<br />
TH ≈ 3.5TL. The emissi<strong>on</strong> areas are roughly c<strong>on</strong>sistent<br />
with the size <str<strong>on</strong>g>of</str<strong>on</strong>g> an NS.<br />
3. The above two items hold for both the persistent emissi<strong>on</strong>,<br />
and short bursts detected from a few objects [15].<br />
4. As shown <strong>in</strong> Fig. 7 (top), the spectral hardness ratio<br />
(1–60 keV flux ratios between the hard and s<str<strong>on</strong>g>of</str<strong>on</strong>g>t comp<strong>on</strong>ents)<br />
has been discovered to anti-correlates with τ<br />
[6]. This is a discovery <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetar evoluti<strong>on</strong>.<br />
5. The hard comp<strong>on</strong>ent has an extremely flat slope, Γ =<br />
1.7−0.4, which becomes even harder (flatter) towards<br />
more aged objects. This is visualized <strong>in</strong> Fig. 6, and<br />
summarized <strong>in</strong> Fig. 7 (bottom).
2 -2 -1<br />
keV (ph cm s keV -1 )<br />
0.1<br />
0.01<br />
0.001<br />
1<br />
1E1547-54 (τ=1.4 kyr)<br />
4U 0142+61 (τ=70 kyr)<br />
10<br />
Energy (keV)<br />
100<br />
Figure 6: Suzaku νFν spectra <str<strong>on</strong>g>of</str<strong>on</strong>g> two magnetars. Green is<br />
1E1547−54 <strong>in</strong> activity [4], while red is 4U 0142+61 [6].<br />
How Magnetars Work?<br />
Based <strong>on</strong> the Suzaku and HETE-2 results, let us present<br />
a speculative <strong>in</strong>terpretati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars. First, we believe<br />
that they really have MF strengths exceed<strong>in</strong>g Bcr<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> eq.(13), even though the estimates <str<strong>on</strong>g>of</str<strong>on</strong>g> eq.(12) could be<br />
very crude. This belief is based <strong>on</strong> the fact that their twocomp<strong>on</strong>ent<br />
X-ray spectra and burst activity (Fig. 4) are really<br />
unique am<strong>on</strong>g various celestial X-ray sources. Thus,<br />
magnetars must be <strong>in</strong> a unique physical c<strong>on</strong>diti<strong>on</strong>, and the<br />
proposed str<strong>on</strong>g-MF scenario [22] is most appropriate.<br />
Next, we found many similarities between persistent and<br />
burst emissi<strong>on</strong>s. Therefore, the persistent emissi<strong>on</strong> is likely<br />
to be formed by a large number <str<strong>on</strong>g>of</str<strong>on</strong>g> micro-bursts [15], just<br />
as the solar cor<strong>on</strong>a may be heated by micro-flares. The<br />
bursts themselves could be a result <str<strong>on</strong>g>of</str<strong>on</strong>g> sporadic magnetic<br />
rec<strong>on</strong>necti<strong>on</strong> [22], either <strong>in</strong> the magnetosphere or stellar<br />
<strong>in</strong>terior. Then, Fig. 7 (top) can be understood as a gradual<br />
decl<strong>in</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic activity, <strong>in</strong> agreement with Fig. 5.<br />
Third, the s<str<strong>on</strong>g>of</str<strong>on</strong>g>t comp<strong>on</strong>ent can be <strong>in</strong>terpreted as thermal<br />
radiati<strong>on</strong> from the NS surface, heated by the magnetic<br />
energy release. The two blackbodies may represent two<br />
phot<strong>on</strong> polarizati<strong>on</strong>s (O-mode and X-mode) with respect<br />
to the str<strong>on</strong>g MF, which have different electr<strong>on</strong>-scatter<strong>in</strong>g<br />
cross secti<strong>on</strong>s [10, 3]. We expect the emissi<strong>on</strong> to be hence<br />
str<strong>on</strong>gly polarized; this will be tested by the wold’s first<br />
X-ray polarimetric missi<strong>on</strong>, GEMS, developed under US-<br />
Japan collaborati<strong>on</strong> and scheduled for launch <strong>in</strong> 2014.<br />
F<strong>in</strong>ally, the enigmatic hard comp<strong>on</strong>ent may be a result<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> splitt<strong>in</strong>g <strong>in</strong> the super-critical MF. In the magnetosphere,<br />
the str<strong>on</strong>g Lorentz field [eq.(8)] will accelerate<br />
electr<strong>on</strong>s (and positr<strong>on</strong>s), which will emit gamma-ray phot<strong>on</strong>s.<br />
These phot<strong>on</strong>s cannot escape out, s<strong>in</strong>ce they would<br />
<strong>in</strong>teract with the MF and split <strong>in</strong>to two lower-energy phot<strong>on</strong>s<br />
[10, 6]. A cascade <str<strong>on</strong>g>of</str<strong>on</strong>g> this process will produce a hard<br />
X-ray c<strong>on</strong>t<strong>in</strong>uum extend<strong>in</strong>g down to ∼ 10 keV with a very<br />
flat Γ. In additi<strong>on</strong>, this cascade will stop at relatively high<br />
energies <strong>in</strong> older magnetars, because <str<strong>on</strong>g>of</str<strong>on</strong>g> their weaker MF<br />
(Fig.5). This can expla<strong>in</strong> Fig. 7 (bottom). Detailed <strong>in</strong>vestigati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars <strong>in</strong> the 100–600 keV regi<strong>on</strong> is an important<br />
future task <str<strong>on</strong>g>of</str<strong>on</strong>g> our ASTRO-H.<br />
Figure 7: (Top) Ratios <str<strong>on</strong>g>of</str<strong>on</strong>g> the 1–60 keV lum<strong>in</strong>osities between<br />
the hard and s<str<strong>on</strong>g>of</str<strong>on</strong>g>t comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars, mesured<br />
with Suzaku, shown as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> their characteristic age<br />
τ [6]. Red and blue <strong>in</strong>dicate different subsets. (Bottom)<br />
The phot<strong>on</strong> <strong>in</strong>dex Γ <str<strong>on</strong>g>of</str<strong>on</strong>g> the hard comp<strong>on</strong>ent shown <strong>in</strong> the<br />
same way [6]. Green po<strong>in</strong>ts are take from literature.<br />
REFERENCES<br />
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QCD ORIGIN OF STRONG MAGNETIC FIELD IN COMPACT STARS ∗<br />
Abstract<br />
T. Tatsumi † , Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Kyoto University, Kyoto 606-8502, Japan<br />
Some magnetic properties <str<strong>on</strong>g>of</str<strong>on</strong>g> quark matter and a microscopic<br />
orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the str<strong>on</strong>g magnetic field <strong>in</strong> compact stars<br />
are discussed; ferromagnetic order is discussed with the<br />
Fermi liquid theory and possible appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> sp<strong>in</strong> density<br />
wave is suggested with<strong>in</strong> the NJL model. Implicati<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> these magnetic properties are briefly discussed for compact<br />
stars.<br />
INTRODUCTION AND MOTIVATION<br />
Nowadays there have been many works about the QCD<br />
phase diagram. Here, we are c<strong>on</strong>centrated <strong>in</strong> magnetic<br />
properties <str<strong>on</strong>g>of</str<strong>on</strong>g> quark matter and their implicati<strong>on</strong>s <strong>on</strong> compact<br />
star phenomena. Let’s beg<strong>in</strong> with a simple questi<strong>on</strong>:<br />
what is and where can we expect magnetism <strong>in</strong> QCD. At<br />
low densities we may expect the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> pi<strong>on</strong> c<strong>on</strong>densati<strong>on</strong><br />
<strong>in</strong> hadr<strong>on</strong>ic matter, where the classical pi<strong>on</strong> field<br />
develops , followed by the specific sp<strong>in</strong>-isosp<strong>in</strong> order <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
nucle<strong>on</strong>s [1]. In quark matter we shall see a n<strong>on</strong>-uniform<br />
phase (called dual chiral density wave (DCDW) phase),<br />
accompany<strong>in</strong>g the restorati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> chiral symmetry, where<br />
the pseudoscalar c<strong>on</strong>densate ⟨¯qiγ5τ3q⟩ ̸= 0 as well as the<br />
scalar c<strong>on</strong>densate spatially oscillates [2]. Accord<strong>in</strong>gly the<br />
magnetizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> quark matter also oscillates like sp<strong>in</strong> density<br />
wave (SDW) <strong>in</strong> c<strong>on</strong>densed matter physics. Furthermore<br />
we may expect a ferromagnetic phase at some density<br />
regi<strong>on</strong> [3].<br />
On the other hand such magnetic properties should have<br />
some implicati<strong>on</strong>s <strong>on</strong> the compact star phenomena. In particular<br />
it has been well known that there is a str<strong>on</strong>g magnetic<br />
field <strong>in</strong> compact stars. The orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> such str<strong>on</strong>g<br />
field is not clear even now, and it has been a l<strong>on</strong>g-stand<strong>in</strong>g<br />
problem s<strong>in</strong>ce the first discovery <str<strong>on</strong>g>of</str<strong>on</strong>g> pulsars <strong>in</strong> early seventies.<br />
The recent discovery <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars seems to revive<br />
the problem aga<strong>in</strong> [4]. Their magnetic field amounts to<br />
O(1015G) from the P − ˙ P diagram. At present many people<br />
believe the <strong>in</strong>heritance <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field from the<br />
progenitor ma<strong>in</strong>-sequence stars or dynamo scenario due to<br />
the electr<strong>on</strong> current. We c<strong>on</strong>sider here a microscopic orig<strong>in</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic field by exam<strong>in</strong><strong>in</strong>g a possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> sp<strong>on</strong>taneous<br />
sp<strong>in</strong> polarizati<strong>on</strong> <strong>in</strong> quark matter.<br />
Next, we c<strong>on</strong>sider a possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> a n<strong>on</strong>-uniform state<br />
<strong>in</strong> the vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g> the chiral transiti<strong>on</strong>. Recently there have<br />
d<strong>on</strong>e many works about the n<strong>on</strong>-uniform state at moderate<br />
∗ Work partially supported by the Grant-<strong>in</strong>-Aid for the Global COE<br />
Program “The Next Generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Spun from Universality and<br />
Emergence” from the M<strong>in</strong>istry <str<strong>on</strong>g>of</str<strong>on</strong>g> Educati<strong>on</strong>, Culture, Sports, Science<br />
and Technology (MEXT) <str<strong>on</strong>g>of</str<strong>on</strong>g> Japan and the Grant-<strong>in</strong>-Aid for Scientific Research<br />
(C) (16540246, 20540267).<br />
† tatsumi@ruby.scphys.kyoto-u.ac.jp<br />
densities. We shall see that this phase exhibits an <strong>in</strong>terest<strong>in</strong>g<br />
magnetic property like SDW.<br />
FERROMAGNETIC TRANSITION<br />
The first study about the ferromagnetism <strong>in</strong> quark matter<br />
has been performed by us<strong>in</strong>g the Bloch idea about the<br />
ferromagnetism <str<strong>on</strong>g>of</str<strong>on</strong>g> it<strong>in</strong>erant electr<strong>on</strong>s [5]. The mechanism<br />
is rather simple due to the Pauli pr<strong>in</strong>ciple: c<strong>on</strong>sider electr<strong>on</strong>s<br />
<strong>in</strong>teract<strong>in</strong>g with each other by the Coulomb <strong>in</strong>teracti<strong>on</strong><br />
<strong>in</strong> the background <str<strong>on</strong>g>of</str<strong>on</strong>g> the uniformly distributed positive<br />
charge to compensate the electr<strong>on</strong> charge. Then the<br />
Fock exchange <strong>in</strong>teracti<strong>on</strong> gives a lead<strong>in</strong>g-order c<strong>on</strong>tributi<strong>on</strong>.<br />
Then the electr<strong>on</strong> pair with the same sp<strong>in</strong> can effectively<br />
avoid the Coulomb <strong>in</strong>teracti<strong>on</strong> to give an attractive<br />
c<strong>on</strong>tributi<strong>on</strong> to the total energy due to the Pauli pr<strong>in</strong>ciple.<br />
As the counter effect such polarized electr<strong>on</strong> system costs<br />
more k<strong>in</strong>etic energy <strong>in</strong>crease. So when the former effect<br />
becomes larger than the latter <strong>on</strong>e, we can expect sp<strong>on</strong>taneous<br />
sp<strong>in</strong> polarizati<strong>on</strong>. A perturbative calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> quark<br />
matter <strong>in</strong>teract<strong>in</strong>g with the <strong>on</strong>e-glu<strong>on</strong>-exchange <strong>in</strong>teracti<strong>on</strong><br />
shows the transiti<strong>on</strong> to ferromagnetic phase at the order <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
nuclear density. Apply<strong>in</strong>g this idea to a star with solar mass<br />
and radius <str<strong>on</strong>g>of</str<strong>on</strong>g> 10km, we can roughly estimate the magnetic<br />
field <str<strong>on</strong>g>of</str<strong>on</strong>g> O(10 13−17 G). Thus we can feel that quark matter<br />
<strong>in</strong>side the core regi<strong>on</strong> may give an orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic field<br />
<strong>in</strong> compact stars.<br />
Recently we have studied the magnetic susceptibility<br />
χM to get more <strong>in</strong>sight about the properties <str<strong>on</strong>g>of</str<strong>on</strong>g> ferromagnetic<br />
transiti<strong>on</strong> with<strong>in</strong> the Fermi-liquid theory [6]. χM is<br />
written <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the quasiparticle <strong>in</strong>teracti<strong>on</strong>s,<br />
χM =<br />
( )2 ¯gDµq N(T )<br />
2 1 + N(T ) ¯ , (1)<br />
f a<br />
where ¯gD ≡ ∫<br />
|k|=kF dΩk/4πgD(k) is the effective gyromagnetic<br />
ratio, N(T ) the effective density <str<strong>on</strong>g>of</str<strong>on</strong>g> states around<br />
the Fermi surface and ¯ f a the sp<strong>in</strong> dependent Landau-<br />
Migdal parameter [7, 8]. The sp<strong>in</strong> susceptibility can be<br />
easily evaluated by us<strong>in</strong>g the OGE <strong>in</strong>teracti<strong>on</strong>. Then we<br />
can see that the Landau-Migdal (LM) parameters <strong>in</strong>volve<br />
<strong>in</strong>frared (IR) divergences <strong>in</strong> gauge theories (QCD/QED).<br />
So we must take <strong>in</strong>to account the screen<strong>in</strong>g effect at least<br />
to obta<strong>in</strong> the mean<strong>in</strong>gful results. We have d<strong>on</strong>e it by calculat<strong>in</strong>g<br />
the quark polarizati<strong>on</strong> operator by the hard-denseloop<br />
(HDL) resummati<strong>on</strong>. As the results, we can see that<br />
the Debye screen<strong>in</strong>g for the l<strong>on</strong>gitud<strong>in</strong>al glu<strong>on</strong>s surely improves<br />
the IR behavior, while the transverse glu<strong>on</strong>s <strong>on</strong>ly<br />
receive the dynamic screen<strong>in</strong>g due to the Landau damp<strong>in</strong>g.
Zero temperature case<br />
There are still left the divergences <strong>in</strong> the LM parameters<br />
at T = 0, but they cancel each other to give a f<strong>in</strong>ite χM. F<strong>in</strong>ally<br />
magnetic susceptibility is given as a sum <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>tributi<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the bare <strong>in</strong>teracti<strong>on</strong> and the static screen<strong>in</strong>g<br />
effect,<br />
(χM/χPauli) −1<br />
0 = 1 − Cf g 2 µ<br />
12π 2 E 2 F kF<br />
− 1<br />
2 (E2 F + 4mEF − 2m 2 )κ ln 2<br />
κ<br />
[<br />
m(2EF + m) −<br />
]<br />
, (2)<br />
with κ = m2 D /2k2 F <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the Debye mass, m2D ≡<br />
g2 µkF /2π2 2<br />
Nc , and Cf = −1<br />
. Thus the screen<strong>in</strong>g effect<br />
2Nc<br />
gives the g4 ln g2 term.<br />
To dem<strong>on</strong>strate the screen<strong>in</strong>g effect, we show <strong>in</strong> Fig. 4.3<br />
the magnetic susceptibility. We assume a flavor-symmetric<br />
quark matter, ρu = ρd = ρs = ρB/3, and take the QCD<br />
coupl<strong>in</strong>g c<strong>on</strong>stant as αs ≡ g2 /4π = 2.2 and the strange<br />
quark mass ms = 300MeV <strong>in</strong>ferred from the MIT bag<br />
model. Note that the screen<strong>in</strong>g effect is qualitatively dif-<br />
χ M /χ Pauli<br />
20<br />
15<br />
10<br />
5<br />
0<br />
-5<br />
-10<br />
-15<br />
-20<br />
α S=2.2<br />
ms=300MeV<br />
mu=md=0<br />
0 0.5 1 1.5 2<br />
kF [1/fm]<br />
Figure 1: Magnetic susceptibility at T = 0. Screen<strong>in</strong>g<br />
effects are shown <strong>in</strong> comparis<strong>on</strong> with the simple OGE<br />
case: the solid curve shows the result with the simple OGE<br />
without screen<strong>in</strong>g, while the dashed and dash-dotted <strong>on</strong>es<br />
shows the screen<strong>in</strong>g effect with Nf = 1 (<strong>on</strong>ly s quark)and<br />
Nf = 2 + 1 (u, d, s quarks), respectively.<br />
ferent, depend<strong>in</strong>g <strong>on</strong> the number <str<strong>on</strong>g>of</str<strong>on</strong>g> flavor Nf . The Debye<br />
mass is given by all the flavors,<br />
m 2 D = ∑<br />
flavors<br />
g 2<br />
2π 2 kF,f EF,f , (3)<br />
so that the κ ln(2/κ) term changes its sign for κ =<br />
m2 D /2k2 F > 2. Thus we can see the screen<strong>in</strong>g flavors sp<strong>on</strong>taneous<br />
magnetizati<strong>on</strong> <strong>in</strong> large Nf .<br />
N<strong>on</strong>-Fermi-liquid effect at f<strong>in</strong>ite temperature<br />
We c<strong>on</strong>sider the low temperature case, T/µ ≪ 1, but<br />
usual low-temperature expansi<strong>on</strong> can not be applied, s<strong>in</strong>ce<br />
the quasiparticles exhibits an anomalous behavior near the<br />
Fermi surface. S<strong>in</strong>ce the l<strong>on</strong>gitud<strong>in</strong>al glu<strong>on</strong>s are short<br />
ranged due to the Debye screen<strong>in</strong>g, their c<strong>on</strong>tributi<strong>on</strong>s are<br />
almost temperature <strong>in</strong>dependent. Thus the ma<strong>in</strong> c<strong>on</strong>tributi<strong>on</strong><br />
to the temperature dependence comes from the transverse<br />
glu<strong>on</strong>s. Careful c<strong>on</strong>siderati<strong>on</strong>s about the quasiparticle<br />
energy show that quark matter behaves like marg<strong>in</strong>al<br />
Fermi liquid, where the Fermi velocity and the renormalizati<strong>on</strong><br />
factor vanish at the Fermi surface [9]. Such behavior<br />
is brought about by the transverse glu<strong>on</strong>s. The magnetic<br />
susceptibility is then given as<br />
(χM /χPauli) −1 = (χM /χPauli) −1<br />
0<br />
+ π2<br />
6k 4 F<br />
(<br />
2E 2 F − m 2 + m4<br />
E 2 F<br />
)<br />
T 2<br />
( )<br />
Λ<br />
T<br />
+ Cf g2uF (2k<br />
72<br />
4 F + k2 F m2 + m4 )<br />
k4 F E2 T<br />
F<br />
2 ln<br />
+ O(g 2 T 2 ). (4)<br />
with uF ≡ vF /EF , where we can see the T 2 ln T term appears<br />
as a novel n<strong>on</strong>-Fermi-liquid effect, besides the usual<br />
T 2 term, It should be <strong>in</strong>terest<strong>in</strong>g to compare this term with<br />
other <strong>on</strong>es <strong>in</strong> specific heat or the superc<strong>on</strong>duct<strong>in</strong>g gap energy.<br />
Furthermore, such logarithmic behavior also resembles<br />
the <strong>on</strong>e by the sp<strong>in</strong> fluctuati<strong>on</strong>s or paramagn<strong>on</strong>s. F<strong>in</strong>ally<br />
the phase diagram is presented <strong>in</strong> Fig. 2, where we<br />
can also asses the importance <str<strong>on</strong>g>of</str<strong>on</strong>g> the n<strong>on</strong>-Fermi-liquid effect.<br />
T [MeV]<br />
70<br />
60<br />
50<br />
40<br />
30<br />
20<br />
10<br />
Full.<br />
w/o dynamical scr.<br />
w/o static scr.<br />
w/o any scr.<br />
0<br />
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8<br />
kF [1/fm]<br />
Figure 2: Magnetic phase diagram <strong>in</strong> the densitytemperature<br />
plane. The open (filled) circle <strong>in</strong>dicates the<br />
Curie temperature at kF = 1.1(1.6) fm −1 while the<br />
squares show those without the T 2 ln T term.<br />
F<strong>in</strong>ally we present a phase diagram <strong>in</strong> Fig. 2, where we<br />
can estimate the Curie temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> several tens <str<strong>on</strong>g>of</str<strong>on</strong>g> MeV.<br />
We can also see how the n<strong>on</strong>-Fermi-liquid effect works for<br />
the ferromagnetic transiti<strong>on</strong>.<br />
MAGNETISM AND CHIRAL SYMMETRY<br />
Recently there have been appeared many studies about<br />
the n<strong>on</strong>-uniform states <strong>in</strong> QCD, stimulated by the development<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the studies about the exact soluti<strong>on</strong>s <strong>in</strong> 1+1 dimensi<strong>on</strong>al<br />
models [10]. The formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the n<strong>on</strong>-uniform
states <strong>in</strong> quark matter have been studied <strong>in</strong> relati<strong>on</strong> to chiral<br />
transiti<strong>on</strong> [11]. The appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> density waves or<br />
crystall<strong>in</strong>e structures has been an <strong>in</strong>terest<strong>in</strong>g possibility at<br />
moderate densities. Note that the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> the n<strong>on</strong>uniform<br />
phase is not special, but rather familiar <strong>in</strong> c<strong>on</strong>densed<br />
matter physics. In some cases it may exhibit an <strong>in</strong>terest<strong>in</strong>g<br />
magnetic property; the sp<strong>in</strong> density wave (SDW)<br />
discussed by Overhauser is a typical example. In the previ-<br />
ψψ<br />
1<br />
0<br />
-1<br />
0<br />
Z<br />
-1<br />
0<br />
1<br />
ψiγ5τ3ψ<br />
Figure 3: Sketch <str<strong>on</strong>g>of</str<strong>on</strong>g> DCDW, where pseudoscalar density as<br />
well as scalar density oscillates al<strong>on</strong>g z directi<strong>on</strong>.<br />
ous paper [2] we have discussed the possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> a density<br />
wave, where pseudoscalar density as well as scalar density<br />
oscillate <strong>in</strong> harm<strong>on</strong>y al<strong>on</strong>g <strong>on</strong>e directi<strong>on</strong>, which is called<br />
dual chiral density wave (DCDW).<br />
⟨ ¯ ψψ⟩ = ∆ cos(θ(r)),<br />
⟨ ¯ ψiγ5τ3ψ⟩ = ∆ s<strong>in</strong>(θ(r)). (5)<br />
The chiral angle θ(r) is taken <strong>in</strong> the <strong>on</strong>e dimensi<strong>on</strong>al form,<br />
θ(r) = q · r. The amplitude ∆ generates the dynamical<br />
mass M, M = −2G∆, while θ produces the axial-vector<br />
field for quarks, τ3γ5γ · ∇θ/2 = τ3γ5γ · q/2. The s<strong>in</strong>gleparticle<br />
(positive) energy is then given by<br />
E ± p = [E 2 p + q 2 /4 ± q √ p 2 z + M 2 ] 1/2 , (6)<br />
with Ep = √ p 2 + M 2 , depend<strong>in</strong>g <strong>on</strong> the sp<strong>in</strong> degree <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
freedom. Accord<strong>in</strong>gly the Fermi sea is split <strong>in</strong>to two deformed<br />
<strong>on</strong>es: <strong>on</strong>e is deformed <strong>in</strong> the prolate shape and the<br />
other <strong>in</strong> the oblate shape.<br />
DCDW enjoys many <strong>in</strong>terest<strong>in</strong>g features. First, the symmetry<br />
break<strong>in</strong>g pattern is Tˆp × U Q 3 5 (1) → U ˆp+Q 3 5 , which<br />
may be 1+1 dimensi<strong>on</strong>al analog <str<strong>on</strong>g>of</str<strong>on</strong>g> Skyrmi<strong>on</strong>. Then the<br />
Nambu-Goldst<strong>on</strong>e bos<strong>on</strong> (”phas<strong>on</strong>”) has a hybrid nature <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
”pi<strong>on</strong>” and ”ph<strong>on</strong><strong>on</strong>”. Sec<strong>on</strong>dly, a direct evaluati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
magnetizati<strong>on</strong> gives ⟨σ12⟩ ∝ cos(q · r), which means a<br />
k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> SDW. In this case quark matter can be regarded as<br />
a k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> liquid crystal endowed with two-dimensi<strong>on</strong>al ferromagnetic<br />
order and <strong>on</strong>e dimensi<strong>on</strong>al anti-ferromagnetic<br />
order. Note that magnetic field is globally vanished <strong>in</strong> this<br />
phase, but locally very str<strong>on</strong>g.<br />
”Nest<strong>in</strong>g” mechanism<br />
Here we discuss the mechanism for the formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
DCDW. There seems to be some c<strong>on</strong>fusi<strong>on</strong>s about it. In<br />
the references [12] authors emphasized the nest<strong>in</strong>g effect<br />
(or Overhauser effect) for the essential mechanism <str<strong>on</strong>g>of</str<strong>on</strong>g> chiral<br />
4<br />
3.5<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
-q/2<br />
q/2+M<br />
q/2<br />
q/2-M<br />
q/2<br />
-0.5<br />
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2<br />
pz<br />
E+(pz)<br />
E-(pz)<br />
Figure 4: Energy spectra for p⊥ = 0 for q/2 > M. Solid<br />
(magenta) curves show the <strong>on</strong>e for massive quarks, while<br />
dashed (blue) curves for massless <strong>on</strong>es, |pz ± q/2|.<br />
density waves, but there is little discussi<strong>on</strong> about DCDW or<br />
other <strong>in</strong>homogeneous phases. For 1+1 dimensi<strong>on</strong>al case,<br />
we can immediately see that q is given as q = 2µ for given<br />
chemical potential µ [10]. This is simply because the energy<br />
spectrum (6) is reduced to E ± <br />
<br />
p → √ p2 z + M 2 <br />
<br />
± q/2<br />
and q is decoupled from pz. Recall that the outstand<strong>in</strong>g relati<strong>on</strong><br />
q = 2pF is held <strong>in</strong> the usual density wave like CDW<br />
or SDW <strong>in</strong> the <strong>on</strong>e dimensi<strong>on</strong>al system, due to the nest<strong>in</strong>g<br />
effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the Fermi surface. We can see that the similar<br />
mechanism works <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> DCDW, but <strong>in</strong> somewhat<br />
different manner from the usual <strong>on</strong>e. For the case,<br />
M > q/2, E ± p are <strong>on</strong>ly shifted ±q/2 from the free particle<br />
energy, so that formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> DCDW depends <strong>on</strong> the <strong>in</strong>teracti<strong>on</strong><br />
strength like <strong>in</strong> the St<strong>on</strong>er model. However, numerical<br />
calculati<strong>on</strong> shows this is not the case: q/2 > M is always<br />
held <strong>in</strong> the DCDW phase. In Fig. 4 we sketch the energy<br />
levels <str<strong>on</strong>g>of</str<strong>on</strong>g> the s<strong>in</strong>gle quark energy for the case, q/2 > M.<br />
Note that mass is generated by the <strong>in</strong>teracti<strong>on</strong> with DCDW<br />
<strong>in</strong> this case. So E ± p can be regarded as a c<strong>on</strong>sequence <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
switch<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>teracti<strong>on</strong> with DCDW between massless<br />
quarks with relative momentum q. For massless quarks, the<br />
two levels cross each other at pz = 0 for any q. Once the<br />
<strong>in</strong>teracti<strong>on</strong> with DCDW is present, mass is generated and<br />
two levels avoid the cross<strong>in</strong>g with the energy gap, 2M, at<br />
pz = 0 (magenta curves <strong>in</strong> Fig. 4). So if we choose q = 2µ<br />
and fill the levels up to pF = µ, there is always the energy<br />
ga<strong>in</strong> due to the <strong>in</strong>teracti<strong>on</strong> with DCDW. In the three dimensi<strong>on</strong>al<br />
case, the simple relati<strong>on</strong> is no more held, but we can<br />
expect some rem<strong>in</strong>iscence. Actually we can numerically<br />
check that the similar relati<strong>on</strong> is held <strong>in</strong> the three dimensi<strong>on</strong>al<br />
case. In the vic<strong>in</strong>ity <str<strong>on</strong>g>of</str<strong>on</strong>g> the critical end po<strong>in</strong>t we have<br />
seen that the chiral correlati<strong>on</strong> functi<strong>on</strong> χ(q) diverges at f<strong>in</strong>ite<br />
q <str<strong>on</strong>g>of</str<strong>on</strong>g> q ∼ 2pF , but the effective mass is almost vanished<br />
<strong>in</strong> this situati<strong>on</strong>.<br />
Thus we can understand DCDW with q/2 > M <strong>in</strong><br />
terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the ”nest<strong>in</strong>g” effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the FErmi surface, while<br />
q smoothly <strong>in</strong>creases from zero for RKC. Their situati<strong>on</strong> is<br />
very different from our case: the opposite relati<strong>on</strong>, |q/2
M|, is realized <strong>in</strong> their case, and the level diagram looks<br />
very different from Fig. (4).<br />
Deformed DCDW<br />
Here we generalize the orig<strong>in</strong>al DCDW by tak<strong>in</strong>g <strong>in</strong>to<br />
account the symmetry break<strong>in</strong>g effect with the current<br />
quark mass mc ∝ m 2 π. Us<strong>in</strong>g a variati<strong>on</strong>al method, we can<br />
show that the functi<strong>on</strong>al form <str<strong>on</strong>g>of</str<strong>on</strong>g> the chiral angle <strong>in</strong> DCDW<br />
is deformed, satisfy<strong>in</strong>g the s<strong>in</strong>e-Gord<strong>on</strong> (SG) equati<strong>on</strong>, and<br />
thereby the allowed regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> DCDW is extended. The stable<br />
soluti<strong>on</strong> is then given <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the Jacobian elliptic<br />
functi<strong>on</strong> with modulus k,<br />
θ = π + 2am (m ∗ πz/k, k) , (7)<br />
with the effective pi<strong>on</strong> mass <strong>in</strong> medium, m ∗ π. To recover<br />
the orig<strong>in</strong>al DCDW <strong>in</strong> the chiral limit, we must require the<br />
follow<strong>in</strong>g relati<strong>on</strong>,<br />
qk = m ∗ ππ/K (8)<br />
with the complete elliptic <strong>in</strong>tegral <str<strong>on</strong>g>of</str<strong>on</strong>g> the first k<strong>in</strong>d K. Note<br />
q/m* π<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0 0.2 0.4 0.6 0.8 1<br />
k<br />
Figure 5: Relati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the modulus k and the parameter q.<br />
k → 0 <strong>in</strong> the chiral limit.<br />
that we can recover the SG equati<strong>on</strong> <strong>in</strong> ref. [13] and m ∗ π →<br />
mπ <strong>in</strong> the 1+1 dimensi<strong>on</strong>al case.<br />
CONCLUDING REMARKS<br />
We have discussed two k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic properties <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
quark matter separately, but a unified or comprehensive<br />
descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetism should be desired. Furthermore<br />
when we are <strong>in</strong>terested <strong>in</strong> magnetism at moderate densities,<br />
we must take <strong>in</strong>to account some n<strong>on</strong>-perturbative effects<br />
explicitly. It may be <strong>on</strong>e way to use the effective models <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
QCD, e.g. NJL model, to this end.<br />
Ferromagnetic order may have a direct implicati<strong>on</strong> <strong>on</strong><br />
magnetic evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> compact stars, but SDW order should<br />
also have some implicati<strong>on</strong>s. DCDW may catalyse the elementary<br />
processes by provid<strong>in</strong>g extra momentum; for example<br />
it allows the quark β-decay process as neutr<strong>in</strong>o emissi<strong>on</strong><br />
dur<strong>in</strong>g the thermal evoluti<strong>on</strong>. The magnetizati<strong>on</strong> is<br />
globally vanished there, but its fluctuati<strong>on</strong>, ⟨M 2 ⟩, becomes<br />
large. Accord<strong>in</strong>gly the local str<strong>on</strong>g magnetic field may <strong>in</strong>duce<br />
new QED processes.<br />
For the present it needs more studies about the properties<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the n<strong>on</strong>-uniform states and relati<strong>on</strong>s am<strong>on</strong>g them.<br />
In particular, the comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> DCDW and the real k<strong>in</strong>k<br />
crystal <strong>in</strong> [11] is important, s<strong>in</strong>ce they are typical structures<br />
<strong>in</strong> QCD, reflect<strong>in</strong>g the different symmetries, U(1) vs Z2.<br />
More studies are needed <strong>in</strong>clud<strong>in</strong>g the symmetry break<strong>in</strong>g<br />
effect, thermal effect or model dependence.<br />
The exist<strong>in</strong>g regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetism may be overlapped<br />
with color superc<strong>on</strong>ductivity.It is then <strong>in</strong>terest<strong>in</strong>g to elucidate<br />
the mutual relati<strong>on</strong> <strong>in</strong> quark matter. Some works<br />
have been already d<strong>on</strong>e [14], but more studies are needed,<br />
<strong>in</strong>clud<strong>in</strong>g unc<strong>on</strong>venti<strong>on</strong>al mechanism <str<strong>on</strong>g>of</str<strong>on</strong>g> pair<strong>in</strong>g.<br />
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Recent progress and prospects <strong>on</strong> laser-plasma<br />
accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles*<br />
Kazuhisa Nakajima #†<br />
High Energy Accelerator Research Organizati<strong>on</strong> (<strong>KEK</strong>) 1-1 Oho, Tsukuba 305-0081, Japan<br />
Abstract<br />
Recent progress <strong>in</strong> laser-driven plasma-based<br />
accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles such as electr<strong>on</strong>s and<br />
i<strong>on</strong>s is overviewed <strong>in</strong> theoretical and experimental aspects.<br />
In particular laser-plasma accelerati<strong>on</strong> physics such as<br />
laser wakefield accelerati<strong>on</strong> (LWFA) <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and i<strong>on</strong><br />
accelerati<strong>on</strong> mechanism is highlighted, show<strong>in</strong>g recent<br />
achievements <str<strong>on</strong>g>of</str<strong>on</strong>g> laser plasma accelerator technologies<br />
that produce high-energy, high-quality beams required for<br />
compact particle beam and radiati<strong>on</strong> sources.<br />
INTRODUCTION<br />
In this decade, worldwide experimental and theoretical<br />
researches <strong>on</strong> laser-driven plasma-based accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
charged particles have achieved great progress <strong>in</strong> highenergy,<br />
high-quality electr<strong>on</strong> beams <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> GeVclass<br />
energy and a 1%-level energy spread [1-6], whereas<br />
laser-driven producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong> beams such as prot<strong>on</strong>s and<br />
carb<strong>on</strong>s is underdeveloped, harness<strong>in</strong>g Petawatt-class<br />
ultra-<strong>in</strong>tense lasers and ultra-th<strong>in</strong> foil targets. These highenergy<br />
high-quality particle beams make it possible to<br />
open the door for a wide range <str<strong>on</strong>g>of</str<strong>on</strong>g> applicati<strong>on</strong>s <strong>in</strong> research,<br />
medical and <strong>in</strong>dustrial uses.<br />
Here recent progress <strong>in</strong> laser-plasma accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
charged particles <strong>in</strong>clud<strong>in</strong>g electr<strong>on</strong>- and i<strong>on</strong>-accelerati<strong>on</strong><br />
is overviewed from the aspects <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>jecti<strong>on</strong> or particle<br />
generati<strong>on</strong>, accelerati<strong>on</strong> process and resultant beam<br />
properties, which are strictly determ<strong>in</strong>ed by accelerati<strong>on</strong><br />
mechanism or laser-plasma <strong>in</strong>teracti<strong>on</strong> such as the bubble<br />
mechanism for electr<strong>on</strong>s and radiati<strong>on</strong> pressure<br />
accelerati<strong>on</strong> for i<strong>on</strong>s.<br />
Although there is no practical applicati<strong>on</strong> to date,<br />
underdeveloped are various applicati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> laser plasma<br />
accelerators such as a compact THz or coherent X-ray<br />
radiati<strong>on</strong> source and radiati<strong>on</strong> therapy driven by laseraccelerated<br />
electr<strong>on</strong>s [7]. On the other hand, a promis<strong>in</strong>g<br />
applicati<strong>on</strong> project <str<strong>on</strong>g>of</str<strong>on</strong>g> laser-driven prot<strong>on</strong> and i<strong>on</strong> beams<br />
to the future hadr<strong>on</strong> therapy is implemented worldwide.<br />
In the future laser-plasma accelerators may come <strong>in</strong>to<br />
be<strong>in</strong>g as a novel versatile tool for develop<strong>in</strong>g fields such<br />
as space science where a compact and cost-effective tool<br />
is required as well as <strong>in</strong>herent applicati<strong>on</strong> to energyfr<strong>on</strong>tier<br />
particle accelerators.<br />
___________________________________________<br />
* Work supported by Ch<strong>in</strong>ese Academy <str<strong>on</strong>g>of</str<strong>on</strong>g> Sciences Visit<strong>in</strong>g<br />
Pr<str<strong>on</strong>g>of</str<strong>on</strong>g>essorship for Senior <str<strong>on</strong>g>Internati<strong>on</strong>al</str<strong>on</strong>g> Scientists.<br />
# Visit<strong>in</strong>g affiliati<strong>on</strong>s : Shanghai Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Optics and F<strong>in</strong>e<br />
Mechanics, Ch<strong>in</strong>ese Academy <str<strong>on</strong>g>of</str<strong>on</strong>g> Sciences, Shanghai 201800, P.R.<br />
Ch<strong>in</strong>a; Shanghai Jiao T<strong>on</strong>g University, Shanghai 200240, P.R. Ch<strong>in</strong>a.<br />
† nakajima@post.kek.jp<br />
LASER WAKEFIELD ACCELERATION<br />
OF ELECTRONS<br />
LWFA <strong>in</strong> the l<strong>in</strong>ear regime<br />
In underdense plasma an ultra<strong>in</strong>tense laser pulse excites<br />
a large-amplitude plasma wave with frequency p =<br />
(4e 2 ne/me) 1/2 and electric field <strong>on</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ ne 1/2<br />
V/cm for the electr<strong>on</strong> rest mass mec 2 and plasma density<br />
ne cm -3 due to the p<strong>on</strong>deromotive force expell<strong>in</strong>g plasma<br />
electr<strong>on</strong>s out <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse and the space charge force<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> immovable plasma i<strong>on</strong>s restor<strong>in</strong>g expelled electr<strong>on</strong>s <strong>on</strong><br />
the back <str<strong>on</strong>g>of</str<strong>on</strong>g> the i<strong>on</strong> column rema<strong>in</strong><strong>in</strong>g beh<strong>in</strong>d the laser<br />
pulse. S<strong>in</strong>ce the phase velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma wave is<br />
approximately equal to the group velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser<br />
pulse vg/c = (1- p 2 /0 2 ) 1/2 ~1 for the laser frequency 0<br />
and the accelerat<strong>in</strong>g field <str<strong>on</strong>g>of</str<strong>on</strong>g> ~ 1 GeV/cm for the plasma<br />
density ~ 10 18 cm -3 , electr<strong>on</strong>s trapped <strong>in</strong>to the plasma<br />
wave are likely to be accelerated up to ~ 1 GeV energy <strong>in</strong><br />
a 1 cm plasma. More accurately <strong>in</strong> the l<strong>in</strong>ear regime for<br />
the normalized vector potential def<strong>in</strong>ed by<br />
2 1<br />
2 18 2<br />
2<br />
a 0 0. 85 I<br />
/10 Wcm m<br />
,<br />
where I is the laser <strong>in</strong>tensity and = 2c/0 the laser<br />
wavelength, the energy ga<strong>in</strong> is given by [8]<br />
2 2<br />
E 1.<br />
3mec<br />
a0<br />
nc<br />
ne<br />
,<br />
2<br />
18 -3<br />
1<br />
P TWr<br />
m<br />
n 10 cm GeV<br />
35<br />
0<br />
e<br />
for the peak laser power P TW focused <strong>on</strong>to the spot<br />
radius r0 m, assum<strong>in</strong>g that the plasma wave is efficiently<br />
excited at p ~ cL for the pulse durati<strong>on</strong> L, and that<br />
electr<strong>on</strong>s are accelerated over the dephas<strong>in</strong>g length given<br />
by Ldp ~ p(p 2 /0 2 ) = p(nc/ne), where nc = /(re 2 ) ⋍<br />
1.115 × 10 21 cm -3 (/m) -2 is the cut<str<strong>on</strong>g>of</str<strong>on</strong>g>f density, re =<br />
e 2 /mec 2 the classical electr<strong>on</strong> radius. The accelerated<br />
electr<strong>on</strong>s overrun the accelerat<strong>in</strong>g field toward the<br />
decelerat<strong>in</strong>g field bey<strong>on</strong>d the dephas<strong>in</strong>g length.<br />
Quasi-m<strong>on</strong>oenergetic accelerati<strong>on</strong> <strong>in</strong> the<br />
n<strong>on</strong>l<strong>in</strong>ear regime<br />
The lead<strong>in</strong>g experiments [9] that successfully<br />
dem<strong>on</strong>strated the producti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> quasi-m<strong>on</strong>oenergetic<br />
electr<strong>on</strong> beams with narrow energy spread have been<br />
elucidated <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> self-<strong>in</strong>jecti<strong>on</strong> and accelerati<strong>on</strong><br />
mechanism <strong>in</strong> the bubble regime [10,11]. In these<br />
experiments, electr<strong>on</strong>s are self-<strong>in</strong>jected <strong>in</strong>to a n<strong>on</strong>l<strong>in</strong>ear<br />
wake, referred to as a “bubble”, i.e. a cavity <str<strong>on</strong>g>of</str<strong>on</strong>g> plasma<br />
electr<strong>on</strong>s c<strong>on</strong>sist<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> a spherical i<strong>on</strong> column surrounded<br />
with a narrow electr<strong>on</strong> sheath, formed beh<strong>in</strong>d the laser<br />
pulse <strong>in</strong>stead <str<strong>on</strong>g>of</str<strong>on</strong>g> a periodic plasma wave <strong>in</strong> the l<strong>in</strong>ear
egime. As analogous to a c<strong>on</strong>venti<strong>on</strong>al RF cavity <strong>in</strong>side<br />
which electromagnetic energy is res<strong>on</strong>antly c<strong>on</strong>f<strong>in</strong>ed at<br />
the matched frequency to accelerate externally <strong>in</strong>jected<br />
particles, <strong>in</strong>duc<strong>in</strong>g a current flow <strong>in</strong> a sk<strong>in</strong> depth <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
metal surface, plasma electr<strong>on</strong>s radially expelled by the<br />
radiati<strong>on</strong> pressure <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser form a sheath with<br />
thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma sk<strong>in</strong> depth 1/kp =<br />
c/p outside the i<strong>on</strong> sphere rema<strong>in</strong><strong>in</strong>g “unshielded”<br />
beh<strong>in</strong>d the laser pulse mov<strong>in</strong>g at relativistic velocity so<br />
that the cavity shape should be determ<strong>in</strong>ed by balanc<strong>in</strong>g<br />
the Lorentz force <str<strong>on</strong>g>of</str<strong>on</strong>g> the i<strong>on</strong> sphere exerted <strong>on</strong> the electr<strong>on</strong><br />
sheath with the p<strong>on</strong>deromotive force <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse.<br />
This estimates the bubble radius RB matched to the laser<br />
spot radius w0 , approximately as ,<br />
for which a best spherical shape <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble is created.<br />
This c<strong>on</strong>diti<strong>on</strong> is reformulated as<br />
where Pc = 17(0/p) 2 GW is the critical power for the<br />
relativistic self-focus<strong>in</strong>g[11].<br />
The l<strong>on</strong>gitud<strong>in</strong>al electric field <strong>in</strong>side the bubble is<br />
obta<strong>in</strong>ed as<br />
, where =z-vBt is<br />
the coord<strong>in</strong>ate <strong>in</strong> the frame <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble mov<strong>in</strong>g at the<br />
velocity vB [10]. One can see that the maximum<br />
accelerat<strong>in</strong>g field is given by e|Ez|max = (1/2)mec 2 kp 2 RB at<br />
the back <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble and the focus<strong>in</strong>g force is act<strong>in</strong>g <strong>on</strong><br />
an electr<strong>on</strong> <strong>in</strong>side the bubble. Assum<strong>in</strong>g the bubble phase<br />
velocity is given by vB ~ vg-vetch~c[1-(1/2+1)(p/0) 2 ],<br />
where vetch ~c(p/0) 2 is the velocity at which the laser<br />
fr<strong>on</strong>t etches back due to the local pump depleti<strong>on</strong>, the<br />
dephas<strong>in</strong>g length leads to<br />
Ldp ~ c/(c-vB)RB ~ (2/3) (0/p) 2 RB = (2/3) (nc/ne)RB.<br />
Hence the electr<strong>on</strong> <strong>in</strong>jected at the back <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble can<br />
be accelerated up to the energy<br />
1<br />
2 2 nc<br />
E e Ez<br />
Ldp<br />
mec<br />
a<br />
max<br />
0 .<br />
2<br />
3 ne<br />
Us<strong>in</strong>g the matched bubble radius c<strong>on</strong>diti<strong>on</strong>, the energy<br />
ga<strong>in</strong> is approximately given by<br />
2<br />
P nc<br />
<br />
E mec<br />
,<br />
Pr<br />
ne<br />
<br />
where Pr = me 2 c 5 /e 2 = 8.72 GW [12].<br />
The 2D or 3D particle-<strong>in</strong>-cell simulati<strong>on</strong>s c<strong>on</strong>firm that<br />
quasi-m<strong>on</strong>oenergetic electr<strong>on</strong> beams are produced due to<br />
self-<strong>in</strong>jecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> plasma electr<strong>on</strong>s at the back <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
bubble from the electr<strong>on</strong> sheath outside the i<strong>on</strong> sphere as<br />
the laser <strong>in</strong>tensity <strong>in</strong>creases to the <strong>in</strong>jecti<strong>on</strong> threshold. As<br />
expelled electr<strong>on</strong>s flow<strong>in</strong>g the sheath are <strong>in</strong>itially<br />
decelerated backward <strong>in</strong> a fr<strong>on</strong>t half <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble and<br />
then accelerated <strong>in</strong> a back half <str<strong>on</strong>g>of</str<strong>on</strong>g> it toward the<br />
propagati<strong>on</strong> axis by the accelerat<strong>in</strong>g and focus<strong>in</strong>g forces<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble i<strong>on</strong>s, their trajectories c<strong>on</strong>centrate at the<br />
back <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble to form a str<strong>on</strong>g local density peak <strong>in</strong><br />
the electr<strong>on</strong> sheath and a spiky accelerat<strong>in</strong>g field.<br />
Eventually the electr<strong>on</strong> is trapped <strong>in</strong>to the bubble when its<br />
velocity reaches the group velocity vg <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser pulse.<br />
Theoretical analysis <strong>on</strong> the trapp<strong>in</strong>g threshold gives kpRB<br />
≥ (2nc/ne) 1/2 [13]. This trapp<strong>in</strong>g c<strong>on</strong>diti<strong>on</strong> leads to<br />
, while the trapp<strong>in</strong>g cross secti<strong>on</strong> ≃<br />
(2/kp 3 d)(ln kpRB/8 1/2 ) -1 [10] with the sheath width d<br />
1 3<br />
2 3<br />
,<br />
imposes kpRB ≥ 2.8, i.e. for the matched bubble<br />
radius. Once an electr<strong>on</strong> bunch is trapped <strong>in</strong> the bubble,<br />
load<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> trapped electr<strong>on</strong>s reduces the wakefield<br />
amplitude below the trapp<strong>in</strong>g threshold and stops further<br />
<strong>in</strong>jecti<strong>on</strong>. C<strong>on</strong>sequently the trapped electr<strong>on</strong>s undergo<br />
accelerati<strong>on</strong> and bunch<strong>in</strong>g process with<strong>in</strong> a separatrix <strong>on</strong><br />
the phase space <str<strong>on</strong>g>of</str<strong>on</strong>g> the bubble wakefield. This is a simple<br />
scenario for produc<strong>in</strong>g high-quality m<strong>on</strong>oenergetic<br />
electr<strong>on</strong> beams <strong>in</strong> the bubble regime. However, <strong>in</strong> most <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
laser-plasma experiments aforementi<strong>on</strong>ed c<strong>on</strong>diti<strong>on</strong>s and<br />
scenarios are not always fulfilled.<br />
In the experiment for the plasma density ne =(1 ~ 2)×<br />
10 19 cm -3 , observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the self-<strong>in</strong>jecti<strong>on</strong> threshold <strong>on</strong><br />
the normalized laser <strong>in</strong>tensity gives after<br />
account<strong>in</strong>g for self-focus<strong>in</strong>g and self-compressi<strong>on</strong> that<br />
occur dur<strong>in</strong>g laser pulse propagati<strong>on</strong> <strong>in</strong> the plasma. In<br />
terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser peak power<br />
)2, the self-<strong>in</strong>jecti<strong>on</strong> threshold for the power (P/Pc )th<br />
≈ 12.6 as the laser spot size reduces to the plasma<br />
wavelength due to the relativistic self-focus<strong>in</strong>g[14]. In the<br />
experiment at ne =(3 ~ 5)×10 18 cm -3 , the self-<strong>in</strong>jecti<strong>on</strong><br />
threshold is (P/Pc )th = 3, corresp<strong>on</strong>d<strong>in</strong>g to [6].<br />
Our 2-D PIC simulati<strong>on</strong>s <strong>on</strong> the self-<strong>in</strong>jecti<strong>on</strong> threshold<br />
show that for the uniform density plasma such as a gas jet<br />
or a gas cell <str<strong>on</strong>g>of</str<strong>on</strong>g> ne =(1.7 ~ 5)×10 18 cm -3 , the self-<strong>in</strong>jecti<strong>on</strong><br />
occurs at and for the preformed plasma channel<br />
such as discharge capillary <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma density ne = 2×<br />
10 18 cm -3 with the density depth nch/ne = 0.3, the<br />
threshold is .<br />
C<strong>on</strong>trolled laser wakefield accelerator<br />
For many applicati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> laser wakefield accelerators,<br />
stability and c<strong>on</strong>trollability <str<strong>on</strong>g>of</str<strong>on</strong>g> the beam performance such<br />
as energy, energy spread, emittance and charge are<br />
<strong>in</strong>dispensable as well as compact and robust features <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the system. In c<strong>on</strong>trast to the c<strong>on</strong>venti<strong>on</strong>al accelerators<br />
composed <str<strong>on</strong>g>of</str<strong>on</strong>g> various complex-functi<strong>on</strong>ed systems, the<br />
performance <str<strong>on</strong>g>of</str<strong>on</strong>g> laser plasma accelerators is str<strong>on</strong>gly<br />
correlated to the <strong>in</strong>jecti<strong>on</strong> mechanism <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> beams<br />
as well as the laser performance. To date, the external<br />
<strong>in</strong>jecti<strong>on</strong> <strong>in</strong>to laser wakefields from the c<strong>on</strong>venti<strong>on</strong>al RF<br />
<strong>in</strong>jector [15] or the stag<strong>in</strong>g c<strong>on</strong>cept, which is c<strong>on</strong>ceivable<br />
<strong>on</strong> the analogy <str<strong>on</strong>g>of</str<strong>on</strong>g> the high-energy RF accelerators, has not<br />
been always successful for generat<strong>in</strong>g <strong>in</strong>tense highquality<br />
electr<strong>on</strong> beams that could be useful for<br />
applicati<strong>on</strong>s. Hence, besides the self-<strong>in</strong>jecti<strong>on</strong>, the optical<br />
<strong>in</strong>jecti<strong>on</strong> scheme with two collid<strong>in</strong>g pulses is highlighted.<br />
The optical <strong>in</strong>jecti<strong>on</strong> scheme for manipulat<strong>in</strong>g electr<strong>on</strong><br />
beams <strong>in</strong> a phase space <str<strong>on</strong>g>of</str<strong>on</strong>g> laser wakefield accelerati<strong>on</strong><br />
with fs-synchr<strong>on</strong>izati<strong>on</strong> and MeV-energy resp<strong>on</strong>se utilize<br />
an <strong>in</strong>jecti<strong>on</strong> pulse split from the same drive pulse with fs<br />
durati<strong>on</strong>, cross<strong>in</strong>g the drive pulse at some angle <strong>in</strong> the<br />
plasma. When cross<strong>in</strong>g each other, the phase space <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
wakefields excited by the drive pulse overlaps with the<br />
phase space <str<strong>on</strong>g>of</str<strong>on</strong>g> beat waves generated by mix<strong>in</strong>g the drive<br />
pulse and the <strong>in</strong>jecti<strong>on</strong> pulse. As a result <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
p<strong>on</strong>deromotive force <str<strong>on</strong>g>of</str<strong>on</strong>g> the beat wave boosts plasma<br />
electr<strong>on</strong>s and locally <strong>in</strong>jects them <strong>in</strong>to the separatrix <str<strong>on</strong>g>of</str<strong>on</strong>g>
the wakefields. In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> head-<strong>on</strong> collisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> two<br />
counter-propagat<strong>in</strong>g laser pulses at the angle <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
p<strong>on</strong>deromotive force<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the <strong>in</strong>jecti<strong>on</strong> beat wave oscillat<strong>in</strong>g with the wavelength<br />
0/2 locally accelerates the plasma electr<strong>on</strong>s to be <strong>in</strong>jected<br />
<strong>in</strong>to the wakefield bucket, where k0 = 2/0 is the laser<br />
wave number, ɑ0, and ɑ1 the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> the drive pulse<br />
and the <strong>in</strong>jecti<strong>on</strong> pulse, respectively, and the Lorentz<br />
factor <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma electr<strong>on</strong>, i.e. 1 for the cold<br />
plasma. On the c<strong>on</strong>trary to the self-<strong>in</strong>jecti<strong>on</strong> with a s<strong>in</strong>gle<br />
drive pulse, this force is <strong>in</strong>dependently c<strong>on</strong>trollable by<br />
chang<strong>in</strong>g the <strong>in</strong>jecti<strong>on</strong> pulse <strong>in</strong>tensity and/or its<br />
polarizati<strong>on</strong> with respect to that <str<strong>on</strong>g>of</str<strong>on</strong>g> the drive pulse as well<br />
as the <strong>in</strong>jecti<strong>on</strong> positi<strong>on</strong>, where two pulses collides.<br />
Therefore the energy and the charge can be c<strong>on</strong>trolled<br />
with<strong>in</strong> some extent, associat<strong>in</strong>g with evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
energy spread due to the beam load<strong>in</strong>g and the <strong>in</strong>jecti<strong>on</strong><br />
volume <str<strong>on</strong>g>of</str<strong>on</strong>g> the phase space.<br />
These effects are successfully dem<strong>on</strong>strated with good<br />
stability by the experiments carried out below the self<strong>in</strong>jecti<strong>on</strong><br />
threshold <str<strong>on</strong>g>of</str<strong>on</strong>g> the drive pulse <strong>in</strong>tensity. The<br />
experiment <str<strong>on</strong>g>of</str<strong>on</strong>g> ref. [16] with , and the<br />
pulse durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 30 fs for both pulses shows an almost<br />
l<strong>in</strong>ear c<strong>on</strong>trol <str<strong>on</strong>g>of</str<strong>on</strong>g> the m<strong>on</strong>oenergetic beam energy from 50<br />
MeV to 250 MeV by chang<strong>in</strong>g the collid<strong>in</strong>g positi<strong>on</strong> over<br />
the 2-mm gas jet at the plasma density <str<strong>on</strong>g>of</str<strong>on</strong>g> ne = 7.5×10 18<br />
cm -3 , c<strong>on</strong>sequently chang<strong>in</strong>g the accelerati<strong>on</strong> length <strong>in</strong><br />
the average accelerat<strong>in</strong>g field <str<strong>on</strong>g>of</str<strong>on</strong>g> Ez = 270 GV/m, which is<br />
close to an estimate <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave break<strong>in</strong>g field for the<br />
cold plasma, Ewb ≈ 0.96ne 1/2 ~ 263 GV/m. The experiment<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> ref. [17] dem<strong>on</strong>strates the collid<strong>in</strong>g optical <strong>in</strong>jecti<strong>on</strong> at<br />
the cross<strong>in</strong>g angle <str<strong>on</strong>g>of</str<strong>on</strong>g> 135 °<strong>in</strong> the 1-mm gas jet with<br />
(Pdrive = 3 TW), (P<strong>in</strong>j = 0.14 TW) and<br />
70 fs pulse durati<strong>on</strong>, result<strong>in</strong>g <strong>in</strong> the energy E =15 MeV<br />
and the energy spread E/E =7.8% at ne = 3.95×10 19 cm -<br />
3 free from the self-<strong>in</strong>jecti<strong>on</strong> as well as the head-<strong>on</strong><br />
collid<strong>in</strong>g <strong>in</strong>jecti<strong>on</strong> at 180° with (Pdrive = 10<br />
TW), (P<strong>in</strong>j = 0.6 TW) and 40 fs pulse durati<strong>on</strong>,<br />
result<strong>in</strong>g <strong>in</strong> the energy E =134 MeV and the energy<br />
spread E/E =3.5% at ne = 1 × 10 19 cm -3 . These<br />
experiments suggest a very compact system <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
electr<strong>on</strong> beam source <strong>in</strong>clud<strong>in</strong>g the laser and the<br />
accelerator <strong>on</strong> a table-top size with high quality and high<br />
stability.<br />
The trapped electr<strong>on</strong>s <strong>in</strong>side the bubble generate<br />
electromagnetic fields and modify the bubble wakefields.<br />
As a result, the trail<strong>in</strong>g electr<strong>on</strong> bunch undergoes less<br />
accelerated field that limits the charge and produces<br />
energy spread. The analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> the beam load<strong>in</strong>g <strong>in</strong> the<br />
bubble regime gives the energy absorbed per unit length<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the beam is given as<br />
Q<br />
eE<br />
4 k<br />
16 3<br />
s s 10 cm<br />
0.<br />
047<br />
pRB<br />
1nC<br />
mec<br />
p<br />
ne<br />
where Qs is the total charge and Es the accelerat<strong>in</strong>g<br />
wakefield at the phase positi<strong>on</strong> where the bunch charge<br />
starts, assum<strong>in</strong>g the density distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the bunch<br />
charge has a trapezoidal shape so that the energy spread<br />
<strong>in</strong>side the bunch is m<strong>in</strong>imized [18]. This equati<strong>on</strong> implies<br />
the trade-<str<strong>on</strong>g>of</str<strong>on</strong>g>f between the total charge that can be<br />
accelerated and the accelerat<strong>in</strong>g gradient, i.e. the<br />
accelerated energy. With , the charge is<br />
proporti<strong>on</strong>al to ne -1/2 .<br />
Toward high-energy accelerati<strong>on</strong> bey<strong>on</strong>d GeV<br />
S<strong>in</strong>ce the energy ga<strong>in</strong> scales as , it would be<br />
necessary for the multi-GeV accelerati<strong>on</strong> to decrease the<br />
operat<strong>in</strong>g plasma density from the 10 18 cm -3 range to the<br />
10 17 cm -3 range and <strong>in</strong>crease the accelerator length, i.e.<br />
plasma channel length, up to several tens cm. Recent<br />
experiments dem<strong>on</strong>strated GeV-class quasim<strong>on</strong>oenergetic<br />
electr<strong>on</strong> beams with a cm-scale gas jet or a capillary<br />
plasma waveguide, rely<strong>in</strong>g <strong>on</strong> self-<strong>in</strong>jecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> plasma<br />
electr<strong>on</strong>s:<br />
E=1 GeV, E/E=2.5%(rms), =1.6mrad, Q=35pC<br />
for P= 40TW, L = 37 fs, a0 =1.4, ne = 4.3×10 18 cm -3<br />
with 3.3-cm gas-fill capillary [1].<br />
E = 0.5 GeV, E/E = 2.5 % (FWHM), = 0.3mrad,<br />
Q > 0.3 pC for P=18TW, L = 42 fs, a0 = 0.84, ne =<br />
8.4×10 18 cm -3 with 15-mm gas-fill capillary [2].<br />
E = 0.59 GeV, E/E = 1.2 % (FWHM), = 0.59<br />
mrad, Q > 10 fC for P=24TW, L = 27 fs, a0 = 1.7, ne<br />
= 1.9×10 18 cm -3 with 4-cm ablative capillary [3].<br />
E = 0.52 GeV, E/E = 5 % (FWHM), = 5.4 mrad,<br />
Q=70 pC for P=32TW, L=80fs, a0=0.8, ne= 1.8×10 18<br />
cm -3 with 3-cm gas-fill capillary [4].<br />
E = 0.8 GeV, E/E = 12 % (FWHM), = 3.6 mrad,<br />
Q=90pC <strong>in</strong> average for P=180TW, L =55fs, a0=3.9,<br />
ne= 5.7×10 18 cm -3 with 8-mm gas jet [5].<br />
E=0.72 GeV, E/E = 14 % (FWHM), = 2.9 mrad,<br />
Q~100pC for P=65TW, L =60fs, a0=2.8, ne= 3×10 18<br />
cm -3 with 8-mm gas jet [6].<br />
One f<strong>in</strong>ds that high-energy and high-quality electr<strong>on</strong><br />
beams with less charge are accelerated with cm-scale<br />
capillary plasma waveguides <strong>in</strong> the quasi l<strong>in</strong>ear regime (a0<br />
< 2), while high-energy and high-charge electr<strong>on</strong> beams<br />
with less quality are produced with gas jets rely<strong>in</strong>g <strong>on</strong><br />
self-guid<strong>in</strong>g <strong>in</strong> the bubble regime (a0 > 2). Recent<br />
progress <str<strong>on</strong>g>of</str<strong>on</strong>g> PW-class lasers boosts the accelerati<strong>on</strong> energy<br />
bey<strong>on</strong>d 1 GeV. Two experiments are attempted with a cmscale<br />
gas cell and a capillary, respectively, show<strong>in</strong>g n<strong>on</strong>m<strong>on</strong>oenergetic<br />
spectra with a clear cut<str<strong>on</strong>g>of</str<strong>on</strong>g>f energy Emax:<br />
Emax=1.45 GeV for P= 110TW, L = 60 fs, a0 =3.8, ne<br />
= 1.3×10 18 cm -3 with 1.3-cm gas cell c<strong>on</strong>ta<strong>in</strong><strong>in</strong>g 97%<br />
He and 3% CO2 mixed gas [19].<br />
Emax=1.8 GeV for P= 72TW, L = 50 fs, a0 =2.9, ne =<br />
2.1×10 18 cm -3 with 4-cm ablative capillary made <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
polycarb<strong>on</strong>ate [20].<br />
Although both cases corresp<strong>on</strong>d to the bubble regime for<br />
a0 high enough to cause self-<strong>in</strong>jecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s, the<br />
i<strong>on</strong>izati<strong>on</strong>-<strong>in</strong>duced trapp<strong>in</strong>g mechanism [21] due to<br />
oxygen impurities enhances the electr<strong>on</strong> <strong>in</strong>jecti<strong>on</strong> <strong>in</strong>to the<br />
bubble c<strong>on</strong>t<strong>in</strong>uously over the plasma length. For the<br />
multi-GeV LWFA operat<strong>in</strong>g <strong>in</strong> the plasma density lower<br />
than 10 18 cm -3 , a dedicated <strong>in</strong>jector would be essential to<br />
produce high-quality electr<strong>on</strong> beams with high-stability.
With extremely high-peak power lasers bey<strong>on</strong>d PW,<br />
based <strong>on</strong> a naïve scal<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> atta<strong>in</strong>able energies <strong>in</strong> a s<strong>in</strong>gle<br />
stage, it is possible to design ultrahigh-energy laserplasma<br />
accelerators bey<strong>on</strong>d 10 GeV <strong>in</strong> a much smaller<br />
size than that <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>venti<strong>on</strong>al accelerators. However,<br />
various requirements for applicati<strong>on</strong>s such as the beam<br />
qualities and the electrical power would strictly limit<br />
acceptable parameters <strong>on</strong> the accelerator performance.<br />
Recent 3D-PIC simulati<strong>on</strong>s [22] show that 10-PW and 3-<br />
PW lasers cannot produce high-energy ( 10 23 W/cm 2 with a l<strong>in</strong>early<br />
polarized laser pulse <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 kJ energy that could accelerate<br />
GeV quasi-m<strong>on</strong>oenergetic i<strong>on</strong>s with high c<strong>on</strong>versi<strong>on</strong><br />
efficiency (> 40%) [27], though such experiments are far<br />
bey<strong>on</strong>d current technology. Circularly polarized laser<br />
pulses <str<strong>on</strong>g>of</str<strong>on</strong>g> which the p<strong>on</strong>deromotive force has no<br />
oscillat<strong>in</strong>g comp<strong>on</strong>ent can push forward electr<strong>on</strong>s steadily,<br />
suppress<strong>in</strong>g foil heat<strong>in</strong>g and expansi<strong>on</strong>. The absence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
hot electr<strong>on</strong>s suppresses TNSA and allows RPA even at<br />
currently achievable laser <strong>in</strong>tensities (I < 10 22 W/cm 2 ).<br />
The <strong>on</strong>e-dimensi<strong>on</strong>al (1D) model <str<strong>on</strong>g>of</str<strong>on</strong>g> RPA is modeled<br />
by the follow<strong>in</strong>g relativistic equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> for the<br />
foil [25]:<br />
2 2 2<br />
dp 2I<br />
1<br />
2I<br />
p c p<br />
<br />
2 2 2<br />
dt c 1<br />
c p c p<br />
where is the foil velocity, p the areal momentum<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the foil and = m<strong>in</strong>il the areal mass <str<strong>on</strong>g>of</str<strong>on</strong>g> the foil with the<br />
i<strong>on</strong> mass mi, the i<strong>on</strong> density ni, and the foil thickness l.<br />
The soluti<strong>on</strong> is given by<br />
with<br />
. One obta<strong>in</strong>s<br />
the f<strong>in</strong>al velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the foil<br />
, where<br />
and<br />
with<br />
the <strong>in</strong>itial electr<strong>on</strong> density n0, the cut<str<strong>on</strong>g>of</str<strong>on</strong>g>f density nc, the<br />
laser wavelength , the pulse durati<strong>on</strong> <strong>in</strong> units <str<strong>on</strong>g>of</str<strong>on</strong>g> laser<br />
period and the normalized vector potential<br />
for a circularly polarized laser pulse. The f<strong>in</strong>al k<strong>in</strong>etic<br />
energy results <strong>in</strong><br />
. Hence the i<strong>on</strong> f<strong>in</strong>al energy scales as and<br />
. As an example, for a 100 fs circularly polarized laser
pulse with I = 10 21 Wcm -2 at = 0.8 m and a 150 nm<br />
thick target with density, m<strong>on</strong>oenergetic i<strong>on</strong>s<br />
with the k<strong>in</strong>etic energy ~ 230 MeV per nucle<strong>on</strong> may be<br />
accelerated.<br />
The optimal c<strong>on</strong>diti<strong>on</strong> for produc<strong>in</strong>g i<strong>on</strong>s with a narrow<br />
spectral peak is obta<strong>in</strong>ed from the equilibrium between<br />
the electrostatic and radiati<strong>on</strong> pressure that cause a str<strong>on</strong>g<br />
expulsi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s and produce a density distributi<strong>on</strong><br />
c<strong>on</strong>sist<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> depleti<strong>on</strong> regi<strong>on</strong> with density<br />
, thickness d and the compressed electr<strong>on</strong> layer<br />
with density np0, thickness ls where all electr<strong>on</strong>s pile at the<br />
rear surface <str<strong>on</strong>g>of</str<strong>on</strong>g> the foil with the reflectivity :<br />
, where<br />
is the total<br />
radiati<strong>on</strong> pressure and<br />
is<br />
the electrostatic pressure, obta<strong>in</strong>ed from the peak<br />
electrostatic field and the charge<br />
c<strong>on</strong>servati<strong>on</strong> . This c<strong>on</strong>diti<strong>on</strong><br />
results <strong>in</strong> , which determ<strong>in</strong>es the optimal foil<br />
thickness<br />
.<br />
The beam quality <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong> beams produced by RPA is<br />
expected to improve drastically <strong>on</strong> that <str<strong>on</strong>g>of</str<strong>on</strong>g> TNSAproduced<br />
i<strong>on</strong>s. The PIC simulati<strong>on</strong> for the <strong>in</strong>teracti<strong>on</strong><br />
between the circularly polarized laser pulse <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
, the pulse durati<strong>on</strong> and a 150 nm<br />
thick foil at the prot<strong>on</strong> density <str<strong>on</strong>g>of</str<strong>on</strong>g> shows a<br />
m<strong>on</strong>oenergetic spectral peak <str<strong>on</strong>g>of</str<strong>on</strong>g> prot<strong>on</strong>s at 485 MeV and<br />
an excellent l<strong>on</strong>gitud<strong>in</strong>al emittance <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
, which is three orders magnitude<br />
smaller than that <str<strong>on</strong>g>of</str<strong>on</strong>g> TNSA [25].<br />
Despite <str<strong>on</strong>g>of</str<strong>on</strong>g> many simulati<strong>on</strong> works, a few experiments<br />
have attempted to dem<strong>on</strong>strate the i<strong>on</strong> accelerati<strong>on</strong> driven<br />
by RPA because <str<strong>on</strong>g>of</str<strong>on</strong>g> difficulty <str<strong>on</strong>g>of</str<strong>on</strong>g> produc<strong>in</strong>g laser pulses<br />
with extremely high-c<strong>on</strong>trast ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> the ma<strong>in</strong> pulse<br />
<strong>in</strong>tensity over the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> ASE (amplified<br />
sp<strong>on</strong>taneous emissi<strong>on</strong>) pedestal and prepulses as well as<br />
high <strong>in</strong>tensities (I 2 > 10 20 Wcm -2 m 2 ) to avoid target<br />
heat<strong>in</strong>g prior to arrival <str<strong>on</strong>g>of</str<strong>on</strong>g> the ma<strong>in</strong> pulse. The first pro<str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>of</str<strong>on</strong>g>-pr<strong>in</strong>ciple<br />
experiment <str<strong>on</strong>g>of</str<strong>on</strong>g> RPA was dem<strong>on</strong>strated by<br />
irradiat<strong>in</strong>g a circularly polarized laser pulse <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
(a0 =3.5) <strong>on</strong> diam<strong>on</strong>dlike carb<strong>on</strong> (DLC)<br />
foils <str<strong>on</strong>g>of</str<strong>on</strong>g> thickness 2.9 – 40 nm at normal <strong>in</strong>cidence and<br />
ultrahigh c<strong>on</strong>trast (~10 11 ), produced by the double-plasma<br />
mirror technique [29]. The results showed a peak <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
maximum i<strong>on</strong> energy per nucle<strong>on</strong> at the optimum<br />
thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> 5.3 nm that corresp<strong>on</strong>ds to the c<strong>on</strong>diti<strong>on</strong><br />
and a broad quasim<strong>on</strong>oenergetic peak <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
(2.5MeV/u) <strong>in</strong> the carb<strong>on</strong> C 6+ spectra for circular<br />
polarizati<strong>on</strong>.<br />
CONCLUSIONS<br />
Recent progress <strong>in</strong> laser-plasma accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charged<br />
particles are overviewed from the aspects <strong>on</strong> laser<br />
wakefield accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s <strong>in</strong> the l<strong>in</strong>ear and<br />
n<strong>on</strong>l<strong>in</strong>ear regime, so-called “bubble” regime,<br />
characterized by quasi m<strong>on</strong>oenergetic beam producti<strong>on</strong><br />
due to the self- and the c<strong>on</strong>trolled <strong>in</strong>jecti<strong>on</strong> as well as the<br />
beam load<strong>in</strong>g, and i<strong>on</strong> accelerati<strong>on</strong> mechanisms such as<br />
TNSA and RPA. Based <strong>on</strong> up-to-date achievements, the<br />
design parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 GeV s<strong>in</strong>gle-stage LWFA are<br />
presented, aim<strong>in</strong>g at a currently advocated goal.<br />
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Abstract<br />
Fly<strong>in</strong>g Mirror as a tool to access ultra-high fields ∗<br />
M. Kando, A. S. Pirozhkov, T. Zh. Esirkepov, T. Nakamura, J. Koga, H. Kotaki<br />
Y. Hayashi, S. V. Bulanov<br />
JAEA, Kizugawa, Kyoto 619-0215, Japan<br />
Thanks to the recent progress <str<strong>on</strong>g>of</str<strong>on</strong>g> laser technology, there<br />
are grow<strong>in</strong>g <strong>in</strong>terests to explore ultra-high fields (electromagnetic<br />
fields) by focus<strong>in</strong>g <strong>in</strong>tense, ultra-short laser<br />
pulses down to a few micr<strong>on</strong> sizes. Presented here is a<br />
study to possibility reach (or boost) such ultra-high fields<br />
us<strong>in</strong>g a new c<strong>on</strong>cept employ<strong>in</strong>g the <strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>tense<br />
laser pulses with plasma. The c<strong>on</strong>cept uses break<strong>in</strong>g waves<br />
excited by ultra-short, <strong>in</strong>tense laser pulses <strong>in</strong> plasma. We<br />
present example parameters to reach the Schw<strong>in</strong>ger field<br />
and review the recent experimental progress <str<strong>on</strong>g>of</str<strong>on</strong>g> the fly<strong>in</strong>g<br />
mirror c<strong>on</strong>cept.<br />
INTRODUCTION<br />
S<strong>in</strong>ce the <strong>in</strong>novati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the chirped pulse amplificati<strong>on</strong><br />
(CPA) technique [1], the peak power <str<strong>on</strong>g>of</str<strong>on</strong>g> lasers has been <strong>in</strong>creas<strong>in</strong>g<br />
and the focused irradiance has reached as high as<br />
10 22 W/cm 2 [2, 3]. The extreme light <strong>in</strong>frastructure (ELI)<br />
project[4] is be<strong>in</strong>g promoted and the goal is to reach 10 26<br />
W/cm 2 . The unprecedented irradiances allow us to explore<br />
a new regime <str<strong>on</strong>g>of</str<strong>on</strong>g> physics, which has previously not been<br />
accessible experimentally. Theoretically there are challeng<strong>in</strong>g<br />
tasks which can be d<strong>on</strong>e <strong>in</strong> the high electromagnetic<br />
fields. One example is to explore the so-called quantum<br />
electrodynamics critical field or the Schw<strong>in</strong>ger field,<br />
at which vacuum breaks down and therefore n<strong>on</strong>-virtual<br />
electr<strong>on</strong>-positr<strong>on</strong> pairs are created from vacuum. Assum<strong>in</strong>g<br />
a direct-current (DC) electric field the QED critical<br />
field Ec = 1.3 × 10 18 V/m is obta<strong>in</strong>ed. The associated<br />
laser irradiance is Ic = 2.3 × 10 29 W/cm 2 , which is 7 orders<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude higher than the world record and still<br />
3 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude higher than that at the planned ELI<br />
project. Can we use the present laser technology to reach<br />
the Schw<strong>in</strong>ger field?<br />
The answer might be ’Yes’. There is theoretical work <strong>in</strong>dicat<strong>in</strong>g<br />
lower<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the limit <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>-positr<strong>on</strong> creati<strong>on</strong><br />
from vacuum[5, 6, 7]. Here <strong>in</strong> additi<strong>on</strong> to the work, we recall<br />
the proposal that an plasma device –relativistic fly<strong>in</strong>g<br />
mirror– can be used to <strong>in</strong>tensify the laser pulse as shown <strong>in</strong><br />
Fig. 1[8]. Fly<strong>in</strong>g mirrors are electr<strong>on</strong> density cusps propagat<strong>in</strong>g<br />
almost at the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light <strong>in</strong> tenuous plasma.<br />
Such electr<strong>on</strong> density cusps are formed when plasma wake<br />
waves excited by <strong>in</strong>tense, ultrashort laser pulses are break<strong>in</strong>g.<br />
The fly<strong>in</strong>g mirror reflects an <strong>in</strong>com<strong>in</strong>g laser, and the<br />
reflected pulse is upshifted due to the double Doppler effect<br />
∗ Work <strong>in</strong> part supported by JAEA and KAKENHI No. 20244065<br />
and is compressed as well. In additi<strong>on</strong>, the fly<strong>in</strong>g mirror<br />
may focus the laser pulse. Thus, the fly<strong>in</strong>g mirror can be<br />
a novel device that enhances the laser focused irradiance<br />
drastically. In this paper, we discuss the possibility to access<br />
ultra-high electromagnetic fields employ<strong>in</strong>g the fly<strong>in</strong>g<br />
mirror.<br />
THEORY<br />
We shall c<strong>on</strong>sider a mirror mov<strong>in</strong>g at the speed vM =<br />
βMc, where βM denotes the ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> the mirror speed to the<br />
speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light c. The reflected frequency <strong>in</strong> the laboratory<br />
frame <str<strong>on</strong>g>of</str<strong>on</strong>g> reference is expressed as<br />
ωr = ωs<br />
1 + βM cos θ<br />
, (1)<br />
1 − βM cos θr<br />
where ωs is the frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>cident source pulse and<br />
π −θ and θr are angles <strong>in</strong> the laboratory frame <str<strong>on</strong>g>of</str<strong>on</strong>g> reference<br />
between the wave propagati<strong>on</strong> directi<strong>on</strong> and the mirror velocity<br />
for the <strong>in</strong>cident and reflected waves, respectively. If<br />
the <strong>in</strong>cident angle satisfies the c<strong>on</strong>diti<strong>on</strong> −π/2 < θ < π/2,<br />
the frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> the reflected pulse is upshifted. The maximum<br />
upshift is achieved when θ = θr = 0. In this case the<br />
reflected frequency is approximately equal to ωr ≈ 4γ 2 M ωs<br />
for γM ≫ 1, where γM = 1/ √ 1 − β 2 M .<br />
The electric field amplitude <strong>in</strong> the reflected wave is ex-<br />
pressed as<br />
Er = R 1/2 Es<br />
( ωr<br />
ωs<br />
)<br />
, (2)<br />
where R is the reflectivity <str<strong>on</strong>g>of</str<strong>on</strong>g> the mov<strong>in</strong>g mirror <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the phot<strong>on</strong> number. For a mirror with a paraboloidal shape,<br />
the reflected light can be focused down to a spot equal to<br />
λs/(2γM). In this case the focused irradiance is given by<br />
Ir = 64Rγ 6 M<br />
( D<br />
λs<br />
) 2<br />
Is. (3)<br />
Here Is is the irradiance <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>cident source pulse <strong>on</strong><br />
the mirror with a waist <str<strong>on</strong>g>of</str<strong>on</strong>g> D. As expla<strong>in</strong>ed <strong>in</strong> the previous<br />
secti<strong>on</strong> fly<strong>in</strong>g mirrors are created dur<strong>in</strong>g the <strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<strong>in</strong>tense, utra-short laser pulses with tenuous plasma. The<br />
velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> the fly<strong>in</strong>g mirror is equal to the phase velocity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma wave, i.e. βMc = βphc. The phase velocity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma wave is approximately equal to the group velocity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser <strong>in</strong> the plasma βphc = c[1−(ωpe/ω) 2 ] 1/2 ,<br />
where ωpe = (4πnee 2 /m) 1/2 , m and e are the electr<strong>on</strong><br />
mass and charge, and ω is the laser angular frequency.
Plasma<br />
Fly<strong>in</strong>g mirrors<br />
Driver pulse<br />
Reflected pulse<br />
Source pulse<br />
Figure 1: C<strong>on</strong>ceptual scheme <str<strong>on</strong>g>of</str<strong>on</strong>g> the fly<strong>in</strong>g mirror. The driver pulse generates electr<strong>on</strong> density modulati<strong>on</strong>s (wake waves).<br />
The counter-propagat<strong>in</strong>g source pulse is reflected, frequency-upshifted, compressed, and focused by the fly<strong>in</strong>g mirror.<br />
The dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> the reflectivity <strong>on</strong> the parameters <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the laser-plasma <strong>in</strong>teracti<strong>on</strong> is <str<strong>on</strong>g>of</str<strong>on</strong>g> the key <strong>in</strong>terest, and has<br />
been analyzed <strong>in</strong> Refs. [8] and [9] for different c<strong>on</strong>figurati<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> break<strong>in</strong>g n<strong>on</strong>l<strong>in</strong>ear waves. In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> a str<strong>on</strong>g<br />
s<strong>in</strong>gularity, the electr<strong>on</strong> density modulati<strong>on</strong> <strong>in</strong> the break<strong>in</strong>g<br />
wave may be described by the Dirac delta functi<strong>on</strong>. In this<br />
case the reflectivity <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong> number is<br />
(<br />
ωpe<br />
Rδ ≈<br />
ωs cos2 ) 2<br />
1<br />
. (4)<br />
(θ/2)<br />
2γph<br />
Us<strong>in</strong>g the approximati<strong>on</strong> γph ≈ ωd/ωpe, where ne is the<br />
plasma density, and assum<strong>in</strong>g that ωd = ωs and θ = 0, the<br />
reflectivity is simplified to<br />
Rδ ≈ 1<br />
2γ3 . (5)<br />
ph<br />
When the electr<strong>on</strong> density has a cusp form <strong>in</strong> the vic<strong>in</strong>ity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a s<strong>in</strong>gularity expressed as n(X) ∝ X −2/3 (cusp), the<br />
reflectivity is<br />
R2/3 = 4 −7/3 3 −1/3 Γ 2 (<br />
(1/3)<br />
ωpe<br />
ωs cos 2 (θ/2)<br />
) 8/3<br />
1<br />
γ 4/3<br />
ph<br />
(6)<br />
where Γ(x) is the Euler gamma functi<strong>on</strong>. Substitut<strong>in</strong>g<br />
γph ≈ ωd/ωpe, ωd = ωs and θ = 0, Eq. (6) reduces to<br />
R2/3 ≈ 0.2<br />
γ4 . (7)<br />
ph<br />
By us<strong>in</strong>g the best reflectivity described <strong>in</strong> Eq. (5), the<br />
irradiance obta<strong>in</strong>ed <strong>in</strong> the fly<strong>in</strong>g mirror scheme, Eq. (3) is<br />
expressed as<br />
Ir = 32γ 3 ph<br />
( D<br />
λs<br />
EXAMPLES<br />
) 2<br />
Is. (8)<br />
Here we show numerical examples to obta<strong>in</strong> the<br />
Schw<strong>in</strong>ger limit (10 29 W/cm 2 ) with the fly<strong>in</strong>g mirror c<strong>on</strong>cept.<br />
We c<strong>on</strong>sider a laser system that does not exist but<br />
is possible to c<strong>on</strong>struct with<strong>in</strong> exist<strong>in</strong>g laser technologies.<br />
The assumpti<strong>on</strong>s used here are as follows. A source laser<br />
irradiance is set to be weakly relativistic Is = 10 17 W/cm 2<br />
<strong>in</strong> order to avoid possible modificati<strong>on</strong>s to the fly<strong>in</strong>g mirrors.<br />
Both driver and source laser pulses have the same<br />
,<br />
wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g> λd = λs=0.8 µm and the same pulse durati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> τ=20 fs. We assume that the driver irradiance is<br />
equal to the <strong>on</strong>e-dimensi<strong>on</strong>al wave break<strong>in</strong>g limit a 2 0/2 =<br />
γph, where a0 = eE/(mcω)=0.86×10 −9 λ[µm]I[W/cm 2 ]<br />
is the normalized field amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> a laser with the electric<br />
field <str<strong>on</strong>g>of</str<strong>on</strong>g> E , the angular frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> ω, the wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
λ, and the irradiance <str<strong>on</strong>g>of</str<strong>on</strong>g> I.<br />
From Eq. (3) the free parameters seem to be γph or<br />
D. We choose γph (plasma density), therefore D is determ<strong>in</strong>ed.<br />
Shown <strong>in</strong> Table 1 are examples when γph =<br />
100, 200.<br />
Table 1: Example parameters to achieve a focused irradiance<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> 10 29 W/cm 2 . ne is the plasma density, Is and Id are<br />
irradiances <str<strong>on</strong>g>of</str<strong>on</strong>g> the source and driver laser pulses. D and wd<br />
are diameters <str<strong>on</strong>g>of</str<strong>on</strong>g> the fly<strong>in</strong>g mirror and the driver laser. If<br />
is the focused irradiance <str<strong>on</strong>g>of</str<strong>on</strong>g> the reflected pulse. Ex denotes<br />
the pulse energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the driver (source) pulse.<br />
Parameters Units Case I Case II<br />
γph 100 200<br />
ne cm −3 2×10 17 4×10 16<br />
Is W/cm 2 1×10 17 1×10 17<br />
D µm 140 50<br />
Id W/cm 2 2×10 20 2×10 18<br />
wd µm 280 100<br />
Ir W/cm 2 1×10 29 1×10 29<br />
Ed kJ 5.4 1.3<br />
Es mJ 320 40<br />
DISCUSSIONS<br />
In the previous secti<strong>on</strong> we assume that all the functi<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the fly<strong>in</strong>g mirror work very well as described <strong>in</strong> Eq. (8).<br />
We notice this assumpti<strong>on</strong> is too optimistic and need further<br />
<strong>in</strong>vestigati<strong>on</strong> to implement it <strong>in</strong> a device.<br />
First, let us check the experimental progress <str<strong>on</strong>g>of</str<strong>on</strong>g> the fly<strong>in</strong>g<br />
mirror. A basic c<strong>on</strong>cept that fly<strong>in</strong>g mirrors can reflect laser<br />
pulses and upshift the <strong>in</strong>cident source laser frequency was<br />
verified <strong>in</strong> an experiment [10, 11]. In the experiment, 180<br />
mJ, 76 fs, 0.8 µm laser pulses were focused <strong>on</strong>to helium<br />
gas-jet targets and 10 mJ, ∼ 100 fs laser pulses were used as<br />
source laser pulses at the angle <str<strong>on</strong>g>of</str<strong>on</strong>g> 135 ◦ . The reflected light
Reflected light <strong>in</strong>tensity (μJ/sr/nm)<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
12 14 16 18 20 22<br />
Wavelength (nm)<br />
Figure 2: Observed spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the reflected laser light <strong>in</strong><br />
the fly<strong>in</strong>g mirror experiment [12].<br />
was measured with a graz<strong>in</strong>g-<strong>in</strong>cidence flat-field spectrograph<br />
<strong>in</strong> the forward (0 ◦ ) directi<strong>on</strong>. The observed spectra<br />
distributed <strong>in</strong> the range <str<strong>on</strong>g>of</str<strong>on</strong>g> 7–15 nm. The upshift factors<br />
were 50–114.<br />
Although the experiment showed a somewhat coherent<br />
effect, the reflected phot<strong>on</strong> number was smaller than the<br />
theoretical expectati<strong>on</strong>. In the 2nd experiment[12], a head<strong>on</strong><br />
collisi<strong>on</strong>s setup <str<strong>on</strong>g>of</str<strong>on</strong>g> the driver and source pulses was employed<br />
thanks to the improvement <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser system and<br />
a more powerful laser was used. The driver and source<br />
laser had peak powers <str<strong>on</strong>g>of</str<strong>on</strong>g> 15 TW(400 mJ/27 fs) and 1.2<br />
TW(42 mJ/ 34 fs), respectively. Reflected source pulses<br />
were observed with an imag<strong>in</strong>g spectrograph that covered<br />
the wavelength range <str<strong>on</strong>g>of</str<strong>on</strong>g> 12-25 nm at the angle range <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
9 ◦ − 17 ◦ . The result is shown <strong>in</strong> Fig. 2. The reflected phot<strong>on</strong><br />
number <strong>in</strong>creased to half <str<strong>on</strong>g>of</str<strong>on</strong>g> the theoretical cusp model<br />
(see Table 2).<br />
To <strong>in</strong>crease the reflectivity further, we can use a variant<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (5) assum<strong>in</strong>g ωd ̸= ωs and θ = 0,<br />
Rδ ≈<br />
( ) 2<br />
ωd 1<br />
ωs 2γ3 ph<br />
. (9)<br />
If ωd > ωs, we obta<strong>in</strong> a ga<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> (ωd/ωs) 2 . For example,<br />
we can use a frequency doubl<strong>in</strong>g crystal for the driver laser.<br />
Table 2: Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> reflectivities and phot<strong>on</strong> numbers<br />
between the theoretical model and the experiment.<br />
Reflectivity Phot<strong>on</strong> number<br />
Theory 4×10 −5 1.5×10 10<br />
Experiment 3×10 −6 1.1×10 9<br />
Exp. (corrected) 2×10 −5 7.9×10 9<br />
There are no measurement so far <strong>on</strong> the pulse durati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a reflected pulse from fly<strong>in</strong>g mirrors. However, the large<br />
phot<strong>on</strong> number obta<strong>in</strong>ed <strong>in</strong> the experiment implies that the<br />
coherent effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong>s exists. Therefore, we may<br />
Intensity (arb. u.)<br />
-3<br />
-3<br />
0<br />
300 400 500<br />
Figure 3: Observed spectrum <strong>in</strong> a high-resoluti<strong>on</strong> particle<strong>in</strong>-cell<br />
simulati<strong>on</strong>.<br />
be allowed to assume that the pulse durati<strong>on</strong> is compressed<br />
down to the theoretical model predicti<strong>on</strong>.<br />
Most serious and to be c<strong>on</strong>firmed is the spot size <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
reflected pulse. There are many po<strong>in</strong>ts that degrades the<br />
focus spot such as a surface error from an ideal parabola,<br />
surface roughness, alignment error to the fly<strong>in</strong>g mirror,<br />
etc. These po<strong>in</strong>ts have not yet been addressed both experimentally<br />
and theoretically. We have just started to estimate<br />
the roughness <str<strong>on</strong>g>of</str<strong>on</strong>g> the fly<strong>in</strong>g mirror. As a prelim<strong>in</strong>ary<br />
result we have obta<strong>in</strong>ed that the surface roughness<br />
<strong>in</strong> two-dimensi<strong>on</strong>al particle-<strong>in</strong>-cell simulati<strong>on</strong>s is smaller<br />
than the simulati<strong>on</strong> resoluti<strong>on</strong> 0.02 µm. Further analysis<br />
is needed to determ<strong>in</strong>e the focusability <str<strong>on</strong>g>of</str<strong>on</strong>g> the reflected<br />
pulse, especially at low densities (high γph) <strong>in</strong> Table 1. We<br />
also obta<strong>in</strong>ed the upshift factor <str<strong>on</strong>g>of</str<strong>on</strong>g> ωr/ωs ∼500 <strong>in</strong> a highresoluti<strong>on</strong><br />
particle-<strong>in</strong>-cell simulati<strong>on</strong> as shown <strong>in</strong> Fig. 3.<br />
Here the driver laser has an energy <str<strong>on</strong>g>of</str<strong>on</strong>g> 2.3 J and a pulse<br />
durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 20 fs, and the focused irradiance <str<strong>on</strong>g>of</str<strong>on</strong>g> 3.7×10 19<br />
W/cm 2 (a0 = 4.1). The source laser has a wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
2.4 µm <strong>in</strong> this simulati<strong>on</strong>. The plasma density is 1.5×10 19<br />
cm −3 and mesh sizes are ∆x =0.8 nm and ∆y=20 nm. Although<br />
the resolved wavelength is larger than the requirement<br />
<strong>in</strong> Table 1, water-w<strong>in</strong>dow or shorter wavelength Xrays<br />
are generated <strong>in</strong> the simulati<strong>on</strong>.<br />
CONCLUSION<br />
A relativistic fly<strong>in</strong>g mirror is exam<strong>in</strong>ed as a tool to access<br />
ultra-high electromagnetic field irradiance. We carved<br />
out laser and plasma parameters assum<strong>in</strong>g plasma and laser<br />
parameters that are not far from present-day laser technologies.<br />
We reviewed the recent experimental progress <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
fly<strong>in</strong>g mirror. The c<strong>on</strong>cept is <strong>in</strong> progress show<strong>in</strong>g that the<br />
reflectivity is close to the theoretical estimate us<strong>in</strong>g a cusp<br />
model. Still further study is necessary to <strong>in</strong>tensify the focused<br />
laser.
REFERENCES<br />
[1] D. Strickland and G. Mourou, Opt. Comm. 56 (1985) 212.<br />
[2] S.-W. Bahk et al., Opt. Lett. 29 (2004) 2837.<br />
[3] V. Yanovsky et al., Opt. Express 16 (2008) 2109.<br />
[4] http://www.extreme-light-<strong>in</strong>frastructure.eu/<br />
[5] G. V. Dunne, H. Gies, R. Schützhold, Phys. Rev. D 80<br />
(2009) 111301(R).<br />
[6] A. Di Pizazza, E. Lötstedt, A. I. Milste<strong>in</strong>, and C. H. Keitel,<br />
Phys. Rev. Lett. 103 (2009) 170403.<br />
[7] S. S. Bulanov, V. D. Mur, N. B. Narozhny, J. Nees, and V. S.<br />
Popov, Phys. Rev. Lett. 104 (2010) 220404.<br />
[8] S. V. Bulanov, T. Zh. Esirkepov, and T. Tajima, Phys. Rev.<br />
Lett. 91 (2003) 085001.<br />
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[11] A. S. Pirozhkov et al., Phys. Plasmas 14 (2007)123106.<br />
[12] M. Kando et al., Phys. Rev. Lett. 103 (2009) 235003.
4-mirror laser stack<strong>in</strong>g cavity for high <strong>in</strong>tensity polarized phot<strong>on</strong> generati<strong>on</strong><br />
T. Akagi, M. Kuriki, T. Takahashi, S. Miyoshi : Hiroshima Univ.<br />
S. Araki, J. Urakawa, T. Omori, T. Okugi, H. Shimizu, N. Terunuma, Y. Funahashi, Y. H<strong>on</strong>da : <strong>KEK</strong><br />
K. Sakaue, T. Hirose, M. Washio : Waseda Univ.<br />
Abstract<br />
We are develop<strong>in</strong>g a compact light source based <strong>on</strong> the<br />
laser-Compt<strong>on</strong> scatter<strong>in</strong>g. We have performed a phot<strong>on</strong><br />
generati<strong>on</strong> experiment at the <strong>KEK</strong>-ATF us<strong>in</strong>g a Fabry-<br />
Perot type 2-mirror laser pulse stack<strong>in</strong>g cavity[1]. The laser<br />
pulses are accumulated and their power was enhanced by<br />
up to 760 times <strong>in</strong> the 2-mirror Fabry-Perot cavity. In order<br />
further improve performance <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser power enhancement,<br />
a new three dimensi<strong>on</strong>al 4 mirror cavity is be<strong>in</strong>g designed.<br />
In this article, we report status and prospect <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
phot<strong>on</strong> generati<strong>on</strong> experiment, and 3D 4-mirrorcavity design.<br />
INTRODUCTION<br />
X-ray have been used <strong>in</strong> various scientific <strong>in</strong>dustrial and<br />
medical applicati<strong>on</strong>s. Compact and bright light source is<br />
highly desirable for these applicati<strong>on</strong>s. We are develop<strong>in</strong>g<br />
compact light source based <strong>on</strong> the laser-Compt<strong>on</strong> scatter<strong>in</strong>g.<br />
In this scheme, for example, tens <str<strong>on</strong>g>of</str<strong>on</strong>g> MeV phot<strong>on</strong>s can<br />
be generated by collisi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> laser phot<strong>on</strong>s with about 1<br />
GeV electr<strong>on</strong> beam, as shown <strong>in</strong> Fig. 1. To <strong>in</strong>crease the <strong>in</strong>tensity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> generated phot<strong>on</strong>s by laser-Compt<strong>on</strong> scatter<strong>in</strong>g,<br />
<strong>in</strong>creas<strong>in</strong>g <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> laser pulses and focus<strong>in</strong>g at collisi<strong>on</strong><br />
po<strong>in</strong>t by accumulat<strong>in</strong>g them <strong>in</strong> an optical cavity is an<br />
attractive method.<br />
We already achieved laser <strong>in</strong>tensity enhancement <str<strong>on</strong>g>of</str<strong>on</strong>g> 760<br />
and laser waist size is 30µm (1σ) by the 2-mirror Fabry-<br />
Perot cavity. To <strong>in</strong>crease the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> generated phot<strong>on</strong>s<br />
more, it is necessary to use high reflectivity mirrors and<br />
to make lasar waist size smaller. However, it is difficult<br />
to achieve the improvement <str<strong>on</strong>g>of</str<strong>on</strong>g> enhancement and focus<strong>in</strong>g<br />
performance at the same time <strong>in</strong> the 2-mirror Fabry-Peror<br />
cavity. Thus, to achieve the two requirement, now we are<br />
develop<strong>in</strong>g three dimensi<strong>on</strong>al 4-mirror optical cavity.<br />
Figure 1: laser-Compt<strong>on</strong> scatter<strong>in</strong>g<br />
OPTICAL CAVITY<br />
We succeeded to generate gamma-rays by laser-<br />
Compt<strong>on</strong> scatter<strong>in</strong>g with the 2-mirror Fabry-Perot cavity at<br />
the <strong>KEK</strong>-ATF. This secti<strong>on</strong> describes our 2-mirror Fabry-<br />
Perot cavity. We use a 357 MHz mode-locked laser, its<br />
repetiti<strong>on</strong> rate is the same as electr<strong>on</strong> bunch spac<strong>in</strong>g <strong>in</strong> the<br />
ATF. The wavelength and pulse width <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser are 1064<br />
nm and 5 ps <strong>in</strong> the root mean square.<br />
In order to accumulate laser pulses <strong>in</strong> the optical cavity,<br />
the optical cavity has to be <strong>on</strong>-res<strong>on</strong>ance. The length (L) <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the optical cavity has to be<br />
L = n λ<br />
2<br />
where λ is the wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser, and n is a positive<br />
<strong>in</strong>teger. In additi<strong>on</strong>, accmulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> laser pulses from<br />
a mode-locked pulsed laser requires that the length <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
optical cavity is <strong>in</strong>teger times <str<strong>on</strong>g>of</str<strong>on</strong>g> the cavity <strong>in</strong>side the laser<br />
oscillator.<br />
L = mL ′<br />
(2)<br />
where L ′ is the length <str<strong>on</strong>g>of</str<strong>on</strong>g> the cavity <strong>in</strong>side the mode-locked<br />
laser oscillator, and m is a positive <strong>in</strong>teger. In this experiment,<br />
the length <str<strong>on</strong>g>of</str<strong>on</strong>g> optical cavities are L = L ′ = 420mm.<br />
The waist size <str<strong>on</strong>g>of</str<strong>on</strong>g> laser pulses is 30µm (1σ) <strong>in</strong> the optical<br />
cavity, because the curvature radius <str<strong>on</strong>g>of</str<strong>on</strong>g> mirrors is 210.5mm.<br />
The performance <str<strong>on</strong>g>of</str<strong>on</strong>g> the optical cavity is expressed by the<br />
f<strong>in</strong>esse (F), and can be expressed as<br />
F = π√ R<br />
1 − R<br />
R = √ R1R2<br />
and the power enhancement factor (S) <str<strong>on</strong>g>of</str<strong>on</strong>g> laser pulses <strong>in</strong> the<br />
optical cavity can be estimated as<br />
S =<br />
T1<br />
(1 − R) 2<br />
where R1 and T1 are the reflectivity and transmissivity <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the entrance mirror, R2 is the reflectivity <str<strong>on</strong>g>of</str<strong>on</strong>g> the other mirror<br />
respectively. The res<strong>on</strong>ant c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the optical cavity<br />
is m<strong>on</strong>itored by the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> transmitted light from the<br />
cavity, which is at the maximum when the cavity is <strong>on</strong>res<strong>on</strong>ance.<br />
The <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> the transmitted light obta<strong>in</strong>ed<br />
by chang<strong>in</strong>g the length <str<strong>on</strong>g>of</str<strong>on</strong>g> the cavity is shown <strong>in</strong> Fig. 2.<br />
The width <str<strong>on</strong>g>of</str<strong>on</strong>g> res<strong>on</strong>ant peak is 0.36 nm with our 2-mirror<br />
optical cavity, which <strong>in</strong>dicates that the length <str<strong>on</strong>g>of</str<strong>on</strong>g> the optical<br />
cavity has to be c<strong>on</strong>trolled with precisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> smaller enough<br />
than <strong>on</strong>e tens <str<strong>on</strong>g>of</str<strong>on</strong>g> a nanometer. The optical cavity has to<br />
be high f<strong>in</strong>esse for high enhancement factor. However, the<br />
(1)<br />
(3)<br />
(4)<br />
(5)
Figure 6: To reduce the laser waist size, 2-mirror cavity has to be c<strong>on</strong>centric type. On the other hand, 4-mirror cavity is<br />
c<strong>on</strong>focal type.<br />
However, <strong>in</strong> the 4-mirror cavity, effective focal length<br />
(ft, fs) are different <strong>in</strong> tangential plane and sagittal plane,<br />
and the difference causes astigmatism at the focal po<strong>in</strong>t. ft<br />
and fs are expressed as<br />
ft = ρ<br />
cos θ (6)<br />
2<br />
fs = ρ<br />
2 cos θ<br />
where θ is reflecti<strong>on</strong> half angle <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>cave mirror. Because<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> this astigmatism, laser pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile <strong>in</strong>side the 4-mirror cavity<br />
will be ellipse.<br />
To avoid the astigmatism, the cavity has to be three dimensi<strong>on</strong>al<br />
c<strong>on</strong>figurati<strong>on</strong>. 3D 4-mirror optical cavity generally<br />
have a circular-polarizati<strong>on</strong> dependent property due to<br />
the rotati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the image <strong>in</strong> the three-dimensi<strong>on</strong>al optical<br />
path. A new method utiliz<strong>in</strong>g this property to obta<strong>in</strong> a differential<br />
signal from the cavity res<strong>on</strong>ance was proposed[2].<br />
The differential signal can be used to lock an optical cavity<br />
at a res<strong>on</strong>ance peak.<br />
We c<strong>on</strong>ducted experiment to obta<strong>in</strong> differential signal<br />
us<strong>in</strong>g a 3D 4-mirror optical cavity testbench. The setup<br />
is shown <strong>in</strong> Fig. 7. A l<strong>in</strong>ear polarizati<strong>on</strong> wave was <strong>in</strong>jected<br />
to the 4-mirror cavity. Then we measured the output <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
differential amplifier while scann<strong>in</strong>g the cavity length. The<br />
signal observed is shown <strong>in</strong> Fig. 8. The bottom l<strong>in</strong>e is the<br />
signal <str<strong>on</strong>g>of</str<strong>on</strong>g> the photo-diode that measurred the transmissi<strong>on</strong>.<br />
And it shows the po<strong>in</strong>ts that the cavity’s res<strong>on</strong>ance <str<strong>on</strong>g>of</str<strong>on</strong>g> right<br />
and left-handed polarized light. The top and middle l<strong>in</strong>es<br />
are output <str<strong>on</strong>g>of</str<strong>on</strong>g> the differential amplifier. The signal crossed<br />
zero at the po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> res<strong>on</strong>ace. It provides a good differential<br />
signal for lock<strong>in</strong>g the cavity to <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the peaks <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
res<strong>on</strong>ance. Actually, we have succeeded to lock the optical<br />
cavity <strong>on</strong> the res<strong>on</strong>ance peak and switch the circularpolarizati<strong>on</strong><br />
peak. It means the polarizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the generated<br />
phot<strong>on</strong>s by laser-Compt<strong>on</strong> can be switched quickly.<br />
Now, a 3D 4-mirror cavity are be<strong>in</strong>g designed <strong>in</strong> order to<br />
be <strong>in</strong>stalled <strong>in</strong> the ATF dur<strong>in</strong>g summer 2011. Then we will<br />
cunduct the experiment <str<strong>on</strong>g>of</str<strong>on</strong>g> gamma-rays generati<strong>on</strong>.<br />
CONCULSION<br />
In order to <strong>in</strong>crease the number <str<strong>on</strong>g>of</str<strong>on</strong>g> generated phot<strong>on</strong>s by<br />
laser-Compt<strong>on</strong> scatter<strong>in</strong>g, it is necessary to reduce the laser<br />
(7)<br />
Figure 7: Setup to obta<strong>in</strong> the differential signal.<br />
Figure 8: Signal <str<strong>on</strong>g>of</str<strong>on</strong>g> transmissi<strong>on</strong> and difference.<br />
waist size. We performed gamma-rays generati<strong>on</strong> experiment<br />
us<strong>in</strong>g a 2-mirror cavity and observed 10.8 ± 0.1 phot<strong>on</strong>s/bunch.<br />
The laser power enhanced by up to 760 times<br />
and laser waist size is 30µm (1σ).<br />
As the next step, a 4-mirror r<strong>in</strong>g cavity with a threedimensi<strong>on</strong>al<br />
(n<strong>on</strong>-planar) c<strong>on</strong>figrati<strong>on</strong> is be<strong>in</strong>g c<strong>on</strong>structed.<br />
The cavity has unique property such that it <strong>on</strong>ly res<strong>on</strong>ate<br />
with circular polarized light. It allows us to utilize new<br />
method to obta<strong>in</strong> feed back signal for the cavity stabilizati<strong>on</strong><br />
as well as fast polarizati<strong>on</strong> switch<strong>in</strong>g.<br />
REFERENCES<br />
[1] S. Miyoshi et al., Nucl. Inst. Meth. A 623 (2010) 576.<br />
[2] Y. H<strong>on</strong>da et al., Opt. Commun. 282 (2009) 3108.
CURRENT STATUS OF LFEX LASER AND EXA-WATT LASER CONCEPT<br />
AT ILE/OSAKA<br />
J. Kawanaka, LFEX-Team, EXA-Team, and H. Azechi<br />
Institute for Laser Eng<strong>in</strong>eer<strong>in</strong>g, Osaka University, Yamadaoka, Suita 565-0871 Japan<br />
Abstract<br />
The LFEX laser has been developed for the basic<br />
research <str<strong>on</strong>g>of</str<strong>on</strong>g> plasma heat<strong>in</strong>g <strong>in</strong> fast igniti<strong>on</strong> scheme.<br />
Its dem<strong>on</strong>strati<strong>on</strong> ensures the high potential <str<strong>on</strong>g>of</str<strong>on</strong>g> the LFEX<br />
laser and various novel applicati<strong>on</strong> fields has been<br />
str<strong>on</strong>gly discussed, such as lab-astrophysics, particle<br />
physics and so <strong>on</strong>. We are plann<strong>in</strong>g the “Gekko-EXA”<br />
laser, which is a sub-exa-watt ultrahigh peak power laser<br />
where various advanced laser technologies developed <strong>in</strong><br />
the LFEX laser project are used. In this letter, the recent<br />
status <str<strong>on</strong>g>of</str<strong>on</strong>g> the LFEX laser and the rough c<strong>on</strong>ceptual design<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> “Gekko-EXA” are menti<strong>on</strong>ed.<br />
INTRODUCTION<br />
The central igniti<strong>on</strong> for the laser fusi<strong>on</strong> has been<br />
studied. The Nati<strong>on</strong>al Igniti<strong>on</strong> Facility (NIF) <strong>in</strong> Lawrence<br />
Liver More Nati<strong>on</strong>al Laboratory (LLNL) will<br />
dem<strong>on</strong>strate it with a mega-joules class laser [1]. On the<br />
other hand, the fast-igniti<strong>on</strong> has been actively researched<br />
<strong>in</strong> our <strong>in</strong>stitute. It reduces a total laser power to <strong>on</strong>e tenth,<br />
which improves not <strong>on</strong>ly a compactness <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser<br />
system but also a repeatability <str<strong>on</strong>g>of</str<strong>on</strong>g> laser operati<strong>on</strong> and a<br />
stability <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser operati<strong>on</strong>. A high peak power laser<br />
with short pulse durati<strong>on</strong> is necessary as a heat<strong>in</strong>g laser<br />
for the fast igniti<strong>on</strong>. The LFEX (Laser for Fusi<strong>on</strong><br />
EXperiments)-Laser system has been developed to clarify<br />
the heat<strong>in</strong>g mechanism [2]. The plasma heat<strong>in</strong>g<br />
experiments have started by us<strong>in</strong>g the LFEX-Laser. In<br />
Figure 1: Block diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> the LFEX-Laser.<br />
additi<strong>on</strong>, the “Gekko-EXA” Laser with a higher peak<br />
power has been under a c<strong>on</strong>ceptual design to improve the<br />
fusi<strong>on</strong> researches and to open the new high field<br />
researches <strong>in</strong> the plasma physics.<br />
LFEX-LASER<br />
The LFEX laser generates 10 kJ pico-sec<strong>on</strong>ds pulse<br />
energy with four beams by us<strong>in</strong>g a chirped-pulse<br />
amplificati<strong>on</strong> (CPA) technique, which corresp<strong>on</strong>ds<br />
to several peta-watts peak power. The laser system<br />
c<strong>on</strong>sists <str<strong>on</strong>g>of</str<strong>on</strong>g> a fr<strong>on</strong>t end, rod amplifier cha<strong>in</strong>s, a ma<strong>in</strong><br />
amplifier, and rear end (pulse compressi<strong>on</strong>), shown <strong>in</strong> fig.<br />
1. A fr<strong>on</strong>t end is a comb<strong>in</strong>ati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a mode-locked fibre<br />
oscillator, two pulse stretchers and three optical<br />
parametric chirped-pulse amplificati<strong>on</strong> (OPCPA) stages.<br />
The OPCPA stages are used <strong>in</strong>stead <str<strong>on</strong>g>of</str<strong>on</strong>g> a c<strong>on</strong>venti<strong>on</strong>al<br />
regenerative amplifier to improve the temporal pulse<br />
c<strong>on</strong>trast. Much care <str<strong>on</strong>g>of</str<strong>on</strong>g> a beam transport with image relays<br />
are taken not to make the beam quality <strong>in</strong>ferior. The<br />
maximum pulse energy is obta<strong>in</strong>ed up to 100 mJ <strong>in</strong> fig. 2.<br />
The typical pulse durati<strong>on</strong> is 3 ns with a 6 nm spectral<br />
width at a 1053 nm centre wavelength. After four-pass<br />
Nd:glass rod (φ50mm)-amplificati<strong>on</strong> (RA), the amplified<br />
beam is divided <strong>in</strong>to four beams and each beam<br />
experiences two rod-amplifiers to obta<strong>in</strong> about 10 J pulse<br />
energy. The ma<strong>in</strong> amplifier is a four-pass amplifier with a<br />
Nd:glass slab amplifier and a l<strong>on</strong>g image relay. Eight<br />
large slab glasses (46 cm x 81 cm x t4cm) are used for
Amplified Pulse Energy (mJ)<br />
100<br />
Horiz<strong>on</strong>tal (μrad)<br />
10<br />
1<br />
0<br />
12<br />
10<br />
8<br />
6<br />
4<br />
2<br />
0<br />
-2<br />
100<br />
Figure 2: The obta<strong>in</strong>ed pulse energy at the threestages<br />
OPCPA.<br />
PV : 4.144!<br />
RMS : 0.937<br />
200<br />
3rd Path<br />
2nd Path<br />
DFM75<br />
each beam and the slab glass series are set side by side to<br />
arrange 2 x 2 beams spatially. The obta<strong>in</strong>ed pulse energy<br />
is 1.77 kJ/beam. The obta<strong>in</strong>ed spectral width is ga<strong>in</strong>narrowed<br />
to 3.1 nm and the pulse durati<strong>on</strong> is shortened at<br />
1.45 ns. A highly excellent focusibility <strong>on</strong> the target is<br />
str<strong>on</strong>gly required for a heat<strong>in</strong>g laser. Two pairs <str<strong>on</strong>g>of</str<strong>on</strong>g> largeaperture<br />
bimorph deformable mirror (DFM) and Shack-<br />
Hartmann sensor are used for the beam wave fr<strong>on</strong>t c<strong>on</strong>trol<br />
<strong>in</strong> the ma<strong>in</strong> amplifier cha<strong>in</strong>, shown <strong>in</strong> fig. 3. A loop<br />
program is used for automatic optimizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the wave<br />
fr<strong>on</strong>t correcti<strong>on</strong>. The obta<strong>in</strong>ed wave fr<strong>on</strong>t after four-pass<br />
amplificati<strong>on</strong> is 4.56λ (peak-valley) and 0.94λ (rms)<br />
without the deformable mirrors. Us<strong>in</strong>g DFMs, they were<br />
c<strong>on</strong>siderably improved at 0.95λ and 0.18λ, respectively,<br />
1st Path 4th Path<br />
4th Path<br />
M0 DA400<br />
SF400<br />
FR400<br />
M1 IMAP<br />
AA<br />
SF50<br />
RA50<br />
OS50 RA50 SF75 OS75<br />
Numerical Cal.!<br />
&!<br />
Apply PZT voltage<br />
OS125<br />
PCS125<br />
DFM125<br />
Sensor Image<br />
Shack-Haltmann<br />
Numerical Cal.!<br />
&!<br />
Apply PZT voltage<br />
Shack-Haltmann<br />
Sensor Image<br />
Figure 3: A schematic diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> a deformable mirror system and the measured wavefr<strong>on</strong>t (a) without and<br />
(b) with deformable-mirror compensati<strong>on</strong>.<br />
!"μ#$%<br />
-4<br />
Menu<br />
-6<br />
1st<br />
-8<br />
2nd<br />
-10<br />
-12<br />
3rd<br />
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12<br />
Vertical (μrad)<br />
300<br />
(a) (b)<br />
400<br />
Pump Energy (mJ)<br />
500<br />
PV : 0.952!<br />
RMS : 0.177<br />
Figure 4: Po<strong>in</strong>t<strong>in</strong>g stability <str<strong>on</strong>g>of</str<strong>on</strong>g> the LFEX<br />
laser before compressi<strong>on</strong>.<br />
600<br />
2x2<br />
700<br />
Focus<strong>in</strong>g Optics<br />
(2F)<br />
Pulse Compressor (1F)<br />
Beam Transfer<br />
(2F ->1F)<br />
1x4<br />
s<br />
2x2<br />
From the Fr<strong>on</strong>t<br />
Figure 5: Diam<strong>on</strong>d-trace pulse compressor with 16 meter-size<br />
segmented grat<strong>in</strong>gs for the LFEX laser.
which were comparable to those before the ma<strong>in</strong><br />
amplificati<strong>on</strong>. The encircled energy was estimated to be<br />
more than 70% at Fλ=2.5 with the observed far-field<br />
patterns when 10% without the DFM correcti<strong>on</strong>. In<br />
additi<strong>on</strong>, all shots with kilo-joule pulse energy were<br />
below 10 μrad <strong>in</strong> the po<strong>in</strong>t<strong>in</strong>g beam stability before the<br />
compressor <strong>in</strong> fig. 4. The amplified pulse beams go <strong>in</strong>to<br />
Figure 6: Block diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> the “Gekko-EXA” laser.<br />
Figure 7: Arrangement plan <str<strong>on</strong>g>of</str<strong>on</strong>g> the “Gekko-EXA” laser.<br />
the pulse compressor to be arranged with 1 x 4 beams<br />
from 2 x 2. The pulse compressor is our orig<strong>in</strong>al diam<strong>on</strong>d<br />
trace compressor with four large (91 cm x 42 cm) grat<strong>in</strong>gs<br />
for <strong>on</strong>e-beam diffracti<strong>on</strong>, which results <strong>in</strong> sixteen grat<strong>in</strong>gs<br />
for <strong>on</strong>e-beam compressi<strong>on</strong>, shown <strong>in</strong> fig. 5. The obta<strong>in</strong>ed<br />
m<strong>in</strong>imum pulse durati<strong>on</strong> is 2.2 ps. The LFEX laser is<br />
gradually mov<strong>in</strong>g <strong>in</strong>to the plasma experiments. The laser
dem<strong>on</strong>strati<strong>on</strong> encourages four-beams full operati<strong>on</strong> for<br />
the plasma heat<strong>in</strong>g.<br />
GEKKO-EXA LASER<br />
The “Gekko-EXA” laser is our great challenge for the<br />
next generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the high field science, the energetic<br />
beam generati<strong>on</strong> and their applicati<strong>on</strong>s, and is under<br />
c<strong>on</strong>ceptual design. The “Gekko-EXA” has two k<strong>in</strong>ds <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
operati<strong>on</strong> mode, 1 PW at 100 Hz with a s<strong>in</strong>gle beam and<br />
0.2 EW as a s<strong>in</strong>gle shot laser with two bundle beams,<br />
respectively, shown <strong>in</strong> fig. 6. These output power is ultrashort<br />
high-peak power laser with pulse durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 fs.<br />
An effective laser ga<strong>in</strong> over the extremely wide spectral<br />
width up to 500 nm is required for such short pulse<br />
amplificati<strong>on</strong>. The laser system is, therefore, based <strong>on</strong> the<br />
large-aperture optical parametric chirped-pulse<br />
amplificati<strong>on</strong> (LA-OPCPA).<br />
In the fr<strong>on</strong>t end, the femto-sec<strong>on</strong>d oscillator supplies<br />
seed pulses for both the pump laser and the white light<br />
generati<strong>on</strong> <strong>in</strong> the LA-OPCPA. Because a temporal jitter<br />
between the pump beam and the seed pulse should be<br />
reduced as possible. The oscillator generates few cycle<br />
pulses with the c<strong>on</strong>siderably broad spectral width. A seed<br />
pulse for the pump laser is temporally stretched and<br />
amplified by us<strong>in</strong>g fibre amplifiers after a pulse picker<br />
and a pulse shaper.<br />
A part <str<strong>on</strong>g>of</str<strong>on</strong>g> the amplified pulse is used as a seed pulse <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
a repeatable amplificati<strong>on</strong> system, which based <strong>on</strong> highpower<br />
diode-pumped solid-state lasers (DPSSL) with<br />
cryogenic Yb:YAG ceramics[3] as a laser material. The<br />
repeatable pump source generates green laser pulses with<br />
70 J pulse energy at 100 Hz repetiti<strong>on</strong> rate after sec<strong>on</strong>d<br />
harm<strong>on</strong>ic generati<strong>on</strong>. The other part <str<strong>on</strong>g>of</str<strong>on</strong>g> the amplified<br />
pulse is divided <strong>in</strong>to four beams and each <str<strong>on</strong>g>of</str<strong>on</strong>g> them seeded<br />
<strong>in</strong>to the s<strong>in</strong>gle-shot-based Nd:glass amplifier system.<br />
Pulse energy <strong>in</strong>creases up to 3 kJ per beam. A bundle <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
four beams supplies 12 kJ (ω) <strong>in</strong> 1 ns and is frequencydoubled<br />
at 8.4 kJ. Two bundles are prepared as pump<br />
sources.<br />
On the other hand, the generated white coherent light is<br />
used as a seed pulse for the OPCPA system. Several<br />
hundreds mili-joules pulse energy is obta<strong>in</strong>ed after fibre<br />
and small-power (SP-) OPCPA stages. Then, <strong>in</strong> the LA-<br />
OPCPA with partially deuterated DKDP crystals, kilojoule<br />
class pulse energy is obta<strong>in</strong>ed with a more than 200<br />
nm spectral width (FWHM). Us<strong>in</strong>g high-damagethreshold<br />
grat<strong>in</strong>gs and chirped mirrors <strong>in</strong> large aperture,<br />
0.1 EW (1 kJ/ 10 fs) pulse beam is generated and two set<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the pump beam and the seed pulse results <strong>in</strong> 0.2 EW.<br />
There are some key issues <strong>in</strong> “Gekko-EXA” laser<br />
design. Spectral dispersi<strong>on</strong> compensati<strong>on</strong> is c<strong>on</strong>siderably<br />
important to improve the temporal pulse c<strong>on</strong>trast ratio up<br />
to 10 12 to reduce pre-pulses and a pedestal.<br />
Technologically, there are two significant optics to be<br />
developed, a grat<strong>in</strong>g and a large aperture chirped mirror,<br />
which have both high optical damage strength <strong>in</strong> J/cm 2 <strong>in</strong><br />
an ultra-broadband up to 500 nm with a large aperture.<br />
Figure 7 shows an arrangement plan for the “Gekko-<br />
EXA” laser. The fr<strong>on</strong>t end and the 1 PW/ 100 Hz laser<br />
system are <strong>in</strong> another room, which is not shown here. The<br />
kJ-pump-laser is at the next to the “Gekko” laser system.<br />
In the additi<strong>on</strong>al planed build<strong>in</strong>g, the booster amplifiers<br />
for 12 kJ, the sec<strong>on</strong>d harm<strong>on</strong>ic generator and the kJ-class<br />
OPCPA system are <strong>on</strong> the same floor. Two pulse<br />
compressors are <strong>in</strong> fr<strong>on</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> the target chamber.<br />
REFERENCE<br />
[1] G. M. Heestand, C. A. Haynam, P. J. Wegner, M. W.<br />
Bowers, S. N. Dixit, G. V. Erbert, M. A. Henesian, M.<br />
R. Hermann, K. S. Jancaitis, K. Knittel, T. Kohut, J. D.<br />
L<strong>in</strong>dl, K. R. Manes, C. D. Marshall, N. C. Mehta, J.<br />
Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C.<br />
D. Orth, R. Patters<strong>on</strong>, R. A. Sacks, R. Saunders, M. J.<br />
Shaw, M. Spaeth, S. B. Sutt<strong>on</strong>, W. H. Williams, C. C.<br />
Widmayer, R. K. White, P. K. Whitman, S. T. Yang,<br />
and B. M. Van W<strong>on</strong>terghem, “Dem<strong>on</strong>strati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> highenergy<br />
2ω (526.5 nm) operati<strong>on</strong> <strong>on</strong> the Nati<strong>on</strong>al<br />
Igniti<strong>on</strong> Facility Laser System,” Applied Optics, Vol.<br />
47 Issue 19, pp.3494-3499 (2008).<br />
[2] N. Miyanaga, H. Azechi, K. A. Tanaka, T. Kanabe, T.<br />
Jitsuno, J. Kawanaka, Y. Fujimoto, R. Kodama, H.<br />
Shiraga, K. Knodo, K. Tsubakimoto, H. Habara, K.<br />
Sueda, H. Murakami, N. Morio, S. Matsuo, N.<br />
Sarukura, Y. Izawa, and K. Mima, “Technological<br />
Challenge and Activati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 10-kJ PW Laser LFEX<br />
for Fast Igniti<strong>on</strong> at ILE,” Fr<strong>on</strong>tiers <strong>in</strong> Optics (FiO)<br />
2008 paper: FWQ1.<br />
[3] J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R.<br />
Yasuhara, T. Kawashima, and H. Kan, “Highly<br />
Efficient Cryogenically_Cooled Yb:YAG Laser”,<br />
Laser <strong>Physics</strong> vol. 20, No. 1079-1084 (2010).
Abstract<br />
X-ray Emissi<strong>on</strong> from Magnetars and Its Physical Interpretati<strong>on</strong><br />
T. Enoto ∗ , KIPAC, Stanford University, CA, 94305-4085, USA †<br />
K. Makishima, Dep. <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, The University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo ‡ , and Suzaku Magnetar Team<br />
Recent astr<strong>on</strong>omy has been accumulat<strong>in</strong>g evidence that a<br />
subset <str<strong>on</strong>g>of</str<strong>on</strong>g> neutr<strong>on</strong> stars has an ultra str<strong>on</strong>g magnetic filed, ∼<br />
10 10−11 T. These enigmatic sources are called magnetars,<br />
and their field is believed to be str<strong>on</strong>ger than those (∼ 10 8<br />
T) <str<strong>on</strong>g>of</str<strong>on</strong>g> normal pulsars by 2–3 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitudes. S<strong>in</strong>ce<br />
such a field exceeds the QED critical field, 4.4 × 10 9 T,<br />
magnetars are a promis<strong>in</strong>g laboratory <strong>in</strong> the universe for<br />
the high-field physics. Us<strong>in</strong>g the Japanese astrophysical<br />
satellite Suzaku, we performed X-ray observati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> ∼9<br />
magnetars. We found a systematic trend <str<strong>on</strong>g>of</str<strong>on</strong>g> the wide-band<br />
X-ray spectra, which is c<strong>on</strong>sidered to be related with the<br />
str<strong>on</strong>g field physics (e.g., phot<strong>on</strong> splitt<strong>in</strong>g).<br />
MAGNETIC FIELD OF NEUTRON STARS<br />
Celestial objects, especially compact objects such as<br />
black holes and neutr<strong>on</strong> stars (NSs), are ideal laboratories<br />
to exam<strong>in</strong>e the extreme physics, which cannot be atta<strong>in</strong>ed<br />
through the ground experiments. For example, astr<strong>on</strong>omical<br />
observati<strong>on</strong>s have been revealed that NSs exhibit a<br />
str<strong>on</strong>g magnetic field (B ∼ 10 8 T), an <strong>in</strong>tense phot<strong>on</strong> field<br />
(its maximum lum<strong>in</strong>osity at L ∼ 10 38 erg s −1 ), and a high<br />
gravity field (its gravitati<strong>on</strong>al redshift at z ∼ 0.2). Thanks<br />
to recent sensitive detectors, we acquired an observati<strong>on</strong>al<br />
approach to study these exotic physics.<br />
Figure 1: An X-ray image <str<strong>on</strong>g>of</str<strong>on</strong>g> the Crab Nebula obta<strong>in</strong>ed by<br />
the Chandra observatory [1]. The Crab pulsar locates at<br />
the center <str<strong>on</strong>g>of</str<strong>on</strong>g> the Nebula.<br />
NSs are born after a gravitati<strong>on</strong>al collapse <str<strong>on</strong>g>of</str<strong>on</strong>g> a massive<br />
star. S<strong>in</strong>ce their mass and radii are estimated to be<br />
M ∼ 1.4 − 2.1M⊙ and R ∼ 10 km, this compact star is<br />
∗ JSPS (Japan Society for the Promoti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Science) Fellow, e-mail:<br />
enoto@stanford.edu<br />
† Address: Stanford University, Kavli Institute for Particle Astrophysics<br />
& Cosmology, <strong>Physics</strong> Astrophysics Build<strong>in</strong>g, 452 Lomita Mall,<br />
MC 4085, Stanford, CA 94305-4085, USA<br />
‡ Address: Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>,School <str<strong>on</strong>g>of</str<strong>on</strong>g> Science, The University<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo, 7-3-1 H<strong>on</strong>go, Bunkyo-ku, Tokyo, 113-0033, Japan<br />
believed to be supported by a degeneracy pressure <str<strong>on</strong>g>of</str<strong>on</strong>g> neutr<strong>on</strong>s.<br />
Electromagnetic waves from NSs have been detected<br />
by various wavelengths with their stellar rotati<strong>on</strong>s. Therefore,<br />
most NSs are called “pulsars”. For example, Figure<br />
1 shows an X-ray image <str<strong>on</strong>g>of</str<strong>on</strong>g> the Crab Nebula, which was<br />
formed at a historically recorded supernova <strong>in</strong> 1054. At the<br />
center <str<strong>on</strong>g>of</str<strong>on</strong>g> this nebula, the Crab pulsar is rotat<strong>in</strong>g at its period<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> P ∼ 33 ms and its period derivative <str<strong>on</strong>g>of</str<strong>on</strong>g> ˙ P ∼ 4.2×10−13 s s−1 . So far, pulsati<strong>on</strong>s and their derivatives have been<br />
measured from more than 1500 normal NSs, which are<br />
plotted <strong>on</strong> the P - ˙ P diagram <strong>in</strong> Figure 2, ma<strong>in</strong>ly distributed<br />
around P ∼ 0.4 s and ˙ P ∼ 10−15 s s−1 . S<strong>in</strong>ce most<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> normal pulsars exhibit a l<strong>on</strong>g-term tim<strong>in</strong>g stability, they<br />
are though to be isolated NSs1 Pdot (s/s)<br />
10 −8<br />
10 −9<br />
10 −10<br />
10 −11<br />
10 −12<br />
10 −13<br />
10 −14<br />
10 −15<br />
10 −16<br />
10 −17<br />
10 −18<br />
10 −19<br />
10 −20<br />
10 −21<br />
10 13 G<br />
10 12 G<br />
10 11 G<br />
10 10 G<br />
10 9 G<br />
10 14 G<br />
Bcr<br />
10 15 G<br />
10−3 10<br />
0.01 0.1 1 10<br />
−22<br />
Period (s)<br />
Figure 2: (a) A P - ˙<br />
P diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> pulsars, <strong>in</strong>clud<strong>in</strong>g<br />
rotati<strong>on</strong>-powered pulsars (black), SGRs (red stars),<br />
and AXPs (blue circles) after [2], http://www.atnf.csiro.<br />
au/research/pulsar/psrcat/. Dotted l<strong>in</strong>es <str<strong>on</strong>g>of</str<strong>on</strong>g> negative slopes<br />
represent c<strong>on</strong>stant magnetic field grids.<br />
Follow<strong>in</strong>g a standard pulsar model [3], emissi<strong>on</strong> from<br />
normal isolated NSs can be powered by their rotati<strong>on</strong>al<br />
energy loss. Let us assume the loss <str<strong>on</strong>g>of</str<strong>on</strong>g> the rotati<strong>on</strong> energy,<br />
d/dt(0.5IP 2 ) is eventually c<strong>on</strong>verted <strong>in</strong>to a magnetic<br />
dipole radiati<strong>on</strong> (e.g., electromagnetic wave), where<br />
I = 10 45 g cm 2 is a momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>ertia <str<strong>on</strong>g>of</str<strong>on</strong>g> the NS. Then,<br />
1 “Isolated” means that the NS is not a b<strong>in</strong>ary system (accreti<strong>on</strong>powered<br />
pulsars), where mass accreti<strong>on</strong> from a compani<strong>on</strong> star make the<br />
pulsati<strong>on</strong> fluctuate.
Radio Pulsar Number<br />
250<br />
200<br />
150<br />
100<br />
50<br />
0<br />
0<br />
7 8 9 10 11 12 13 14 15 16<br />
Magnetic Field Strength [Log B(Gauss)]<br />
Figure 3: Magnetic field distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> normal (radio) pulsars<br />
(blue, peak at ∼ 10 12 G) from [2], accreti<strong>on</strong>-powered<br />
pulsars (green), and magnetars (red, peak at ∼ 10 15 G) [4].<br />
1 T = 10 4 G.<br />
the surface magnetic field <str<strong>on</strong>g>of</str<strong>on</strong>g> the NS can be estimated to be<br />
B = 3.2 × 10 15√ (P ˙<br />
P ) T. (1)<br />
The evaluated fields are shown <strong>in</strong> Figure 3 and Fig. 2 as<br />
dashed l<strong>in</strong>es. Typical value <str<strong>on</strong>g>of</str<strong>on</strong>g> normal NSs, ∼ 10 8 T, is a<br />
basis <str<strong>on</strong>g>of</str<strong>on</strong>g> the current emissi<strong>on</strong> model from pulsars, and the<br />
model expla<strong>in</strong>s the observati<strong>on</strong>s so far.<br />
ULTRA-STRONG FIELD OF MAGNETARS<br />
Dur<strong>in</strong>g a last few decades, a new subclass <str<strong>on</strong>g>of</str<strong>on</strong>g> isolated<br />
NSs is emerg<strong>in</strong>g <strong>on</strong> the upper right corner <strong>on</strong> the P - ˙ P<br />
diagram, mostly observed <strong>on</strong>ly <strong>in</strong> the X-ray band. From<br />
their peculiar properties, the <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> them are named S<str<strong>on</strong>g>of</str<strong>on</strong>g>t<br />
Gamma Repeaters (SGRs) and the others are Anomalous<br />
X-ray Pulsars (AXPs). Both <str<strong>on</strong>g>of</str<strong>on</strong>g> them are slowly rotat<strong>in</strong>g<br />
X-ray pulsars (P ∼ 2 − 10 s), show<strong>in</strong>g large period derivatives<br />
( ˙ P ∼ 10−12 − 10−9 s s−1 ). Us<strong>in</strong>g the above estimati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field, SGRs and AXPs should have an<br />
ultra-str<strong>on</strong>g magnetic field, B ∼ 1010−11 T (Fig. 3). This<br />
field strength is str<strong>on</strong>ger than those <str<strong>on</strong>g>of</str<strong>on</strong>g> normal pulsars by 2–<br />
3 orders <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude, and even exceeds the QED critical<br />
field, Bcr = m2 ec3 /¯he = 4.4 × 109 T, where the Landau<br />
level <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong>s <strong>in</strong> the magnetic field becomes equivalent<br />
with the rest mass <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong>. If SGRs and AXPs have<br />
<strong>in</strong>deed the ultra-str<strong>on</strong>g field, they are promis<strong>in</strong>g targets to<br />
exam<strong>in</strong>e the QED physics. As shown <strong>in</strong> Fig. 3, SGRs and<br />
AXPs have a different field distributi<strong>on</strong> from those <str<strong>on</strong>g>of</str<strong>on</strong>g> normal<br />
rotati<strong>on</strong> powered pulsars, theay are correctively called<br />
“magnetars” [5, 6, 7].<br />
There are accumulated evidence that magnetars have an<br />
ultra-str<strong>on</strong>g field. First <str<strong>on</strong>g>of</str<strong>on</strong>g> all, bright X-ray lum<strong>in</strong>osity<br />
(Lx ∼ 1035 erg s−1 ) <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars cannot be expla<strong>in</strong>ed<br />
by their sp<strong>in</strong>-down power (L ∼ 1033 erg s−1 ). This is<br />
a prom<strong>in</strong>ent difference from normal pulsar, and thus, the<br />
emissi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetars are believed to be susta<strong>in</strong>ed by their<br />
huge stored magnetic energy. Sec<strong>on</strong>dly, magnetars exhibit<br />
various burst activities, which cannot be seen from normal<br />
pulsars so far. Sporadic burst emissi<strong>on</strong> occurres with a<br />
8<br />
6<br />
4<br />
2<br />
Magnetar & Accreti<strong>on</strong> Pulsar Number<br />
Figure 4: A light curve <str<strong>on</strong>g>of</str<strong>on</strong>g> the giant flare from SGR 1806-20<br />
<strong>in</strong> the 20–100 keV energy range, recorded by the RHESSI<br />
satellite <strong>on</strong> 2004 December 27. The <strong>in</strong>set shows a precursor<br />
which was recorded 142 s before the giant flare [8].<br />
keV 2 cm -2 s -1 keV -1<br />
1 mCrab<br />
1806-20 (2007 Oct.)<br />
1900+14 (2006 Apr.)<br />
1547-54 (2009 (2009 Jan.) Jan.)<br />
1841-04 (2006 (2006 Apr.) Apr.)<br />
1708-40 (2009 (2009 Aug.) Aug.)<br />
0501+45 (2008 (2008 Aug.) Aug.)<br />
0142+61 (2007 (2007 Aug.) Aug.)<br />
2259+58 2259+58 (2009 (2009 (2009 (2009 (2009 May) May) May) May) May)<br />
1 10 100<br />
Energy (keV)<br />
Figure 5: X-ray spectra (νFν form) <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetars observed<br />
by Suzaku with abbreviated names [12]. Individual<br />
spectra are shown with <str<strong>on</strong>g>of</str<strong>on</strong>g>fsets, and are arranged <strong>in</strong> order<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>creas<strong>in</strong>g magnetic field from bottom (weak) to top<br />
(str<strong>on</strong>g). Blue horiz<strong>on</strong>tal l<strong>in</strong>es <strong>in</strong>dicate a 1 mCrab <strong>in</strong>tensity.<br />
short time scale (a few hundread ms – a few s). One prom<strong>in</strong>ent<br />
example is a giant flare from SGRs, as dem<strong>on</strong>strated <strong>in</strong><br />
Figure 4. A current hypothesis speculates that these bursts<br />
are emissi<strong>on</strong> from trapped fire balls <strong>on</strong> the magnetar surface,<br />
presumably related with magnetic activities (e.g., rec<strong>on</strong>necti<strong>on</strong>s).<br />
Interest<strong>in</strong>gly, the lum<strong>in</strong>osities <str<strong>on</strong>g>of</str<strong>on</strong>g> these burst
sometimes exceed the Edd<strong>in</strong>gt<strong>on</strong> lum<strong>in</strong>osity (a maximum<br />
permitted lum<strong>in</strong>osity), and this excess is c<strong>on</strong>sidered to orig<strong>in</strong>ate<br />
from a suppressi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Thoms<strong>on</strong> cross secti<strong>on</strong> <strong>in</strong><br />
the high magnetic field. Thirdly, there is marg<strong>in</strong>al evidence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> prot<strong>on</strong> cyclotr<strong>on</strong>s <strong>in</strong> the magnetar X-ray spectra, which<br />
suggests B ∼ 10 15 G [9].<br />
X-RAY EMISSION OF MAGNETARS<br />
Persistent X-ray emissi<strong>on</strong> from magnetars have been extensively<br />
observed <strong>in</strong> the ∼0.2–10 keV. In this energy band,<br />
a thermal emissi<strong>on</strong> with its temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> kT ∼ 0.5 keV is<br />
c<strong>on</strong>sidered to be emitted from the stellar surface. However,<br />
through a new hard X-ray imag<strong>in</strong>g with the INTEGRAL<br />
satellite, some persistently bright magnetars were discovered<br />
to emit a dist<strong>in</strong>ct hard-tail comp<strong>on</strong>ent which emerges<br />
above ∼10 keV with an extremely hard phot<strong>on</strong> <strong>in</strong>dex <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Γh ∼ 1 [10]. This unusual new comp<strong>on</strong>ent, though not yet<br />
observed from all magnetars, is expected to provide a h<strong>in</strong>t<br />
to the high field physics <strong>in</strong> magnetars.<br />
Us<strong>in</strong>g the Suzaku X-ray satellite [11], we performed<br />
broad-band (0.8–70 keV) spectral analyses <str<strong>on</strong>g>of</str<strong>on</strong>g> the persistent<br />
X-ray emissi<strong>on</strong> from 9 magnetars [12]. As shown <strong>in</strong><br />
Figure 5, the s<str<strong>on</strong>g>of</str<strong>on</strong>g>t thermal comp<strong>on</strong>ent was detected from all<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> them (green l<strong>in</strong>es), while the hard-tail comp<strong>on</strong>ent, dom<strong>in</strong>at<strong>in</strong>g<br />
above ∼10 keV, was detected at ∼1 mCrab 2 <strong>in</strong>tensity<br />
from 7 <str<strong>on</strong>g>of</str<strong>on</strong>g> them (red l<strong>in</strong>es). In additi<strong>on</strong>, as shown <strong>in</strong> this<br />
figure, the hard-tail comp<strong>on</strong>ent has a tendency to become<br />
str<strong>on</strong>ger but s<str<strong>on</strong>g>of</str<strong>on</strong>g>ter towards sources with larger magnetic<br />
fields. To quantitatively evaluate this trend, we employ the<br />
1–60 keV fluxes <str<strong>on</strong>g>of</str<strong>on</strong>g> s<str<strong>on</strong>g>of</str<strong>on</strong>g>t-thermal and hard-tail comp<strong>on</strong>ents,<br />
Fs and Fh, respectively, and then, calculate the hardness<br />
ratio (HR) between these two comp<strong>on</strong>ents, ξ ≡ Fh/Fs. As<br />
shown <strong>in</strong> Figure 6 (top), the HR is found to be tightly correlated<br />
with the magnetic field B as<br />
ξ = (0.09 ± 0.07) × (B/Bcr) 1.2±0.2<br />
with a correlati<strong>on</strong> coefficient <str<strong>on</strong>g>of</str<strong>on</strong>g> 0.873, over the range from<br />
ξ ∼ 10 to ξ ∼ 0.1. On the other hand, as shown <strong>in</strong> Figure 6<br />
(bottom), the phot<strong>on</strong> <strong>in</strong>dex becomes s<str<strong>on</strong>g>of</str<strong>on</strong>g>ter toward str<strong>on</strong>ger<br />
field pulsars with ξ becom<strong>in</strong>g larger.<br />
Although several scenarios have been proposed [13, 14,<br />
15, 16], the emissi<strong>on</strong> mechanism <str<strong>on</strong>g>of</str<strong>on</strong>g> the hard X-rays has<br />
not yet been resolved. One <str<strong>on</strong>g>of</str<strong>on</strong>g> the biggest difficulties is<br />
how to expla<strong>in</strong> the extremely hard Γh ∼ 1 with its spectral<br />
trend depend<strong>in</strong>g <strong>on</strong> B. A possible candidate <str<strong>on</strong>g>of</str<strong>on</strong>g> the emissi<strong>on</strong><br />
process is phot<strong>on</strong>-splitt<strong>in</strong>gs [17, 18]. In ultra-str<strong>on</strong>g<br />
magnetic fields exceed<strong>in</strong>g Bcr, electr<strong>on</strong>-positr<strong>on</strong> pair cascades<br />
are suppressed, while the phot<strong>on</strong> splitt<strong>in</strong>g may be<br />
dom<strong>in</strong>ant. In this case, gamma-rays from the surface may<br />
repeatedly split <strong>in</strong>to lower energy phot<strong>on</strong>s. This process<br />
can also expla<strong>in</strong> the differences <strong>in</strong> Γh am<strong>on</strong>g magnetars, <strong>in</strong><br />
such a way that higher fields objects will allow the phot<strong>on</strong>splitt<strong>in</strong>g<br />
cascade to proceed down to lower energies, and<br />
hence to make the c<strong>on</strong>t<strong>in</strong>uum s<str<strong>on</strong>g>of</str<strong>on</strong>g>ter.<br />
2 1 mCrab is <strong>on</strong>e-thousandth <str<strong>on</strong>g>of</str<strong>on</strong>g> the Crab Nebula <strong>in</strong>tensity, which is a<br />
standard candle <str<strong>on</strong>g>of</str<strong>on</strong>g> the astr<strong>on</strong>omy.<br />
(2)<br />
Hardness Ratio ξ = F h /F s<br />
Hardness Ratio ξ = F h /F s<br />
10<br />
1<br />
0.1<br />
10<br />
1<br />
0.1<br />
2259+58<br />
(b)<br />
0142+61<br />
0501+45<br />
(2009)<br />
0501+45<br />
(2008)<br />
0142+61<br />
1547-54<br />
1900+14<br />
1708-40<br />
1841-04<br />
10<br />
Magnetic Field (G)<br />
14 1015 0501+45<br />
1806-20<br />
1841-04<br />
1900+14<br />
1806-20<br />
0 0.5 1 1.5 2<br />
Phot<strong>on</strong> <strong>in</strong>dex Γ h<br />
1708-40<br />
1547-54<br />
Figure 6: (top) A correlati<strong>on</strong> between the HR ξ and the<br />
magnetic field B [12]. Green solid l<strong>in</strong>e represents the best<br />
fit <str<strong>on</strong>g>of</str<strong>on</strong>g> equati<strong>on</strong> (2). SGRs and AXPs are shown <strong>in</strong> red and<br />
blue, respectively. (bottom) The HR ξ as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong><br />
<strong>in</strong>dices Γh <str<strong>on</strong>g>of</str<strong>on</strong>g> the hard-tail comp<strong>on</strong>ent [12].<br />
REFERENCES<br />
[1] Hester, J. J., et al. 2002, Astrophys. J. Let., 577, L49<br />
[2] Manchester, R. N., et al., 2005, VizieR Onl<strong>in</strong>e Data Catalog<br />
[3] Goldreich, P., & Julian, W. H. 1969, Astrophys. J., 157, 869<br />
[4] Enoto T., et al., 2009, Suzaku <str<strong>on</strong>g>C<strong>on</strong>ference</str<strong>on</strong>g> 2009 Proceedng<br />
[5] Duncan, R. C., & Thomps<strong>on</strong>, C. 1992, ApJL, 392, L9<br />
[6] Woods, P. M., & Thomps<strong>on</strong>, C. 2006, Compact stellar X-ray<br />
sources, 547<br />
[7] Mereghetti, S. 2008, Astr<strong>on</strong>. and Astrophy. Rev., 15, 225<br />
[8] Hurley, K., et al. 2005, Nature, 434, 1098<br />
[9] Ibrahim, A. I., et al., 2003, ApJL, 584, L17<br />
[10] Kuiper, L., et al., 2006, Astrophysical Journal, 645, 556<br />
[11] Mitsuda, K., et al. 2007, PASJ, 59, 1<br />
[12] Enoto, T., et al., 2010, ApJL, 722, L162<br />
[13] Heyl, J. S., & Hernquist, L. 2005, MNRAS, 362, 777<br />
[14] Thomps<strong>on</strong>, C., & Beloborodov, A. M. 2005, ApJ, 634, 565<br />
[15] Beloborodov, A. M., & Thomps<strong>on</strong>, C. 2007, ApJ, 657, 967<br />
[16] Bar<strong>in</strong>g, M. G., & Hard<strong>in</strong>g, A. K. 2001, ApJ, 547, 929<br />
[17] Bar<strong>in</strong>g, M. G., & Hard<strong>in</strong>g, A. K. 2001, ApJ., 547, 929<br />
[18] Enoto, T., et al. 2010, Ph.D thesis, The University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo
Abstract<br />
THE NIELSEN-OLESEN INSTABILITIES IN THE GLASMA<br />
H. Fujii, Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo, Komaba, Tokyo<br />
K. Itakura, Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Particle and Nuclear Studies (IPNS), <strong>KEK</strong>, Ibaraki<br />
A. Iwazaki, Int’l Ec<strong>on</strong>omics and Politics, Nishogakusha University, Kashiwa, Chiba<br />
In the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> the Color Glass C<strong>on</strong>densate, str<strong>on</strong>g<br />
color electric and magnetic fields are expected to appear <strong>in</strong><br />
the early transient stage <str<strong>on</strong>g>of</str<strong>on</strong>g> ultrarelativistic heavy-i<strong>on</strong> collisi<strong>on</strong>s.<br />
We show that this c<strong>on</strong>figurati<strong>on</strong> with l<strong>on</strong>gitud<strong>in</strong>ally<br />
polarized str<strong>on</strong>g gauge fields has the Nielsen-Olesen<br />
<strong>in</strong>stability[1], and discuss its relevance to the <strong>in</strong>itial stage<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the heavy i<strong>on</strong> collisi<strong>on</strong>s[2, 3, 4].<br />
STRONG COLOR FIELD IN GLASMA<br />
In ultrarelativistic heavy-i<strong>on</strong> collisi<strong>on</strong>s (HIC), two heavy<br />
nuclei at nearly the light speed smash <strong>in</strong>to each other to<br />
generate a dense medium <str<strong>on</strong>g>of</str<strong>on</strong>g> liberated quarks and glu<strong>on</strong>s.<br />
Study<strong>in</strong>g properties <str<strong>on</strong>g>of</str<strong>on</strong>g> this extremely dense medium is <strong>on</strong>e<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the ma<strong>in</strong> subjects <strong>in</strong> subnuclear physics.<br />
The Color Glass C<strong>on</strong>densate picture<br />
Each <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>cident nuclei is seen as an assembly <str<strong>on</strong>g>of</str<strong>on</strong>g> part<strong>on</strong>s<br />
(quarks and glu<strong>on</strong>s) with l<strong>on</strong>gitud<strong>in</strong>al momentum fracti<strong>on</strong><br />
x = p/( √ s/2) <strong>in</strong> HIC at very large center-<str<strong>on</strong>g>of</str<strong>on</strong>g>-mass energy<br />
√ s. At high energies, particle producti<strong>on</strong>s are dom<strong>in</strong>ated<br />
by collisi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the part<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> small x such as x √ s/2 ∼<br />
mπ with mπ the pi<strong>on</strong> mass because these small-x part<strong>on</strong>s,<br />
predom<strong>in</strong>antly glu<strong>on</strong>s, are abundant through QCD<br />
bremsstrahlung processes from the larger-x colored part<strong>on</strong>s<br />
with<strong>in</strong> the nuclear wavefuncti<strong>on</strong> and because the QCD<br />
cross secti<strong>on</strong> is larger at the smaller momentum scale.<br />
Such dense small-x glu<strong>on</strong> comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> the nuclear<br />
wavefuncti<strong>on</strong> should be described as classical gauge fields<br />
generated from larger-x color source, and the effective<br />
theory for small-x degrees <str<strong>on</strong>g>of</str<strong>on</strong>g> hadr<strong>on</strong>ic wavefuncti<strong>on</strong> is<br />
known as the Color Glass C<strong>on</strong>densate (CGC) framework.<br />
The field strength is characterized by a momentum scale,<br />
Q 2 s(x), called saturati<strong>on</strong> scale, which emerges from n<strong>on</strong>l<strong>in</strong>ear<br />
effects <str<strong>on</strong>g>of</str<strong>on</strong>g> QCD and estimated empirically as about<br />
1 (GeV/c) 2 <strong>in</strong> HIC at √ s = 200 GeV. Keep <strong>in</strong> m<strong>in</strong>d that<br />
any hadr<strong>on</strong>s (nuclei) are color s<strong>in</strong>glet objects and the (transversely<br />
polarized) color fields exist <strong>on</strong>ly with<strong>in</strong> the hadr<strong>on</strong><br />
as fluctuati<strong>on</strong>s whose lifetime is Lorentz-el<strong>on</strong>gated.<br />
S<strong>in</strong>ce the particle producti<strong>on</strong>s are dom<strong>in</strong>ated by the<br />
small-x part<strong>on</strong>s <strong>in</strong> the high-energy limit, nucleus-nucleus<br />
collisi<strong>on</strong>s then may be effectively modeled as CGC-CGC<br />
collisi<strong>on</strong>s where the classical fields coupled with the charge<br />
sources <strong>on</strong> the light-c<strong>on</strong>e <strong>in</strong>tersect with each other. It is<br />
shown that cross<strong>in</strong>g the two CGCs with each other results<br />
<strong>in</strong> l<strong>on</strong>gitud<strong>in</strong>ally polarized color electric and magnetic<br />
fields due to n<strong>on</strong>-Abelian nature <str<strong>on</strong>g>of</str<strong>on</strong>g> QCD. The term<br />
x –<br />
τ = c<strong>on</strong>st<br />
t<br />
η=c<strong>on</strong>st<br />
A µ<br />
=pure gauge Aµ =pure gauge<br />
(1) (2)<br />
CGC (1)<br />
QGP<br />
Glasma<br />
CGC (2)<br />
Figure 1: CGC-CGC collisi<strong>on</strong>. Str<strong>on</strong>g fields c<strong>on</strong>f<strong>in</strong>ed <strong>in</strong><br />
each nucleus before the collisi<strong>on</strong>, and boost-<strong>in</strong>variantly extend<br />
between the two reced<strong>in</strong>g nuclei after the collisi<strong>on</strong>.<br />
τ = √ t 2 − z 2 and η = (1/2) ln((t + z)/(t − z)).<br />
Glasma[5] was recently co<strong>in</strong>ed for this transient c<strong>on</strong>figurati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g fields produced from the CGC, which evolves<br />
toward the QCD plasma. This is a l<strong>on</strong>gitud<strong>in</strong>al-boost <strong>in</strong>variant<br />
c<strong>on</strong>figurati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> very str<strong>on</strong>g classical fields. See<br />
Fig. 1, where <strong>in</strong>com<strong>in</strong>g nuclei are modeled as two CGCs.<br />
Expand<strong>in</strong>g flux tube<br />
With a simple Abelian Ansatz for a purely magnetic<br />
flux tube with l<strong>on</strong>gitud<strong>in</strong>al polarizati<strong>on</strong>, <strong>on</strong>e can f<strong>in</strong>d an<br />
analytic soluti<strong>on</strong>[3], which stretches al<strong>on</strong>g <strong>in</strong> the z directi<strong>on</strong><br />
and expands transversely, as shown <strong>in</strong> Fig. 2. There<br />
2<br />
1<br />
-2 -1 1 2<br />
-1<br />
-2<br />
Qsr<br />
x +<br />
z<br />
Qsz<br />
Figure 2: Magnetic flux tube pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile at Qsτ = 1, 2.<br />
are <strong>on</strong>ly two n<strong>on</strong>vanish<strong>in</strong>g comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> field strengths,<br />
l<strong>on</strong>gitud<strong>in</strong>al Bz and transverse ET . The time evoluti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the field strengths <strong>in</strong>tegrated over the transverse plane<br />
at z = 0 is plotted <strong>in</strong> Fig. 3. The stress tensor at
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
B 2 z<br />
E 2 T<br />
0.5 1 1.5 2 2.5 3<br />
Qsτ<br />
Figure 3: Evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the field strength as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Qsτ. Evoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a purely electric flux tube is obta<strong>in</strong>ed by<br />
exchang<strong>in</strong>g E and B.<br />
τ = 0+ is found as diag(E, E, E, −E) with energy density<br />
E, which is quite different from the equilibrium form<br />
diag(E, E/3, E/3, E/3). Note that the l<strong>on</strong>gitud<strong>in</strong>al pressure<br />
∝ E 2 T − B2 z is always negative and <strong>on</strong>ly approaches<br />
to zero at later times, while the transverse pressure ∝ B 2 z is<br />
positive.<br />
How will this highly anisotropic c<strong>on</strong>figurati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g<br />
fields evolve <strong>in</strong>to locally thermalized plasma — quark<br />
glu<strong>on</strong> plasma (QGP)? Here we argue that the <strong>in</strong>itial<br />
glasma c<strong>on</strong>figurati<strong>on</strong> with color magnetic fields has unstable<br />
modes, which play a role <strong>in</strong> the system evoluti<strong>on</strong>.<br />
NIELSEN-OLESEN INSTABILITY<br />
S<strong>in</strong>ce late 70s it has been known that a uniform magnetic<br />
field c<strong>on</strong>figurati<strong>on</strong> <strong>in</strong> SU(2) gauge theory is unstable — the<br />
Nielsen-Olesen (N-O) <strong>in</strong>stability[1], which we review here.<br />
We decompose the SU(2) gauge fields <strong>in</strong>to diag<strong>on</strong>al and<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g>f-diag<strong>on</strong>al parts, and assume a homogeneous magnetic<br />
field <strong>in</strong> the 3rd color directi<strong>on</strong>:<br />
L = − 1<br />
4 fµνf µν − 1<br />
2 |Dµφν − Dνφµ| 2 + igf µν φ ∗ µφν<br />
+ 1<br />
4 g2 (φµφ ∗ ν − φνφ ∗ µ) 2 , (1)<br />
where Aµ ≡ A 3 µ, φµ ≡ (A 1 µ + iA 2 µ)/ √ 2 and fµν =<br />
∂µAν − ∂νAµ. D = ∂µ + igAµ is the covariant derivative<br />
w.r.t. Aµ. Regard<strong>in</strong>g Aµ gauge theory, the <str<strong>on</strong>g>of</str<strong>on</strong>g>f-diag<strong>on</strong>al<br />
parts φµ behave as charged massless vector fields. They<br />
form the Landau levels EN = (2N +1)gB (N = 0, 1, · · · )<br />
<strong>in</strong> a c<strong>on</strong>stant magnetic field background B with<strong>in</strong> the l<strong>in</strong>ear<br />
approximati<strong>on</strong> with φ be<strong>in</strong>g small fluctuati<strong>on</strong>s. However,<br />
the most important observati<strong>on</strong> is that the third term<br />
<strong>in</strong> Eq. (1) gives the Zeeman splitt<strong>in</strong>g ±2gB for sp<strong>in</strong> ±, result<strong>in</strong>g<br />
<strong>in</strong> the mode energy<br />
ω 2 = p 2 z + (2N + 1)gB ± 2gB , (2)<br />
where we see that the lowest mode has the mass term with<br />
a wr<strong>on</strong>g sign −gB. This is the N-O <strong>in</strong>stability. The growth<br />
rate <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>stability depends <strong>on</strong> the l<strong>on</strong>gitud<strong>in</strong>al momentum<br />
pz as γ = √ gB − p 2 z. The <strong>in</strong>stability appears even at<br />
pz = 0, which is qualitative difference from the pz dependence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Weibel <strong>in</strong>stability known <strong>in</strong> plasma physics.<br />
Figure 4: Schematic picture for glasma with fluctuati<strong>on</strong>s,<br />
between two reced<strong>in</strong>g nuclei.<br />
GLASMA INSTABILITIES<br />
As previously menti<strong>on</strong>ed, the boost-<strong>in</strong>variant background<br />
never acquires positive l<strong>on</strong>gitud<strong>in</strong>al pressure,<br />
which is necessary for system isotropizati<strong>on</strong>. We c<strong>on</strong>sider<br />
here small fluctuati<strong>on</strong>s φ around the background field <strong>in</strong> the<br />
τ-η coord<strong>in</strong>ates. For simplicity, we neglect the transverse<br />
structure <str<strong>on</strong>g>of</str<strong>on</strong>g> the collisi<strong>on</strong> event and assume transversely a<br />
uniform magnetic background B, even though, more realistically,<br />
the background field should have transverse structures<br />
characterized by the scale Qs, and an electric field <strong>in</strong><br />
the background is also allowed.<br />
By comb<strong>in</strong><strong>in</strong>g the fluctuat<strong>in</strong>g fields φµ(τ, η, x⊥) as<br />
φ ± ≡ 1<br />
√ 2 (φ1 ± iφ2) , (3)<br />
we f<strong>in</strong>d the mode equati<strong>on</strong> for φ ± [2, 3],<br />
(<br />
)<br />
˜φ ± = 0 , (4)<br />
1<br />
τ ∂τ (τ∂τ ˜ φ ± ) +<br />
EN ± 2gB + ν2<br />
τ 2<br />
where ˜ φ is the mode with N specify<strong>in</strong>g the Landau level<br />
and with ν the momentum c<strong>on</strong>jugate to η. There is another<br />
equati<strong>on</strong> for φ ∗ which has the opposite charge to φ.<br />
• First we note that the sp<strong>in</strong>-+ mode with ν = 0 <strong>in</strong> the<br />
lowest Landau level N = 0 has a negative mass −gB<br />
just as <strong>in</strong> the previous secti<strong>on</strong>. There exists the N-O<br />
<strong>in</strong>stability <strong>in</strong> the glasma.<br />
• N<strong>on</strong>zero ν modes are stabilized by the term (ν/τ) 2 .<br />
For ν is dimensi<strong>on</strong>less, physical l<strong>on</strong>gitud<strong>in</strong>al momentum<br />
corresp<strong>on</strong>ds to pz ∼ ν/τ[6], which is large at<br />
small τ with fixed ν. The maximum value νmax for<br />
the <strong>in</strong>stability is determ<strong>in</strong>ed by<br />
νmax = τ √ gB ∼ τQs . (5)<br />
In fact, these po<strong>in</strong>ts were observed <strong>in</strong> a numerical simulati<strong>on</strong><br />
for the glasma[7], although the proporti<strong>on</strong>ality c<strong>on</strong>stant<br />
<strong>in</strong> the sec<strong>on</strong>d po<strong>in</strong>t was found very small there. At<br />
the same time, we notice that the expansi<strong>on</strong> generally delays<br />
the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>stability.
GLASMA INSTABILITIES IN A BOX<br />
Recently, classical statistical simulati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Abelian<br />
gauge theories were performed <strong>in</strong> a box without expansi<strong>on</strong><br />
but with very anisotropic <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong>s[8], motivated by<br />
the idea <str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma. They found the unstable behavior<br />
whose growth rate is shown <strong>in</strong> Fig. 5 as “primary.” Furthermore,<br />
they claimed that there is a sec<strong>on</strong>dary <strong>in</strong>stability<br />
which is triggered by the first (primary) <strong>in</strong>stability.<br />
The pz dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> the growth rate γ <str<strong>on</strong>g>of</str<strong>on</strong>g> the primary<br />
<strong>in</strong>stability looks c<strong>on</strong>sistent with the N-O <strong>in</strong>stability; γ is<br />
n<strong>on</strong>zero at zero pz and decreases as pz <strong>in</strong>creases. On the<br />
other hand, the growth rate γ <str<strong>on</strong>g>of</str<strong>on</strong>g> the sec<strong>on</strong>dary <strong>in</strong>creases<br />
with pz, and extends to the larger pz regi<strong>on</strong>. In Ref. [8], the<br />
sec<strong>on</strong>dary was discussed as n<strong>on</strong>l<strong>in</strong>ear effects <str<strong>on</strong>g>of</str<strong>on</strong>g> the primary<br />
<strong>in</strong>stability <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the re-summed self-energy diagrams.<br />
Here, we attempt to <strong>in</strong>terpret their f<strong>in</strong>d<strong>in</strong>gs from the viewpo<strong>in</strong>t<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the N-O <strong>in</strong>stability scenario[4].<br />
The numerical simulati<strong>on</strong> starts with the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong><br />
very smooth <strong>in</strong> the z directi<strong>on</strong> but randomized <strong>in</strong> the x-y directi<strong>on</strong>s<br />
with the scale ∆ ∼ Qs. The color electric fields<br />
are set to zero at the <strong>in</strong>itial time. This c<strong>on</strong>figurati<strong>on</strong> will<br />
allow tube structures <str<strong>on</strong>g>of</str<strong>on</strong>g> the color magnetic fields al<strong>on</strong>g the<br />
z directi<strong>on</strong>, and therefore the N-O <strong>in</strong>stability is expected<br />
there as the primary[4]. If the magnetic field po<strong>in</strong>t to the<br />
3rd directi<strong>on</strong> <strong>in</strong> the SU(2) color space, the <str<strong>on</strong>g>of</str<strong>on</strong>g>f-diag<strong>on</strong>al<br />
comp<strong>on</strong>ents φ ± w.r.t. the magnetic field are amplified <strong>in</strong><br />
time due to the N-O <strong>in</strong>stability.<br />
Because φ ± are charged fields, they can also <strong>in</strong>duce<br />
the color electric current al<strong>on</strong>g the magnetic field. Although<br />
the directi<strong>on</strong> and magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric current<br />
cannot be determ<strong>in</strong>ed with<strong>in</strong> the l<strong>in</strong>ear analysis, <strong>on</strong>e<br />
may roughly estimate the strength <str<strong>on</strong>g>of</str<strong>on</strong>g> the current as J z ∼<br />
O(g · √ gB · B/g) = O((gB) 3/2 /g) ∼ O(Q3 s/g) with<br />
the momentum scale √ gB ∼ Qs and the field amplitude<br />
φ ± ∼ √ B/g. Accord<strong>in</strong>g to the Ampère law, the color<br />
magnetic field <str<strong>on</strong>g>of</str<strong>on</strong>g> order O(Q2 s/g) will be generated around<br />
this current. Let us exam<strong>in</strong>e the c<strong>on</strong>sequence <str<strong>on</strong>g>of</str<strong>on</strong>g> this azimuthal<br />
magnetic field.<br />
Follow<strong>in</strong>g the same analysis as the N-O <strong>in</strong>stability but <strong>in</strong><br />
the cyl<strong>in</strong>drical coord<strong>in</strong>ates with a c<strong>on</strong>stant background Bθ ,<br />
we f<strong>in</strong>d the mode equati<strong>on</strong> for the <str<strong>on</strong>g>of</str<strong>on</strong>g>f-diag<strong>on</strong>al comp<strong>on</strong>ent<br />
ϕ ± = (φz ± iφr )/ √ 2 as<br />
{<br />
− 1 d d<br />
r<br />
r dr dr + ( pz − gB θ r )2 1<br />
+ − 2gBθ} ϕ<br />
2r2 − (r)<br />
=ω 2 ϕ − (r) , (6)<br />
assum<strong>in</strong>g that ϕ + is small compared to ϕ − and sett<strong>in</strong>g<br />
ϕ + = 0. The parameter pz shifts the potential m<strong>in</strong>imum<br />
outward. Thus, for larger pz the N-O <strong>in</strong>stability appears as<br />
<strong>in</strong> the orig<strong>in</strong>al case, while 1/r 2 term cannot be neglected<br />
for smaller pz — the growth rate is approximately<br />
−ω 2 ≡ γ 2 ∼ gB θ<br />
(<br />
1 − gBθ<br />
4p2 )<br />
z<br />
. (7)<br />
This gives rise to qualitatively the same behavior as γ for<br />
the sec<strong>on</strong>dary shown <strong>in</strong> Fig. 5.<br />
γ / ε 1/4<br />
1.0<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
0.0 0.5 1.0 1.5 2.0 2.5<br />
pz / ε 1/4<br />
prim. 96 3<br />
prim. 128 3<br />
sec. 96 3<br />
sec. 128 3<br />
Figure 5: Growth rates <str<strong>on</strong>g>of</str<strong>on</strong>g> the primary and sec<strong>on</strong>dary <strong>in</strong>stabilities<br />
vs l<strong>on</strong>gitud<strong>in</strong>al momentum pz, observed <strong>in</strong> Ref. [8]<br />
(<strong>in</strong> units <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>itial energy density ε).<br />
OUTLOOK<br />
In ultrarelativistic HIC, a system <str<strong>on</strong>g>of</str<strong>on</strong>g> extremely str<strong>on</strong>g<br />
color fields, glasma, is generated. N<strong>on</strong>-equilibrium dynamics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the glasma itself is very <strong>in</strong>terest<strong>in</strong>g and at the same<br />
time is quite important to understand the producti<strong>on</strong> mechanism<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> QGP. We have discussed the Nielsen-Olesen <strong>in</strong>stabilities<br />
<strong>in</strong> the c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g> HIC, and shown that results <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the numerical simulati<strong>on</strong>s[7, 8] can be understood nicely<br />
from this viewpo<strong>in</strong>t.<br />
The Nielsen-Olesen <strong>in</strong>stability certa<strong>in</strong>ly exists <strong>in</strong> n<strong>on</strong>-<br />
Abelian gauge theories. Its relevance to “thermalizati<strong>on</strong>”<br />
mechanism <strong>in</strong> HIC, however, is under debate. The Weibel<br />
<strong>in</strong>stability is another possibility am<strong>on</strong>g others. Towards understand<strong>in</strong>g<br />
the early stage <str<strong>on</strong>g>of</str<strong>on</strong>g> HIC, <strong>on</strong>e needs obviously to<br />
go bey<strong>on</strong>d the l<strong>in</strong>ear analysis and has to study the n<strong>on</strong>l<strong>in</strong>ear<br />
dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> gauge field. To this end, it is <strong>in</strong>dispensable to<br />
pursue more detailed numerical simulati<strong>on</strong>s together with<br />
theoretical physics <strong>in</strong>sights.<br />
REFERENCES<br />
[1] N.K. Nielsen and P. Olesen, Nucl. Phys. B 144 376 (1978).<br />
[2] A. Iwazaki, Prog. Theor. Phys. 121 809 (2009).<br />
[3] H. Fujii and K. Itakura, Nucl. Phys. A 809 88 (2008).<br />
[4] H. Fujii, K. Itakura and A. Iwazaki, Nucl. Phys. A 828 178<br />
(2009).<br />
[5] T. Lappi and L. McLerran, Nucl. Phys. A 772 200 (2006).<br />
[6] N. Tanji, arXiv:arXiv:1010.4516 [hep-ph].<br />
[7] P. Romatschke and R. Venugopalan, Phys. Rev. Lett. 96<br />
045011 (2006).<br />
[8] J. Berges, S. Scheffler and D. Sexty, Phys. Rev. D 77 034504<br />
(2008).
First order quantum correcti<strong>on</strong> to the Larmor radiati<strong>on</strong> ∗<br />
Gen Nakamura<br />
Department <str<strong>on</strong>g>of</str<strong>on</strong>g> Physical Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan<br />
Abstract<br />
First-order quantum correcti<strong>on</strong> to the Larmor radiati<strong>on</strong> is<br />
<strong>in</strong>vestigated <strong>on</strong> the basis <str<strong>on</strong>g>of</str<strong>on</strong>g> the scalar QED <strong>on</strong> a homogeneous<br />
background <str<strong>on</strong>g>of</str<strong>on</strong>g> a time-dependent electric field, which<br />
is a generalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a recent work by Higuchi and Walker<br />
so as to be extended for an accelerated charged particle <strong>in</strong><br />
a relativistic moti<strong>on</strong>. We obta<strong>in</strong> a simple approximate formula<br />
for the quantum correcti<strong>on</strong> <strong>in</strong> the limit <str<strong>on</strong>g>of</str<strong>on</strong>g> the relativistic<br />
moti<strong>on</strong> when the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle moti<strong>on</strong> is<br />
parallel to that <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric field.<br />
INTRODUCTION<br />
The Larmor radiati<strong>on</strong> is the classical radiati<strong>on</strong> from a<br />
charged particle <strong>in</strong> an accelerated moti<strong>on</strong> [2]. In the recent<br />
paper by Higuchi and Walker [3], the quantum correcti<strong>on</strong><br />
to the Larmor radiati<strong>on</strong> is <strong>in</strong>vestigated <strong>on</strong> the basis<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the scalar quantum electrodynamics (QED). In their<br />
approach, the mode functi<strong>on</strong> for the complex scalar field<br />
is c<strong>on</strong>structed with the Wentzel-Kramers-Brillou<strong>in</strong> (WKB)<br />
approximati<strong>on</strong>, <strong>in</strong> a form expanded with respect to ¯h. In a<br />
series <str<strong>on</strong>g>of</str<strong>on</strong>g> Higuchi and Mart<strong>in</strong>’s work [4, 5, 6] (see also references<br />
there<strong>in</strong>), it has been well understood that the mode<br />
functi<strong>on</strong> reproduces the classical Larmor formula when the<br />
radiati<strong>on</strong> energy is evaluated at the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ¯h 0 . The firstorder<br />
quantum correcti<strong>on</strong> to the classical Larmor radiati<strong>on</strong><br />
is evaluated at the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ¯h <strong>in</strong> Ref. [3], though the <strong>in</strong>vestigati<strong>on</strong><br />
is limited to the n<strong>on</strong>-relativistic moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
charged particle.<br />
In our work, we c<strong>on</strong>sider a simple generalizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Higuchi and Walker’s work [3], <strong>in</strong> order to <strong>in</strong>vestigate the<br />
case a relativistic moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an accelerated charge. Assum<strong>in</strong>g<br />
a homogeneous but time-vary<strong>in</strong>g background <str<strong>on</strong>g>of</str<strong>on</strong>g> electric<br />
field, we derive a formula for the radiati<strong>on</strong> energy <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ¯h, the first-order correcti<strong>on</strong> due to the quantum<br />
effect. This generalized formula is applicable to the<br />
accelerated charge <strong>in</strong> a relativistic moti<strong>on</strong>, and we focus<br />
our <strong>in</strong>vestigati<strong>on</strong> <strong>on</strong> the first-order quantum correcti<strong>on</strong> to<br />
the Larmor radiati<strong>on</strong> <strong>in</strong> the limit <str<strong>on</strong>g>of</str<strong>on</strong>g> the relativistic moti<strong>on</strong>.<br />
FORMULATION<br />
We c<strong>on</strong>sider the scalar QED with the acti<strong>on</strong>,<br />
∫<br />
S = dtd 3 x<br />
×<br />
[<br />
(Dµφ) † D µ φ − m2<br />
¯h 2 φ† φ − 1<br />
∗ This presentati<strong>on</strong> is based <strong>on</strong> Ref.[1]<br />
4µ0<br />
FµνF µν<br />
]<br />
, (1)<br />
φ<br />
Pi<br />
k<br />
Pf<br />
Figure 1: Feynman Diagram for the process.<br />
where Dµ = (∂/∂x µ + ieAµ/¯h), e and m are the charge<br />
and the mass <str<strong>on</strong>g>of</str<strong>on</strong>g> the massive scalar field, respectively, and<br />
µ0 is the magnetic permeability <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum. We work <strong>in</strong><br />
the M<strong>in</strong>kowski spacetime, but c<strong>on</strong>sider the homogeneous<br />
electric background field E(t), which is related to the vector<br />
potential by Aµ = (0, A(t)) and ˙ A(t) = −E(t), where<br />
the dot denotes the differentiati<strong>on</strong> with respect to the time.<br />
The equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the free scalar field yields<br />
( ∂ 2<br />
∂t 2 + (p − eA(t))2 + m 2<br />
¯h 2<br />
γ<br />
φ<br />
)<br />
ϕp(t) = 0, (2)<br />
where ϕp(t) is the coefficient <str<strong>on</strong>g>of</str<strong>on</strong>g> the Fourier expansi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the field, i.e., the mode functi<strong>on</strong>. Us<strong>in</strong>g the mode functi<strong>on</strong>,<br />
which is normalized so as to be ˙ϕ ∗ pϕp − ϕ ∗ p ˙ϕp = i, the<br />
quantized field is c<strong>on</strong>structed as<br />
φ(x) =<br />
√ ¯h<br />
L 3<br />
∑<br />
p<br />
(<br />
ϕp(t)bp + ϕ ∗ −p(t)c †<br />
)<br />
−p e ip·x/¯h , (3)<br />
where L 3 is the volume <str<strong>on</strong>g>of</str<strong>on</strong>g> the space, the creati<strong>on</strong> and annihilati<strong>on</strong><br />
operators satisfy the commutati<strong>on</strong> relati<strong>on</strong>s,<br />
[bp, b †<br />
p ′] = δp,p ′, [bp, bp ′] = [b† p, b †<br />
′] = 0, (4)<br />
and the same relati<strong>on</strong>s hold for cp and c † p. We also quantize<br />
the free electromagnetic field as,<br />
Aµ =<br />
√ µ0¯h<br />
L 3<br />
∑<br />
λ=1,2<br />
∑<br />
k<br />
ɛ λ µ<br />
p<br />
( −ikt e<br />
√ a<br />
2k λ )<br />
k + h.c. e ik·x , (5)<br />
where ɛ λ µ denotes the polarizati<strong>on</strong> vector, and a λ†<br />
k and aλ k<br />
are the creati<strong>on</strong> and annihilati<strong>on</strong> operators which satisfy the<br />
follow<strong>in</strong>g commutati<strong>on</strong> relati<strong>on</strong>,<br />
[a λ k, a λ′ †<br />
k ′ ] = δ λλ′<br />
δk,k ′. (6)<br />
We c<strong>on</strong>sider the process, <strong>in</strong> which <strong>on</strong>e phot<strong>on</strong> is emitted<br />
from a charged particle, as shown <strong>in</strong> Fig. 1. Note that<br />
this process is prohibited without the background electric
field because <str<strong>on</strong>g>of</str<strong>on</strong>g> the Lorentz <strong>in</strong>variance <str<strong>on</strong>g>of</str<strong>on</strong>g> the M<strong>in</strong>kowski<br />
spacetime, which ensures existence <str<strong>on</strong>g>of</str<strong>on</strong>g> the frame that the<br />
charged particle is at rest. However, <strong>on</strong> the electric field<br />
background, we have the radiati<strong>on</strong> energy from the process,<br />
which can be evaluated, as follows. Us<strong>in</strong>g the <strong>in</strong>-<strong>in</strong><br />
formalism [7, 8], we may compute the radiati<strong>on</strong> energy at<br />
the lowest order <str<strong>on</strong>g>of</str<strong>on</strong>g> the coupl<strong>in</strong>g c<strong>on</strong>stant,<br />
E = ∑<br />
∫<br />
λ<br />
∑<br />
∫<br />
−2<br />
= ¯h<br />
λ<br />
d 3 k¯hk〈a λ†<br />
k aλ k〉<br />
d 3 ∫ ∞ ∫ ∞<br />
k¯hkRe dt2<br />
−∞<br />
<br />
× <strong>in</strong>|HI(t1)a λ†<br />
k aλkHI(t2)|<strong>in</strong> dt1<br />
−∞<br />
<br />
, (7)<br />
where we adopted the range <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tegrati<strong>on</strong> from the <strong>in</strong>f<strong>in</strong>ite<br />
past to the <strong>in</strong>f<strong>in</strong>ite future, and |<strong>in</strong>〉 denotes the <strong>in</strong>itial<br />
state, which we choose as <strong>on</strong>e charged particle state with<br />
the momentum pi, i.e., |<strong>in</strong>〉 = b † pi |0〉, and<br />
HI(t) = − ie<br />
∫<br />
d<br />
¯h<br />
3 xA µ<br />
{(<br />
× ∂µ − ie<br />
¯h Āµ<br />
)<br />
φ † φ − φ †<br />
(<br />
∂µ + ie<br />
¯h Āµ<br />
) }<br />
φ .(8)<br />
In order to evaluate the quantum correcti<strong>on</strong>, we c<strong>on</strong>sider<br />
the expansi<strong>on</strong> <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> a power series <str<strong>on</strong>g>of</str<strong>on</strong>g> ¯h. Up to the<br />
order <str<strong>on</strong>g>of</str<strong>on</strong>g> O(¯h), we have<br />
where we def<strong>in</strong>ed<br />
E = E (0) + E (1) + O(¯h 2 ), (9)<br />
E (0) =<br />
(( 2 d x<br />
×<br />
dξ2 e2<br />
(4π) 2ɛ0 )2<br />
∫ ∫<br />
dΩˆ k dξ<br />
(<br />
− ˆk · d2x dξ2 )2)<br />
. (10)<br />
The expressi<strong>on</strong> (10) yields the classical formula <str<strong>on</strong>g>of</str<strong>on</strong>g> the Larmor<br />
radiati<strong>on</strong> from a charged particle. The first-order quantum<br />
correcti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ¯h is described by<br />
E (1) =<br />
e2¯h (4π) 3 ∫ ∫ ∫<br />
dΩˆ k dξ dξ<br />
ɛ0<br />
′ 1<br />
ξ − ξ ′<br />
×<br />
{ (<br />
d d<br />
−<br />
dξ dξ ′<br />
)<br />
d d<br />
dξ dξ ′<br />
[( (ˆk<br />
dx<br />
)(<br />
· ˆk<br />
dx<br />
·<br />
dξ<br />
′<br />
dξ ′<br />
)<br />
− dx dx′<br />
·<br />
dξ dξ ′<br />
)(<br />
ˆk · dx dτ<br />
dt dt + ˆ k · dx′<br />
dt ′<br />
dτ ′<br />
dt ′<br />
+<br />
)]<br />
2 d2<br />
dξ2 d2 dξ ′2<br />
[( (ˆk<br />
dx<br />
)(<br />
· ˆk<br />
dx<br />
·<br />
dξ<br />
′<br />
dξ ′<br />
)<br />
− dx dx′<br />
·<br />
dξ dξ ′<br />
×<br />
)<br />
∫ ξ(t) ′′<br />
′′ dτ<br />
dξ<br />
dξ ′′<br />
( (<br />
1 − ˆk · dx′′<br />
dt ′′<br />
) 2)] }<br />
, (11)<br />
ξ ′ (t ′ )<br />
where we follow the notati<strong>on</strong>s <strong>in</strong> Ref.[1]<br />
APPROXIMATE FORMULAS<br />
In the n<strong>on</strong>-relativistic limit, where the velocity v =<br />
dx/dt is small enough compared with the velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> light,<br />
|v| ≪ 1, Eqs. (10) and (11) reduce to<br />
E (0) e<br />
=<br />
2 ∫<br />
dt ˙v(t) · ˙v(t), (12)<br />
6πɛ0<br />
E (1) =<br />
e 2 ¯h<br />
6π 2 ɛ0m<br />
∫ ∫<br />
dt dt ′<br />
× ¨v(t) · ˙v(t′ ) − ˙v(t) · ¨v(t ′ )<br />
t − t ′ , (13)<br />
respectively. Eq. (13) was found for the first time by<br />
Higuchi and Walker <strong>in</strong> Ref. [3]. In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the periodic<br />
electric field, |E| = E0 s<strong>in</strong> ωt, where E0 is a c<strong>on</strong>stant,<br />
we have the periodic accelerati<strong>on</strong>, | ˙v| = (eE0/m) s<strong>in</strong> ωt.<br />
Then,<br />
dE (0)<br />
dt = e4E2 0<br />
m2 dE (1)<br />
dt = −¯he4 E2 0<br />
m2 s<strong>in</strong> 2 ωt<br />
,<br />
6πɛ0<br />
(14)<br />
ω<br />
.<br />
12πɛ0m<br />
(15)<br />
After tak<strong>in</strong>g an average over a l<strong>on</strong>g time-durati<strong>on</strong>, we have<br />
E (1)<br />
E<br />
¯hω<br />
= − , (16)<br />
(0) mc2 where c is the light velocity, which is restored here. The<br />
quantum effect becomes important when the time scale <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the accelerati<strong>on</strong> multiplied by c is comparable to the Compt<strong>on</strong><br />
wavelength, namely, when the wave-like feature <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
particle appears.<br />
Next, let us c<strong>on</strong>sider the relativistic limit, |pi| ≫<br />
|eA|, m. For simplicity, we c<strong>on</strong>sider the case when the<br />
directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle moti<strong>on</strong> is always parallel to that <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the background electric field, i.e., v ∝ A. Namely, we c<strong>on</strong>sider<br />
the case when the directi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle’s moti<strong>on</strong><br />
and the background electric field are parallel at any moment,<br />
and adopt this directi<strong>on</strong> as the z axis. Then, we may<br />
write A = (0, 0, A(t)), A ˙ = (0, 0, −E(t)), v = (0, 0, v),<br />
and pi = (0, 0, pi). In this case, we have<br />
E (0) = 1 m<br />
6πɛ0<br />
4e4 p6 i<br />
∫<br />
dt<br />
˙<br />
A 2 (t)<br />
(1 − v2 . (17)<br />
) 3<br />
We c<strong>on</strong>sider the case, pi ≫ |eA|, m. We also assume<br />
|A| ∼ | ˙ A/ω| ∼ | Ä/ω2 |, where 1/ω is a time-scale <str<strong>on</strong>g>of</str<strong>on</strong>g> timevary<strong>in</strong>g<br />
background electric field. In this relativistic limit,<br />
we have<br />
E (1) − e4¯h 3(2π) 2 m<br />
ɛ0<br />
2<br />
p5 ∫ ∫<br />
dt dt<br />
i<br />
′<br />
×<br />
1<br />
(1 − ¯v 2 ) 3<br />
Ä(t) ˙ A(t ′ ) − ˙ A(t) Ä(t′ )<br />
t − t ′ . (18)<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the periodic background <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric<br />
field, ˙<br />
A(t) = −E0 s<strong>in</strong> ωt, where E0 is a c<strong>on</strong>stant, we have<br />
dE (0)<br />
dt = e4 m 4<br />
6πɛ0p 6 i<br />
E2 0 cos2 ωt<br />
(1 − v2 , (19)<br />
) 3
dE (1)<br />
dt = ¯he4 m 2<br />
12πɛ0p 5 i<br />
E2 0ω<br />
(1 − v2 . (20)<br />
) 3<br />
After averag<strong>in</strong>g over sufficiently l<strong>on</strong>g time-durati<strong>on</strong>, we<br />
have<br />
E (1)<br />
pi<br />
=<br />
E (0) mc<br />
¯hω<br />
. (21)<br />
mc2 Note that the quantum correcti<strong>on</strong> E (1) is positive, which is<br />
a c<strong>on</strong>trast to the n<strong>on</strong>-relativistic case.<br />
For the radiati<strong>on</strong> from an electr<strong>on</strong> <strong>in</strong> a periodic electric<br />
field, e.g., by a laser field, Eq. (21) is estimated as<br />
E (1)<br />
∼ 2.6 × 10−3<br />
E (0)<br />
×<br />
(<br />
pic<br />
)<br />
GeV<br />
( mc2 )−2 (<br />
0.5MeV<br />
ω<br />
10 15 s −1<br />
)<br />
,(22)<br />
where ω ∼ 10 15 s −1 corresp<strong>on</strong>ds to an X-ray laser. The<br />
quantum effect becomes significant when the electr<strong>on</strong> k<strong>in</strong>etic<br />
energy reaches TeV scale. The above formula is<br />
derived under the c<strong>on</strong>diti<strong>on</strong>, pi ≫ |eA|, m. For a periodic<br />
electric field <str<strong>on</strong>g>of</str<strong>on</strong>g> large amplitude, pi ∼ |eA|, the<br />
c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the relativistic moti<strong>on</strong> cannot be always guaranteed,<br />
because the physical momentum might become<br />
|pi − eA| ∼ m. In this case, it is difficult to express the<br />
quantum correcti<strong>on</strong> <strong>in</strong> a simple analytic form. We need a<br />
more general treatment <strong>in</strong>clud<strong>in</strong>g fully numerical calculati<strong>on</strong>,<br />
because the n<strong>on</strong>-locality plays an important role. Potentially,<br />
there is a lot <str<strong>on</strong>g>of</str<strong>on</strong>g> room for discussi<strong>on</strong> about how to<br />
detect the quantum effect <str<strong>on</strong>g>of</str<strong>on</strong>g> the Larmor radiati<strong>on</strong> experimentally,<br />
but this is out <str<strong>on</strong>g>of</str<strong>on</strong>g> scope <str<strong>on</strong>g>of</str<strong>on</strong>g> the present work.<br />
SUMMARY<br />
In this work, we obta<strong>in</strong>ed the general formula for the<br />
first-order quantum correcti<strong>on</strong> to the Larmor radiati<strong>on</strong> from<br />
a charged particle mov<strong>in</strong>g <strong>in</strong> spatially homogeneous timedependent<br />
electric field. This formula reproduces the same<br />
result as that <strong>in</strong> Ref. [3], <strong>in</strong> the limit <str<strong>on</strong>g>of</str<strong>on</strong>g> a n<strong>on</strong>-relativistic<br />
moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the charged particle. Our result is useful to <strong>in</strong>vestigate<br />
the case <str<strong>on</strong>g>of</str<strong>on</strong>g> a relativistic moti<strong>on</strong>.<br />
By apply<strong>in</strong>g the formula to the cases <str<strong>on</strong>g>of</str<strong>on</strong>g> a periodic accelerati<strong>on</strong>,<br />
it was dem<strong>on</strong>strated that the lead<strong>in</strong>g quantum<br />
effect enhances the radiati<strong>on</strong> <strong>in</strong> the relativistic limit and<br />
that it decreases <strong>in</strong> the n<strong>on</strong>-relativistic limit. This quantum<br />
effect will become important when the <strong>in</strong>cident k<strong>in</strong>etic<br />
electr<strong>on</strong> energy approaches TeV scale with an x-ray laser.<br />
This situati<strong>on</strong> is the same <strong>in</strong> some possible functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> accelerati<strong>on</strong><br />
cases. The details <str<strong>on</strong>g>of</str<strong>on</strong>g> our work can be found <strong>in</strong><br />
Ref.[1].<br />
REFERENCES<br />
[1] K. Yamamoto, G. Nakamura, arXiv:1012.5182, accepted for<br />
publicati<strong>on</strong> <strong>in</strong> Phys. Rev. D<br />
[2] J. D. Jacks<strong>on</strong>, Classical Electrodynamics (Wiley, 1998)<br />
[3] A. Higuchi and P. J. Walker, Phys. Rev. D 80 105019 (2009)<br />
[4] A. Higuchi and G. D. R. Mart<strong>in</strong>, Found. Phys. 35 1149<br />
(2005)<br />
[5] A. Higuchi and G. D. R. Mart<strong>in</strong>, Phys. Rev. D 73 025019<br />
(2006)<br />
[6] A. Higuchi and G. D. R. Mart<strong>in</strong>, Phys. Rev. D 74 125002<br />
(2006)<br />
[7] S. We<strong>in</strong>berg, Phys. Rev. D 72 043514 (2005)<br />
[8] P. Adshead, R. Easther and E. A. Lim, Phys. Rev. D 80<br />
083521 (2009)<br />
[9] H. Nomura, M. Sasaki and K. Yamamoto JCAP 0611 013<br />
(2006)<br />
[10] P. Chen, T. Tajima, Phys. Rev. Lett. 83 256 (1999)<br />
[11] R. Schutzhold, G. Schaller, D. Habs, Phys. Rev. Lett. 97,<br />
121302 (2006) ;Erratum-ibid. 97 139902 (2006)<br />
[12] L.C. B. Crisp<strong>in</strong>o, A. Higuchi, and G. E. A. Matsas, Rev.<br />
Mod. Phys. 80 787 (2008)<br />
[13] S. Iso, Y. Yamamoto, S. Zhang, arXiv:1011.4191<br />
[14] A. Higuchi and P. J. Walker, Phys. Rev. D 79 105023 (2009)<br />
[15] R. Kimura, G. Nakamura, and K. Yamamoto,<br />
arXiv:1101.4699, accepted for publicati<strong>on</strong> <strong>in</strong> Phys.<br />
Rev. D<br />
[16] N. D. Birrell and P. C. W. Davies, Quantum fields <strong>in</strong> curved<br />
space (Cambridge University Press ,1982)
FAST VACUUM DECAY INTO PARTICLE PAIRS<br />
IN STRONG ELECTRIC AND MAGNETIC FIELDS ∗<br />
Y. Hidaka, T. Iritani, and H. Suganuma,<br />
Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Kyoto University, Kitashirakawa Oiwakecho, Sakyo, Kyoto 606-8502, Japan<br />
Abstract<br />
We discuss fermi<strong>on</strong> pair producti<strong>on</strong>s <strong>in</strong> str<strong>on</strong>g electric<br />
and magnetic fields. We po<strong>in</strong>t out that, <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> massless<br />
fermi<strong>on</strong>s, the vacuum persistency probability per unit<br />
time and volume is zero <strong>in</strong> the str<strong>on</strong>g electric and magnetic<br />
fields, while it is f<strong>in</strong>ite when the magnetic field is absent.<br />
The c<strong>on</strong>tributi<strong>on</strong> from the lowest Landau level (LLL) dom<strong>in</strong>ates<br />
this phenomen<strong>on</strong>. We also discuss dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
vacuum decay, us<strong>in</strong>g an effective theory <str<strong>on</strong>g>of</str<strong>on</strong>g> the LLL projecti<strong>on</strong>,<br />
tak<strong>in</strong>g <strong>in</strong>to account the back reacti<strong>on</strong>.<br />
INTRODUCTION<br />
Dynamics <strong>in</strong> str<strong>on</strong>g fields has been an <strong>in</strong>terest<strong>in</strong>g subject<br />
<strong>in</strong> theoretical physics. Recently, this subject is be<strong>in</strong>g<br />
paid attenti<strong>on</strong> also <strong>in</strong> the experimental physics <str<strong>on</strong>g>of</str<strong>on</strong>g> creati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the quark glu<strong>on</strong> plasma. In high-energy heavy-i<strong>on</strong> collisi<strong>on</strong><br />
experiments, at the so-called Glasma stage [1] just<br />
after the collisi<strong>on</strong>, l<strong>on</strong>gitud<strong>in</strong>al color electric and magnetic<br />
fields are expected to be produced <strong>in</strong> the c<strong>on</strong>text <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
color glass c<strong>on</strong>densate <str<strong>on</strong>g>of</str<strong>on</strong>g> order 1–2 GeV <strong>in</strong> RHIC and 5<br />
GeV <strong>in</strong> LHC. In the peripheral collisi<strong>on</strong>, a str<strong>on</strong>g magnetic<br />
field <str<strong>on</strong>g>of</str<strong>on</strong>g> order 100 MeV would be <strong>in</strong>duced. The questi<strong>on</strong> is<br />
how the str<strong>on</strong>g fields decay and the system is thermalized.<br />
In this work, we c<strong>on</strong>centrate <strong>on</strong> how the str<strong>on</strong>g fields decay<br />
<strong>in</strong>to particles. For this purpose, we first briefly review<br />
the Schw<strong>in</strong>ger mechanism <strong>in</strong> the coexistence <str<strong>on</strong>g>of</str<strong>on</strong>g> electric and<br />
magnetic fields. We will po<strong>in</strong>t out that the vacuum immediately<br />
decays <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> massless fermi<strong>on</strong> and n<strong>on</strong>zero<br />
E and B. For simplicity, we c<strong>on</strong>sider the case that the<br />
electric and magnetic fields are covariantly c<strong>on</strong>stant [2],<br />
i.e., [Dµ, E] = [Dµ, B] = 0, where Dµ = ∂µ − igAµ<br />
is the covariant derivative with the gauge field Aµ. The<br />
electric and magnetic fields are def<strong>in</strong>ed as E i = F i0 and<br />
B i = −ɛ ijk Fjk/2 with Fµν = i[Dµ, Dν]/g. This is a generalizati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>stant fields <strong>in</strong> QED, ∂µE = ∂µB = 0, to<br />
the n<strong>on</strong>-Abelian fields. For the covariantly c<strong>on</strong>stant fields,<br />
all the comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> E and B can be diag<strong>on</strong>alized to be<br />
c<strong>on</strong>stant matrices <strong>in</strong> color space by a gauge transformati<strong>on</strong>.<br />
Without loss <str<strong>on</strong>g>of</str<strong>on</strong>g> generality, <strong>on</strong>e can also set E = (0, 0, E)<br />
and B = (0, 0, B) by choos<strong>in</strong>g an appropriate Lorentz<br />
frame and the coord<strong>in</strong>ate axis.<br />
∗ This work was supported by the Grant-<strong>in</strong>-Aid for the Global COE<br />
Program “The Next Generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Spun from Universality and<br />
Emergence” from the M<strong>in</strong>istry <str<strong>on</strong>g>of</str<strong>on</strong>g> Educati<strong>on</strong>, Culture, Sports, Science and<br />
Technology (MEXT) <str<strong>on</strong>g>of</str<strong>on</strong>g> Japan.<br />
SCHWINGER MECHANISM<br />
The vacuum decay <strong>in</strong> an electric field was discussed by<br />
[3, 4]. C<strong>on</strong>sider the vacuum persistency probability, which<br />
is def<strong>in</strong>ed by<br />
|〈Ωout|Ω<strong>in</strong>〉| 2 = exp(−V T w), (1)<br />
where V and T are <strong>in</strong>f<strong>in</strong>ite space volume and time length.<br />
|Ω<strong>in</strong>〉 and |Ωout〉 are the <strong>in</strong>-vacuum and the out-vacuum def<strong>in</strong>ed<br />
at t = −T/2 and t = T/2, respectively. If the vacuum<br />
is unstable, w has a n<strong>on</strong>zero value, while, if the vacuum<br />
is stable, w vanishes. Therefore, w denotes magnitude<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the vacuum decay per unit volume and time. When w is<br />
small, |〈Ωout|Ω<strong>in</strong>〉| 2 ≈ 1 − V T w, so that w is regarded as<br />
the pair producti<strong>on</strong> probability per unit volume and time.<br />
For QCD, the analytic formula <str<strong>on</strong>g>of</str<strong>on</strong>g> w for the quark-pair<br />
creati<strong>on</strong> <strong>in</strong> the covariantly c<strong>on</strong>stant is given by [2]<br />
w =<br />
∞∑<br />
n=1<br />
tr g2 EB<br />
4π 2<br />
1<br />
n coth(πnBE−1 )e −nπm2 / √ g 2 E 2<br />
,<br />
(2)<br />
where m denotes the quark-mass matrix and the trace is<br />
taken over the <strong>in</strong>dices <str<strong>on</strong>g>of</str<strong>on</strong>g> color and flavor. This is a n<strong>on</strong>-<br />
Abelian extensi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the follow<strong>in</strong>g formula for QED [5]:<br />
w =<br />
∞∑<br />
n=1<br />
e 2 EB<br />
4π 2<br />
1<br />
n coth(πnBE−1 )e −nπm2 / √ e 2 E 2<br />
, (3)<br />
with the QED coupl<strong>in</strong>g c<strong>on</strong>stant e(> 0). Note that the<br />
fermi<strong>on</strong> pair creati<strong>on</strong> formalism <strong>in</strong> the covariantly c<strong>on</strong>stant<br />
fields <strong>in</strong> QCD is similar to that <strong>in</strong> QED, so that we hereafter<br />
give the formula for QED, where we set E ≥ 0 and B ≥ 0<br />
by a suitable axis choice and the parity transformati<strong>on</strong>.<br />
In the absence <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field, this formula reduces<br />
to the well-known result,<br />
w =<br />
∞∑<br />
n=1<br />
e 2 E 2<br />
4π 3<br />
1<br />
n 2 e−nπm2 /(eE) . (4)<br />
If the masses are zero, w has a f<strong>in</strong>ite value <str<strong>on</strong>g>of</str<strong>on</strong>g> w =<br />
e 2 E 2 /(24π). The situati<strong>on</strong> changes if the magnetic field<br />
exists. From Eq. (3), w diverges <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic<br />
field. To see this, summ<strong>in</strong>g over all modes <strong>in</strong> Eq. (3),<br />
we obta<strong>in</strong> for small m as<br />
w e2EB 4π2 ln eE<br />
. (5)<br />
πm2 As m → 0, w logarithmically diverges as<br />
w ∝ − ln m → ∞. (6)
Next, let us c<strong>on</strong>sider the orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the divergence <str<strong>on</strong>g>of</str<strong>on</strong>g> w <strong>in</strong><br />
terms <str<strong>on</strong>g>of</str<strong>on</strong>g> effective dimensi<strong>on</strong>al reducti<strong>on</strong> <strong>in</strong> a str<strong>on</strong>g magnetic<br />
field. When a magnetic field exists, the spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the transverse directi<strong>on</strong> is discretized by Landau quantizati<strong>on</strong>.<br />
Actually, the energy spectrum for E=0 is given by<br />
ε = ± √ p 2 z + 2eB(n + 1/2 ∓ sz) + m 2 , (7)<br />
where n = 0, 1, · · · corresp<strong>on</strong>d to the Landau levels,<br />
and sz = ±1/2 is the sp<strong>in</strong>. The system effectively becomes<br />
1+1 dimensi<strong>on</strong>al system with <strong>in</strong>f<strong>in</strong>ite tower <str<strong>on</strong>g>of</str<strong>on</strong>g> massive<br />
state: m 2 n,eff ≡ 2eBn + m2 . For the lowest Landau<br />
level (LLL), n = 0 and s = +1/2, the energy is<br />
ε = ± √ p 2 z + m 2 . This is the spectrum <strong>in</strong> 1+1 dimensi<strong>on</strong>s.<br />
This LLL causes the divergence <str<strong>on</strong>g>of</str<strong>on</strong>g> w as will be shown below.<br />
The divergence <str<strong>on</strong>g>of</str<strong>on</strong>g> w does not mean the divergence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the <strong>in</strong>f<strong>in</strong>ite pair producti<strong>on</strong> per unit space-time. The divergence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> w rather implies that the vacuum always decays<br />
and produces pairs <str<strong>on</strong>g>of</str<strong>on</strong>g> fermi<strong>on</strong>. The questi<strong>on</strong> is where the<br />
vacuum goes. In the coexistence <str<strong>on</strong>g>of</str<strong>on</strong>g> B and E, <strong>on</strong>e can obta<strong>in</strong><br />
the probability <str<strong>on</strong>g>of</str<strong>on</strong>g> the n pairs <str<strong>on</strong>g>of</str<strong>on</strong>g> fermi<strong>on</strong> with LLL as<br />
|〈n pairs|Ω<strong>in</strong>〉| 2<br />
= exp<br />
[<br />
V eB<br />
−<br />
4π2 ( ∫<br />
eET −<br />
dpznpz<br />
)<br />
ln eE<br />
πm2 ]<br />
.<br />
The vacuum persistency probability corresp<strong>on</strong>ds to all<br />
npz’s be<strong>in</strong>g zero <strong>in</strong> Eq. (8), and w is equal to Eq. (5), so<br />
that w diverges at m = 0. At m = 0, this probability is<br />
f<strong>in</strong>ite <strong>on</strong>ly if the follow<strong>in</strong>g equati<strong>on</strong> is satisfied:<br />
∫<br />
eET − dpznpz = 0. (9)<br />
Therefore, the number <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle with the LLL is restricted<br />
by Eq. (9), and l<strong>in</strong>early <strong>in</strong>creases with time. The<br />
higher Landau levels give heavy effective masses <str<strong>on</strong>g>of</str<strong>on</strong>g> order<br />
eB, so that all the c<strong>on</strong>tributi<strong>on</strong>s to the pair producti<strong>on</strong>s<br />
from such modes are suppressed. The total number <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
particle pairs can be calculated:<br />
N = V T e2 E 2<br />
4π 3<br />
πB<br />
E<br />
(8)<br />
πB<br />
coth . (10)<br />
E<br />
At B = 0, N = V T e 2 E 2 /(4π 3 ). The c<strong>on</strong>tributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> LLL<br />
is obta<strong>in</strong>ed as<br />
N = V T e2 E 2<br />
4π 3<br />
πB<br />
, (11)<br />
E<br />
which is equal to tak<strong>in</strong>g coth(πB/E) → 1 <strong>in</strong> Eq. (10). In<br />
Fig. 1, the total number <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle for the full c<strong>on</strong>tributi<strong>on</strong><br />
and LLL c<strong>on</strong>tributi<strong>on</strong> are shown. The LLL dom<strong>in</strong>ates<br />
for B > E, so that the effective model for the LLL works<br />
well for B > E.<br />
THEORY OF STRONG MAGNETIC FIELD<br />
In this secti<strong>on</strong>, we study particle producti<strong>on</strong>s com<strong>in</strong>g<br />
from the LLL for QED taken <strong>in</strong>to account the back reacti<strong>on</strong>.<br />
For this purpose, we c<strong>on</strong>sider LLL projected theory,<br />
N(E,B)/N(E,B=0)<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
LLL<br />
Full<br />
0 0.5 1 1.5<br />
B/E<br />
Figure 1: Ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle number to that at B = 0.<br />
The solid l<strong>in</strong>e denotes the c<strong>on</strong>tributi<strong>on</strong> from the LLL, and<br />
the dotted l<strong>in</strong>e denotes the c<strong>on</strong>tributi<strong>on</strong> from all modes.<br />
that is, the wave functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the fermi<strong>on</strong> is projected to the<br />
LLL state. The wave functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the LLL is<br />
√ ( ) l<br />
2<br />
eB eB<br />
φl(x, y) =<br />
(x + iy)<br />
2πl! 2<br />
l<br />
(<br />
× exp − eB<br />
4 (x2 + y 2 ) (12)<br />
) ,<br />
where l denotes the angular momentum <strong>in</strong> z directi<strong>on</strong> for<br />
the LLL, and the energy is degenerate for l. One can decompose<br />
the fermi<strong>on</strong> field <strong>in</strong>to the l<strong>on</strong>gitud<strong>in</strong>al mode and<br />
the transverse mode <strong>in</strong> a suitable representati<strong>on</strong> as<br />
(∑<br />
ψ(x) = l φl(x,<br />
)<br />
y)ϕl(t, z)<br />
, (13)<br />
0<br />
where ϕl(t, z) is the two comp<strong>on</strong>ent Dirac field <strong>in</strong> 1+1 dimensi<strong>on</strong>s.<br />
Then the fermi<strong>on</strong> acti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> QED <strong>in</strong> 3+1 dimensi<strong>on</strong>s<br />
reduces to that <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Abelian gauge theory <strong>in</strong> 1 + 1<br />
dimensi<strong>on</strong>s:<br />
∫<br />
S = d 4 x ¯ ψ(x)iγ µ Dµψ(x)<br />
∑<br />
∫<br />
dtdz ¯ϕl ′(t, z)i˜γµ D˜ l ′ l<br />
µ ϕl(t, z),<br />
(14)<br />
l,l ′<br />
where ˜γ t and ˜γ z are the gamma matrices <strong>in</strong> 1 + 1 dimensi<strong>on</strong>s<br />
and ˜γ x = ˜γ y = 0. The covariant derivative is def<strong>in</strong>ed<br />
by ˜ Dl′ l<br />
µ = δl′ l∂µ − ieÃl′ l<br />
µ with<br />
à l′ l<br />
µ (t, z) =<br />
∫<br />
dxdyφ ∗ l ′(x, y)φl(x, y)Aµ(x, y, z, t). (15)<br />
à l′ l<br />
µ (t, z) corresp<strong>on</strong>ds to the gauge field <strong>in</strong> U(∞) gauge<br />
theory, s<strong>in</strong>ce Ãl′ l<br />
µ (t, z) is an Hermite matrix, Ã∗l′ l<br />
µ (t, z) =<br />
à ll′<br />
µ (t, z), and the <strong>in</strong>dices l and l ′ run from 0 to ∞. To<br />
simplify the situati<strong>on</strong>, we assume that the At and Az do<br />
not depend <strong>on</strong> the transverse directi<strong>on</strong>s, x and y. Then the l<br />
dependence can be factorized out: Ãl′ l<br />
µ (t, z) = δll ′õ(t, z)<br />
and ϕl(t, z) = ϕ(t, z). The acti<strong>on</strong> <strong>in</strong> Eq. (14) becomes<br />
S eBV⊥<br />
2π<br />
∫<br />
dtdz ¯ϕ(t, z)i˜γ µ Dµϕ(t, ˜ z), (16)
E/E 0<br />
N/N max<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
-6 -4 -2 0<br />
ωt<br />
2 4 6<br />
1<br />
0.5<br />
0<br />
-6 -4 -2 0<br />
ωt<br />
2 4 6<br />
Figure 2: The electric field (upper) and the number <str<strong>on</strong>g>of</str<strong>on</strong>g> pairs<br />
(lower) at z = 0 plotted aga<strong>in</strong>st time. Both values are normalized<br />
by that at maximum values.<br />
where V⊥ is the volume <str<strong>on</strong>g>of</str<strong>on</strong>g> the transverse directi<strong>on</strong>s. This<br />
acti<strong>on</strong> is noth<strong>in</strong>g but that <strong>in</strong> 1+1 QED, i.e., the Schw<strong>in</strong>ger<br />
model, except for the overall factor eBV⊥/(2π). The exact<br />
soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the effective acti<strong>on</strong> for the fermi<strong>on</strong> is known as<br />
Γ(Aµ) = − m2 ∫<br />
γV⊥<br />
dtdz<br />
2<br />
õ(t, z)<br />
×<br />
(<br />
g µν<br />
<br />
− ∂µ<br />
∂ν <br />
∂ 2 <br />
)<br />
Ãν(t, z),<br />
(17)<br />
where denotes for l<strong>on</strong>gitud<strong>in</strong>al directi<strong>on</strong>s, t and z. mγ<br />
denotes the effective phot<strong>on</strong> mass, m 2 γ ≡ e 3 B/(2π 2 ).<br />
Equati<strong>on</strong> (17) is manifestly gauge <strong>in</strong>variant. In the Lorenz<br />
gauge, it reduces to the result by [6]. The mass mγ is <strong>in</strong>duced<br />
by the axial anomaly, <str<strong>on</strong>g>of</str<strong>on</strong>g> which effect is called “dynamical<br />
Higgs effect.” This mass generati<strong>on</strong> is related to<br />
the fact w → ∞ as m → 0. Us<strong>in</strong>g this form, we can<br />
calculate the fermi<strong>on</strong> and axial currents,<br />
j µ (x) = δΓ(A)<br />
δ(eAµ) = −e2 B<br />
2π 2 õ (t, z), (18)<br />
j µ<br />
5 (x) = −ɛµν jν(x), (19)<br />
where we choose the Lorenz gauge, ∂µ õ = 0. The divergence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the axial current leads the axial anomaly <strong>in</strong> 1 + 1<br />
dimensi<strong>on</strong>s except for the overall factor eB/(2π):<br />
∂µj µ<br />
5 (x) = e2 B<br />
2π2 ɛµν∂µ Ãν(t, z) = e2<br />
BE. (20)<br />
2π2 This relati<strong>on</strong> is noth<strong>in</strong>g but axial anomaly <strong>in</strong> 3 + 1 dimensi<strong>on</strong>s.<br />
S<strong>in</strong>ce the effective acti<strong>on</strong> <strong>in</strong> Eq. (17) has a quadratic<br />
form <strong>in</strong> õ, the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> for the phot<strong>on</strong> can be<br />
solved. For example, the electric field <str<strong>on</strong>g>of</str<strong>on</strong>g> z directi<strong>on</strong> is<br />
E = E0 cos(ωt − kzz), (21)<br />
where ω =<br />
√<br />
k2 z + m2 γ. The currents satisfy ej µ =<br />
−ɛ µν∂νE and ej µ<br />
5 = ∂µ E. We show the electric field<br />
and the number density for the spatially homogeneous case,<br />
kz = 0, as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> time <strong>in</strong> Fig. 2. The electric field<br />
oscillates with a frequency ω. In this case, jt = 0, but<br />
jt 5 = 0. The number density <str<strong>on</strong>g>of</str<strong>on</strong>g> pairs is equal to |jt 5|/2.<br />
These results agree with the previous works [5, 7].<br />
The generalizati<strong>on</strong> to n<strong>on</strong>-Abelian theories is straight<br />
forward if the magnetic field is enough str<strong>on</strong>g. The fermi<strong>on</strong><br />
determ<strong>in</strong>ant becomes Wess-Zum<strong>in</strong>o-Witten acti<strong>on</strong>.<br />
SUMMARY AND OUTLOOK<br />
In this work, we have discussed the vacuum decay <strong>in</strong><br />
str<strong>on</strong>g electric and magnetic fields. When the fermi<strong>on</strong> is<br />
massless, the vacuum persistency probability per unit time<br />
and volume becomes zero, and hence w diverges. The orig<strong>in</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the divergence is from discretized spectra <str<strong>on</strong>g>of</str<strong>on</strong>g> transverse<br />
directi<strong>on</strong>s and the lowest Landau level. The LLL<br />
level dom<strong>in</strong>ates for B > E. With the LLL projecti<strong>on</strong>,<br />
we have analytically calculated the effective acti<strong>on</strong> <strong>in</strong> this<br />
situati<strong>on</strong>, and reproduced the previous numerical results.<br />
S<strong>in</strong>ce the effective theory <str<strong>on</strong>g>of</str<strong>on</strong>g> the LLL is solvable, there is<br />
no chaotic behavior nor thermalizati<strong>on</strong>. The thermalizati<strong>on</strong><br />
does not happen <strong>in</strong> the LLL-dom<strong>in</strong>ant process.<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> QCD, the glu<strong>on</strong> is more <strong>in</strong>terest<strong>in</strong>g because<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> its self-<strong>in</strong>teracti<strong>on</strong>. The Landau quantizati<strong>on</strong><br />
causes <strong>in</strong>stability because the helicity <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong> is <strong>on</strong>e. Insert<strong>in</strong>g<br />
sz = 1 <strong>in</strong> Eq. (7), the energy becomes imag<strong>in</strong>ary<br />
when pz < eB, <str<strong>on</strong>g>of</str<strong>on</strong>g> which <strong>in</strong>stability is known as Nielsen-<br />
Olesen <strong>in</strong>stability. The situati<strong>on</strong> is similar to quench<strong>in</strong>g<br />
phenomen<strong>on</strong> <strong>in</strong> general phase transiti<strong>on</strong>, where a temperature<br />
suddenly changes. In such a situati<strong>on</strong>, a phase separati<strong>on</strong><br />
occurs. The same phenomena would happen <strong>in</strong> relativistic<br />
heavy-i<strong>on</strong> collisi<strong>on</strong>s, because the electric and magnetic<br />
fields are suddenly <strong>in</strong>duced by the collisi<strong>on</strong>, and the<br />
perturbative vacuum <str<strong>on</strong>g>of</str<strong>on</strong>g> glu<strong>on</strong> is unstable. Although dynamics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> unstable glu<strong>on</strong> vacuum is very <strong>in</strong>terest<strong>in</strong>g, we<br />
leave this topic <strong>in</strong> the future work.<br />
REFERENCES<br />
[1] T. Lappi and L. McLerran, Nucl. Phys. A 772, 200 (2006).<br />
[2] H. Suganuma, T. Tatsumi, Prog. Theor. Phys. 90, 379 (1993).<br />
[3] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).<br />
[4] J. Schw<strong>in</strong>ger, Phys. Rev. 82, 664 (1951).<br />
[5] N. Tanji, Ann. Phys. 324, 1691 (2009); ibid 325, 2018 (2010).<br />
[6] J. Schw<strong>in</strong>ger Phys. Rev. 128, 2425 (1962); <strong>in</strong> Theoretical<br />
<strong>Physics</strong>, Trieste Lectures 1962 (IAEA, Vienna, 1963) p.89.<br />
[7] A. Iwazaki, Phys. Rev. C 80, 7 (2009).
Abstract<br />
NONCANONICAL LIE PERTURBATION ANALYSIS FOR THE<br />
RELATIVISTIC PONDEROMOTIVE FORCE ∗<br />
N. Iwata † , Y. Kishimoto and K. Imadera, Kyoto University, Kyoto, Japan<br />
An analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> a relativistic particle moti<strong>on</strong> <strong>in</strong> a n<strong>on</strong>uniform<br />
high <strong>in</strong>tensity laser field with l<strong>in</strong>ear polarizati<strong>on</strong><br />
is presented by us<strong>in</strong>g the n<strong>on</strong>can<strong>on</strong>ical Lie perturbati<strong>on</strong><br />
method, which is based <strong>on</strong> the perturbati<strong>on</strong> theory <str<strong>on</strong>g>of</str<strong>on</strong>g> phase<br />
space Lagrangian. Introduc<strong>in</strong>g a smallness parameter ϵ by<br />
the ratio between the excursi<strong>on</strong> length l and the scale length<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field amplitude L, the relativistic p<strong>on</strong>deromotive<br />
force and the corresp<strong>on</strong>d<strong>in</strong>g particle moti<strong>on</strong> are derived<br />
up to the sec<strong>on</strong>d order with respect to ϵ, which is a n<strong>on</strong>local<br />
extensi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the c<strong>on</strong>venti<strong>on</strong>al p<strong>on</strong>deromotive force. Specifically,<br />
the particle is found to exhibit a betatr<strong>on</strong>-like oscillati<strong>on</strong><br />
with a l<strong>on</strong>g period characterized by the curvature <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the laser field amplitude.<br />
INTRODUCTION<br />
Recently, the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g> ultra-short high power lasers<br />
has reached at the level <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 22 W/cm 2 . In this regime,<br />
electr<strong>on</strong>s irradiated by the lasers exhibit highly relativistic<br />
characters. In order to realize such high <strong>in</strong>tensities, the reducti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the pulse width and/or the spot size is necessary.<br />
In such spatially localized laser fields, the p<strong>on</strong>deromotive<br />
force (light pressure) exists <strong>in</strong>evitably and plays an important<br />
role <strong>in</strong> the particle dynamics [1, 2].<br />
The relativistic p<strong>on</strong>deromotive force, which is proporti<strong>on</strong>al<br />
to the field gradient, has been <strong>in</strong>vestigated us<strong>in</strong>g the<br />
averag<strong>in</strong>g method to the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> assum<strong>in</strong>g that<br />
the ratio between the excursi<strong>on</strong> length l and the scale length<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field amplitude L is small, i.e. ϵ ∼ l/L ≪ 1.<br />
However, as the laser field is tightly focused, higher order<br />
perturbati<strong>on</strong>s such as the spatial curvature become important.<br />
Namely, <strong>in</strong> the n<strong>on</strong>-uniform laser fields, besides the<br />
force that simply ejects charged particles from the regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the str<strong>on</strong>g field, the particles suffer an additi<strong>on</strong>al force orig<strong>in</strong>at<strong>in</strong>g<br />
from the curvature <str<strong>on</strong>g>of</str<strong>on</strong>g> the field. Such a higher-order<br />
force may be utilized to c<strong>on</strong>f<strong>in</strong>e the particle by carefully<br />
c<strong>on</strong>troll<strong>in</strong>g the laser field pattern.<br />
In order to <strong>in</strong>vestigate the particle moti<strong>on</strong> <strong>in</strong> complicated<br />
electromagnetic fields, we have <strong>in</strong>troduced the n<strong>on</strong>can<strong>on</strong>ical<br />
Lie perturbati<strong>on</strong> method based <strong>on</strong> the perturbati<strong>on</strong> theory<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> phase space Lagrangian [3, 4, 5], and derived the<br />
p<strong>on</strong>deromotive force up to the first order <str<strong>on</strong>g>of</str<strong>on</strong>g> ϵ [6]. The<br />
method is found to be efficient and powerful <strong>in</strong> determ<strong>in</strong><strong>in</strong>g<br />
the particle moti<strong>on</strong> systematically keep<strong>in</strong>g the Hamilt<strong>on</strong>ian<br />
structure. Motivated by these studies, here, we extend the<br />
analysis to the higher order particle dynamics <strong>in</strong>clud<strong>in</strong>g the<br />
curvature effect.<br />
∗ Work supported by a Grant-<strong>in</strong>-Aid from JSPS (No. 21340171)<br />
† iwata@center.iae.kyoto-u.ac.jp<br />
NONCANONICAL TRANSFORMATION<br />
We c<strong>on</strong>sider the moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a particle with charge q <strong>in</strong><br />
vacuum irradiated by a l<strong>in</strong>early-polarized high-<strong>in</strong>tensity<br />
laser field. The field is assumed to propagate <strong>in</strong> the zdirecti<strong>on</strong><br />
and be localized <strong>in</strong> the transverse x- and ydirecti<strong>on</strong>s.<br />
As discussed <strong>in</strong> <strong>in</strong>troducti<strong>on</strong>, we def<strong>in</strong>e a smallness<br />
parameter, ϵ ∼ l/L, where l and L (see later) are the<br />
transverse excursi<strong>on</strong> length <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle and the transverse<br />
scale length <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field amplitude, respectively.<br />
Here, we normalize the vector potential <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field A<br />
as a ≡ qA/mc 2 , where m is the rest mass <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle<br />
and c is the speed <str<strong>on</strong>g>of</str<strong>on</strong>g> light, and express a as<br />
a(x, y, η) = ax(x, y, η)êx + ϵaz(η)êz, (1)<br />
where ax(x, y, η) ≡ a0x(x, y) s<strong>in</strong> η, az(η) ≡ a0z cos η, êx<br />
and êz are the unit vectors <strong>in</strong> the x- and z-directi<strong>on</strong>s, respectively,<br />
and η ≡ ωt − kzz. Note that az is necessary to<br />
satisfy the Maxwell equati<strong>on</strong> <strong>in</strong> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> ϵ. As we discuss<br />
later, it is found that az does not affect <strong>on</strong> the secular<br />
moti<strong>on</strong> <strong>in</strong> the first order, whereas it does <strong>in</strong> the sec<strong>on</strong>d order<br />
where we neglect the <strong>in</strong>fluence <strong>in</strong> the present analysis for<br />
simplicity. We expand the amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> the vector potential<br />
around the <strong>in</strong>itial particle positi<strong>on</strong> (x, y, z) = (x0, y0, 0) as<br />
a0x (x⊥) = a0x0 + ϵ˜x⊥ · ∂x⊥ a0x (x⊥0)<br />
+ ϵ2<br />
[ ˜x 2 ∂ 2 xa0x (x⊥0) + ˜y 2 ∂ 2 ya0x (x⊥0) ]<br />
2<br />
+ ϵ 2 ˜x˜y∂x∂ya0x (x⊥0) + O ( ϵ 3) , (2)<br />
where a0x0 ≡ a0x(x⊥0), ˜x⊥ ≡ x⊥ − x⊥0 and<br />
∂x⊥a0x (x⊥0) = ∂a0x (x⊥) /∂x⊥|x⊥=x⊥0 .<br />
Here, we <strong>in</strong>troduce the extended phase space expressed<br />
by the can<strong>on</strong>ical variables as z µ = (t; q, pc) =<br />
(t; qx, qy, qz, pcx, pcy, pcz), where the time t is the <strong>in</strong>dependent<br />
variable. The corresp<strong>on</strong>d<strong>in</strong>g covariant vector is given<br />
by γµ = (−h; pc, 0), where h is the relativistic Hamilt<strong>on</strong>ian<br />
expressed as<br />
√<br />
h(q, pc, t) = m2c4 + c2 (pc − mca) 2 . (3)<br />
In this paper, we use Lat<strong>in</strong> <strong>in</strong>dices that run from 1 to 6<br />
whereas Greek from 0 to 6. Us<strong>in</strong>g these notati<strong>on</strong>s, the variati<strong>on</strong>al<br />
pr<strong>in</strong>ciple is expressed as δ ∫ γµdz µ = 0. We call<br />
ˆγ ≡ γµdz µ a fundamental 1-form. The general transformati<strong>on</strong><br />
law from γµ to the new covariant vector Γµ under<br />
arbitrary coord<strong>in</strong>ate transformati<strong>on</strong> z µ → Z µ can be obta<strong>in</strong>ed<br />
by the relati<strong>on</strong> γµdz µ = ΓµdZ µ . As a preparatory<br />
transformati<strong>on</strong>, we first <strong>in</strong>troduce a n<strong>on</strong>can<strong>on</strong>ical coord<strong>in</strong>ate,<br />
z µ = (η; x, y, z, px, py, pη), (4)
where p = pc − mca is the mechanical momentum, x =<br />
q, and pη ≡ pz − γmc where γ is the relativistic factor.<br />
Here, we take η as the <strong>in</strong>dependent variable to move to<br />
the oscillati<strong>on</strong>-center coord<strong>in</strong>ate <strong>in</strong> the later analysis. The<br />
corresp<strong>on</strong>d<strong>in</strong>g covariant vector is then calculated as<br />
γµ = (−K; px + mcax(x⊥, η), py,<br />
pη + ϵmcaz(η), 0, 0, 0), (5)<br />
where K = − (2kpη) −1 [ m2c2 + p2 ⊥ + p2 ]<br />
η is the new<br />
Hamilt<strong>on</strong>ian. By tak<strong>in</strong>g pη as <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the coord<strong>in</strong>ate variables,<br />
we can simplify the zeroth-order Poiss<strong>on</strong> tensor<br />
which determ<strong>in</strong>es the structure <str<strong>on</strong>g>of</str<strong>on</strong>g> the equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong><br />
[3] as that <strong>in</strong> the can<strong>on</strong>ical coord<strong>in</strong>ate. Note that the field a<br />
does not explicitly appear <strong>in</strong> the new Hamilt<strong>on</strong>ian but <strong>in</strong> the<br />
first comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> γµ, that also simplifies the perturbati<strong>on</strong><br />
analysis.<br />
ORBIT ANALYSIS IN LASER FIELDS<br />
In the coord<strong>in</strong>ate given by Eq. (4), the unperturbed particle<br />
orbit z (0)i is obta<strong>in</strong>ed by solv<strong>in</strong>g the zeroth-order equati<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong>, which are derived by the variati<strong>on</strong>al pr<strong>in</strong>-<br />
ciple to the 1-form ˆγ (0) = γ (0)<br />
µ dz (0)µ , as<br />
z (0)i =<br />
( a0x0<br />
kzζ0<br />
1<br />
2kzζ 2 0<br />
(cos η − 1) + x0, y0,<br />
[ a 2 0x0<br />
2<br />
(<br />
η − 1<br />
2<br />
− mca0x0 s<strong>in</strong> η, 0, pη0<br />
)<br />
s<strong>in</strong> 2η<br />
+ ( 1 − ζ 2 ]<br />
)<br />
0 η ,<br />
)<br />
, (6)<br />
under the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong> (x, p, pη) = (x⊥0, 0, 0, 0, pη0)<br />
and pz = pz0 at η = 0. Here, we def<strong>in</strong>ed ζ0 as pη0 ≡<br />
−mcζ0. In this notati<strong>on</strong>, ζ0 = 1 when the <strong>in</strong>itial momentum<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the particle is zero, i.e. pz0 = 0. The particle<br />
exhibits the well-known figure-eight orbit with the<br />
drift moti<strong>on</strong> <strong>in</strong> the z-directi<strong>on</strong> [7]. The excursi<strong>on</strong> length<br />
l is obta<strong>in</strong>ed from the first comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (6) as l =<br />
a0x0/kzζ0.<br />
Next, to <strong>in</strong>vestigate the secular moti<strong>on</strong>, we transform the<br />
coord<strong>in</strong>ate z µ to that <str<strong>on</strong>g>of</str<strong>on</strong>g> the oscillati<strong>on</strong>-center <str<strong>on</strong>g>of</str<strong>on</strong>g> the zerothorder<br />
oscillatory moti<strong>on</strong>, Z µ = (η; X, Y, Z, Px, Py, pη).<br />
The relati<strong>on</strong>ship between the old and new coord<strong>in</strong>ates is<br />
given by zi = Zi + z i(0)<br />
os. , where z i(0)<br />
os. is the oscillatory part<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the zeroth-order orbit. Then, the new covariant vector is<br />
obta<strong>in</strong>ed as<br />
(<br />
Γµ = − κ; Px + mc (ax(X⊥, η) − ax0(η)) ,<br />
)<br />
Py, pη + ϵmcaz(η), 0, 0, 0 , (7)<br />
where ax0(η) ≡ ax(X⊥0, η). Here, κ is the new Hamilt<strong>on</strong>ian<br />
calculated us<strong>in</strong>g the relati<strong>on</strong>ship between the old and<br />
new coord<strong>in</strong>ates, which yields<br />
κ =K + l [Px + mc (ax(X⊥, η) − ax0(η))] s<strong>in</strong> η<br />
+ a0x0<br />
l [pη + ϵmcaz(η)] cos 2η. (8)<br />
4ζ0<br />
The old Hamiot<strong>on</strong>ian K is now written <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the new coord<strong>in</strong>ate variables Px⊥ , pη [<br />
and η as K =<br />
− (mc) 2 + (Px − mcax0(η)) 2 + P 2 y + p2 ]<br />
η /2kzpη. In<br />
the zeroth order, the trajectory is found to be c<strong>on</strong>sistent<br />
with that given by Eq. (6).<br />
To analyze the first-order moti<strong>on</strong>, we perform a<br />
near-identity Lie transformati<strong>on</strong> from the oscillati<strong>on</strong>center<br />
coord<strong>in</strong>ate Z µ to a new <strong>on</strong>e, Z ′µ , as Z ′µ =<br />
exp ( ϵL (1)) Z µ by the operator def<strong>in</strong>ed to act as L (n) f =<br />
g (n)µ ∂µf for a scalar functi<strong>on</strong> f, where g (n)µ is the<br />
nth order Lie generator <str<strong>on</strong>g>of</str<strong>on</strong>g> the transformati<strong>on</strong>. In<br />
the new coord<strong>in</strong>ate, the first-order covariant vector<br />
is simplified as Γ ′(1)<br />
(<br />
µ = V (0)µ Γ (1)<br />
)<br />
µ ; 0, 0 , where<br />
( )<br />
˜X ′ + Y ˜ ′ /2kzp ′ ηL. Here, L =<br />
V (0)µ Γ (1)<br />
µ = m 2 c 2 a 2 0x0<br />
(∂x [log a0x(X⊥0)]) −1 = (∂y [log a0x(X⊥0)]) −1 , ˜ X ′ ≡<br />
X ′ − x0, ˜ Y ′ ≡ Y ′ − y0, and V (0)µ is the unperturbed<br />
flow vector def<strong>in</strong>ed by V (0)0 = 1 and V (0)i (Z µ ) =<br />
dZ (0)i /dZ 0 , respectively. The overl<strong>in</strong>e <strong>in</strong>dicates the average<br />
over <strong>on</strong>e cycle <str<strong>on</strong>g>of</str<strong>on</strong>g> the fast oscillatory moti<strong>on</strong> with period<br />
η. Note that all the terms <strong>in</strong>clud<strong>in</strong>g az are removed<br />
out <strong>in</strong> the averag<strong>in</strong>g process. The new covariant vector up<br />
to the first order is given by Γ ′ µ = Γ (0)<br />
µ +ϵΓ ′(1)<br />
µ . S<strong>in</strong>ce Γ ′(1)<br />
µ<br />
c<strong>on</strong>ta<strong>in</strong>s variables X ′ , Y ′ and p ′ η, the equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong><br />
for P ′ x, P ′ y and Z ′ c<strong>on</strong>ta<strong>in</strong> the terms <str<strong>on</strong>g>of</str<strong>on</strong>g> O(ϵ) as<br />
dZ ′<br />
dη<br />
dP ′ ⊥<br />
dη<br />
= dZ(0)<br />
dη<br />
<br />
<br />
<br />
Z ′µ + m2c2 a2 0x0<br />
p ′2<br />
η<br />
2kz<br />
[ ]<br />
˜X ′ + Y ˜ ′<br />
, (9)<br />
L<br />
mc<br />
=<br />
p ′ mca<br />
η<br />
2 0x0<br />
2kzL ê⊥, (10)<br />
where ê⊥ is the unit vector perpendicular to the z-axis. The<br />
expressi<strong>on</strong> dZ (0) /dη|Z ′µ denotes to replace the coord<strong>in</strong>ate<br />
variables Z (0)µ <strong>in</strong> the zeroth-order equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> with<br />
those <str<strong>on</strong>g>of</str<strong>on</strong>g> Z ′µ . Note that s<strong>in</strong>ce the backward Lie transformati<strong>on</strong>,<br />
Z µ = exp ( −ϵL (1)) Z ′µ , adds <strong>on</strong>ly the oscillatory<br />
comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> the moti<strong>on</strong>, we have the relati<strong>on</strong> ¯ Z ′i = ¯ Zi .<br />
Therefore, the right-hand side <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (10) is the p<strong>on</strong>deromotive<br />
force <strong>in</strong> the orig<strong>in</strong>al oscillati<strong>on</strong>-center coord<strong>in</strong>ate.<br />
From Eq. (10), we can see that az does not affect the firstorder<br />
p<strong>on</strong>deromotive force. We have also c<strong>on</strong>firmed that<br />
the terms proporti<strong>on</strong>al to az appear <strong>in</strong> the first-order oscillatory<br />
part <strong>in</strong> both the x- and z-directi<strong>on</strong>s. This result is<br />
physically reas<strong>on</strong>able s<strong>in</strong>ce the first order oscillati<strong>on</strong> <strong>in</strong> the<br />
z-directi<strong>on</strong> generated by the z-comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the electric<br />
field affects <strong>on</strong> the oscillati<strong>on</strong> <strong>in</strong> the x-directi<strong>on</strong> through<br />
the v × B force.<br />
Next, we analyze the sec<strong>on</strong>d-order particle moti<strong>on</strong>.<br />
Here, we c<strong>on</strong>sider the case where the field is uniform <strong>in</strong> the<br />
y-directi<strong>on</strong>, i.e. ∂ya = 0, for simplicity. We also neglect<br />
the z-comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the vector potential <strong>in</strong> order to see <strong>on</strong>ly<br />
the curvature effect <strong>on</strong> the particle moti<strong>on</strong>. We transform<br />
the coord<strong>in</strong>ate to a new <strong>on</strong>e, Z ′′µ . In the sec<strong>on</strong>d order Lie<br />
transformati<strong>on</strong>, the new covariant vector is given by Γ ′′ µ =<br />
Γ (0)<br />
µ + ϵΓ ′(1)<br />
µ + ϵ2Γ ′′(2)<br />
µ , where Γ ′′(2)<br />
µ =<br />
(<br />
V (0)µ C (2)<br />
µ ; 0, 0<br />
)
with C (2)<br />
µ = Γ (2)<br />
µ − ˆ L (1) Γ (1)<br />
µ + ˆ L (1)2 Γ (0)<br />
µ /2. Then, we have<br />
V (0)µ C (2)<br />
µ = mca2 0x0<br />
4kz<br />
mc<br />
p ′′<br />
[ ( ) (<br />
1 1<br />
+<br />
η L2 R<br />
+ 1<br />
L 2<br />
3 a0x0<br />
l<br />
16 kz<br />
˜X ′′2 + l2<br />
4<br />
)<br />
mc<br />
p ′′<br />
]<br />
η<br />
. (11)<br />
Here, R ≡ ([ ∂2 xa0x(x0) ] ) −1<br />
/a0x0 is the scale length <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the field curvature. S<strong>in</strong>ce the new Hamilt<strong>on</strong>ian, −Γ ′′<br />
0, does<br />
not c<strong>on</strong>ta<strong>in</strong> the variable Z ′′ , the corresp<strong>on</strong>d<strong>in</strong>g comp<strong>on</strong>ent<br />
p ′′<br />
η is found to be c<strong>on</strong>stant. Then, the equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong><br />
<strong>in</strong> the x-directi<strong>on</strong> are reduced to<br />
dX ′′<br />
dη<br />
dP ′′<br />
x<br />
dη<br />
′′ P x<br />
= − , (12)<br />
kzpη0<br />
= −mca0x0 l<br />
2<br />
[<br />
1<br />
L +<br />
( )<br />
1 1<br />
+<br />
L2 R<br />
˜X ′′<br />
]<br />
. (13)<br />
These equati<strong>on</strong>s determ<strong>in</strong>e the particle moti<strong>on</strong> up to the<br />
sec<strong>on</strong>d order <str<strong>on</strong>g>of</str<strong>on</strong>g> ϵ, which varies slowly compared with the<br />
period <str<strong>on</strong>g>of</str<strong>on</strong>g> the fast oscillati<strong>on</strong> appeared <strong>in</strong> the zeroth-order<br />
orbit.<br />
In the case 1/L 2 +1/R ≥ 0, we obta<strong>in</strong> a slow oscillatory<br />
moti<strong>on</strong> given by<br />
P ′′<br />
x = α s<strong>in</strong> θη + P (2)<br />
x0<br />
X ′′ = − α 1<br />
(cos θη − 1) +<br />
mc θζ0kz<br />
cos θη, (14)<br />
P (2)<br />
x0<br />
mca0x0<br />
l<br />
s<strong>in</strong> θη + X′′ 0 ,<br />
θ<br />
(15)<br />
where θ = l √ (1/L 2 + 1/R) /2, α is a c<strong>on</strong>-<br />
stant determ<strong>in</strong>ed by the <strong>in</strong>itial c<strong>on</strong>diti<strong>on</strong> as α =<br />
mca0x0θ ( 1 − l/ ( 2Lθ 2) − 7l/ (8L) ) , X ′′<br />
0 is the <strong>in</strong>itial<br />
particle positi<strong>on</strong> and P (2)<br />
x0 is the sec<strong>on</strong>d-order <strong>in</strong>itial value<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> P ′′<br />
x calculated by the sec<strong>on</strong>d-order backward Lie transformati<strong>on</strong>.<br />
It is remarkably noted that the unbounded secular<br />
moti<strong>on</strong> orig<strong>in</strong>at<strong>in</strong>g from the first-order p<strong>on</strong>deromotive<br />
force given <strong>in</strong> Eq. (10) is changed to the bounded soluti<strong>on</strong>,<br />
Eqs. (14) and (15), by tak<strong>in</strong>g <strong>in</strong>to account the sec<strong>on</strong>d order<br />
curvature terms. This moti<strong>on</strong> corresp<strong>on</strong>ds to a betatr<strong>on</strong><br />
oscillati<strong>on</strong> by which the particle is c<strong>on</strong>f<strong>in</strong>ed <strong>in</strong> the f<strong>in</strong>ite<br />
radial regi<strong>on</strong>. Note that s<strong>in</strong>ce the amplitude factor α and<br />
the period θ are def<strong>in</strong>ed by the local gradient and curvature<br />
at the <strong>in</strong>itial particle positi<strong>on</strong>, they are valid <strong>on</strong>ly <strong>in</strong> the<br />
regi<strong>on</strong> where the variati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the curvature is sufficiently<br />
small dur<strong>in</strong>g <strong>on</strong>e cycle <str<strong>on</strong>g>of</str<strong>on</strong>g> the l<strong>on</strong>g period oscillati<strong>on</strong>s.<br />
In the case 1/L2 + 1/R < 0, Eqs. (12) and (13) yield to<br />
the soluti<strong>on</strong><br />
P ′′<br />
x =<br />
(2)<br />
α + P x0<br />
e<br />
2<br />
θη +<br />
(2)<br />
−α + P x0<br />
e<br />
2<br />
−θη . (16)<br />
This soluti<strong>on</strong> <strong>in</strong>dicates that the particle is rapidly ejected<br />
from the regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> large laser field amplitude. Tak<strong>in</strong>g the<br />
expansi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the right-hand side <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (16) assum<strong>in</strong>g θη ∼<br />
O (ϵ), Eq. (16) leads to<br />
P ′′<br />
x = αθη + P (2)<br />
x0<br />
X ′′ = α<br />
mc<br />
1<br />
kzζ0<br />
, (17)<br />
θ<br />
2 η2 +<br />
P (2)<br />
x0<br />
mc<br />
1<br />
kzζ0<br />
η + X ′′<br />
0 . (18)<br />
This soluti<strong>on</strong> is c<strong>on</strong>sistent with that obta<strong>in</strong>ed <strong>in</strong> Eq. (10)<br />
up to the first order, though the sec<strong>on</strong>d order collecti<strong>on</strong> is<br />
<strong>in</strong>cluded <strong>in</strong> Eqs. (17) and (18).<br />
Here, we have neglected the z-comp<strong>on</strong>ent <str<strong>on</strong>g>of</str<strong>on</strong>g> the vector<br />
potential, az, for simplicity. The <strong>in</strong>clusi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> az may cause<br />
modulati<strong>on</strong> to the amplitude factor α and/or the period θ,<br />
which will be discussed separately.<br />
SUMMARY<br />
We derived a equati<strong>on</strong> system describ<strong>in</strong>g the relativistic<br />
p<strong>on</strong>deromotive force and the related particle dynamics<br />
<strong>in</strong> a transversely-focused l<strong>in</strong>early-polarized laser field<br />
up to the sec<strong>on</strong>d order with respect to ϵ. In the first order,<br />
we obta<strong>in</strong>ed the p<strong>on</strong>deromotive force proporti<strong>on</strong>al to<br />
the field gradient <strong>in</strong> the x- and y-directi<strong>on</strong>s that is essentially<br />
the same as the result <strong>in</strong> Ref. [6]. In the sec<strong>on</strong>d order,<br />
we found that the particle can exhibit a slow period<br />
betatr<strong>on</strong>-like oscillatory moti<strong>on</strong> characterized by the curvature<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the laser field amplitude. This suggests that the<br />
c<strong>on</strong>trol <str<strong>on</strong>g>of</str<strong>on</strong>g> the curvature is important <strong>in</strong> c<strong>on</strong>f<strong>in</strong><strong>in</strong>g the particle<br />
and keep<strong>in</strong>g the laser-particle <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> transversely<br />
localized high-<strong>in</strong>tensity laser fields. The betatr<strong>on</strong>-like oscillati<strong>on</strong><br />
may cause <strong>in</strong>tense radiati<strong>on</strong> that will be discussed<br />
<strong>in</strong> a future paper. The present result up to the first order<br />
and the expansi<strong>on</strong> form, Eqs. (17) and (18), up to the sec<strong>on</strong>d<br />
order are c<strong>on</strong>sistent with those obta<strong>in</strong>ed by perform<strong>in</strong>g<br />
the perturbati<strong>on</strong> expansi<strong>on</strong> directly to the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong>.<br />
However, <strong>in</strong> the present analysis, the n<strong>on</strong>local soluti<strong>on</strong>s,<br />
Eqs. (14), (15) and (16) are obta<strong>in</strong>ed for the first time<br />
through the Lie perturbati<strong>on</strong> approach.<br />
REFERENCES<br />
[1] E. A. Startsev and C. J. McK<strong>in</strong>strie, Phys. Rev. E 55 (1996)<br />
7527.<br />
[2] P. Gibb<strong>on</strong>, ”Short Pulse Laser Interacti<strong>on</strong>s with Matter”, Imperial<br />
College Press, L<strong>on</strong>d<strong>on</strong>, p. 36 (2005).<br />
[3] J. R. Cary and R. G. Littlejohn, Ann. Phys. 151 (1983) 1.<br />
[4] Y. Kishimoto, S. Tokuda and K. Sakamoto, Phys. Plasmas 2<br />
(1995) 1316.<br />
[5] K. Imadera and Y. Kishimoto, accepted for publicati<strong>on</strong> <strong>in</strong><br />
Plasma Fusi<strong>on</strong> Res..<br />
[6] N. Iwata, K. Imadera and Y. Kishimoto, Plasma Fusi<strong>on</strong> Res.<br />
5 (2010) 028.<br />
[7] E. S. Sarachik and G. T. Schappert, Phys. Rev. D 1 (1970)<br />
2738.
PARTICLE BASED INTEGRATED CODE EPIC3D<br />
FOR LASER-MATTER INTERACTION<br />
Y. Kishimoto, Graduate School <str<strong>on</strong>g>of</str<strong>on</strong>g> Energy Science, Kyoto University, Uji, Kyoto 611-0011, Japan<br />
Abstract<br />
A complex plasma state, where neutral atoms and<br />
molecules, i<strong>on</strong>s with different charge states, free<br />
electr<strong>on</strong>s and positr<strong>on</strong>s, various wavelength radiati<strong>on</strong>s<br />
<strong>in</strong>clud<strong>in</strong>g X-rays and γ-rays, etc. coexist is established <strong>in</strong><br />
nature and laboratory through complex atomic and<br />
nuclear processes. In such plasmas, the level <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
complexity is <str<strong>on</strong>g>of</str<strong>on</strong>g> especially high due to the synergetic<br />
<strong>in</strong>terplays am<strong>on</strong>g spatio-temporally different scale<br />
dynamics <strong>in</strong>side and outside the Debye sphere. We refer<br />
to this k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> plasma state as synergetic complexity <strong>in</strong><br />
dist<strong>in</strong>cti<strong>on</strong> from that used <strong>in</strong> c<strong>on</strong>venti<strong>on</strong>al ideal plasmas.<br />
Such plasma can be established <strong>in</strong> high-power laser<br />
matter <strong>in</strong>teracti<strong>on</strong>. In order to <strong>in</strong>vestigate such plasmas,<br />
we have developed a comprehensive particle based<br />
<strong>in</strong>tegrated code EPIC3D, which <strong>in</strong>cludes various atomic<br />
and collisi<strong>on</strong>al relaxati<strong>on</strong> processes self-c<strong>on</strong>sistently.<br />
Based <strong>on</strong> the EPIC3D, we performed the simulati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
laser-cluster <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> parameter regimes relevant to<br />
the experiment and also complex discharge/lightn<strong>in</strong>g<br />
process. Through these simulati<strong>on</strong>s, we successfully<br />
reproduced prom<strong>in</strong>ent structure formati<strong>on</strong>s, which lead to<br />
the key physical understand<strong>in</strong>gs <str<strong>on</strong>g>of</str<strong>on</strong>g> complex plasma state.<br />
INTRODUCTION<br />
Interacti<strong>on</strong> am<strong>on</strong>g various material states, charged<br />
particles, energetic phot<strong>on</strong>s, etc. leads to a complex<br />
plasma state, <strong>in</strong> which multiply charged i<strong>on</strong>s, electr<strong>on</strong>s<br />
and positr<strong>on</strong>s, neutral atoms and molecules, etc coexist.<br />
Such plasmas exhibit the characteristics as an active<br />
medium which is highly n<strong>on</strong>-l<strong>in</strong>er, n<strong>on</strong>-equilibrium and<br />
n<strong>on</strong>-stati<strong>on</strong>ary, and can be seen not <strong>on</strong>ly <strong>in</strong> laboratory, but<br />
also <strong>in</strong> space and universe, such as aurora, lightn<strong>in</strong>g, solar<br />
flares, <strong>in</strong>ter-stellar medium, accreti<strong>on</strong> disks, etc. We refer<br />
to this k<strong>in</strong>d <str<strong>on</strong>g>of</str<strong>on</strong>g> plasma state as synergetic complexity <strong>in</strong><br />
dist<strong>in</strong>cti<strong>on</strong> from that used <strong>in</strong> c<strong>on</strong>venti<strong>on</strong>al ideal plasmas.<br />
Such plasma can be also established <strong>in</strong> high-power laser<br />
matter <strong>in</strong>teracti<strong>on</strong>.<br />
The particle-<strong>in</strong>-cell (PIC) method that solves the<br />
n<strong>on</strong>l<strong>in</strong>ear wave–particle <strong>in</strong>teracti<strong>on</strong> has been widely used<br />
<strong>in</strong> simulat<strong>in</strong>g complex plasma dynamics. However, most<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> them a priori made an assumpti<strong>on</strong> that the plasma is<br />
fully i<strong>on</strong>ized <strong>in</strong> the <strong>in</strong>itial state. For applicati<strong>on</strong>s utiliz<strong>in</strong>g<br />
relatively high-Z materials, complex atomic and<br />
relaxati<strong>on</strong> processes play an important role <strong>in</strong> determ<strong>in</strong><strong>in</strong>g<br />
the <strong>in</strong>teracti<strong>on</strong> and subsequent dynamics. Here, <strong>in</strong> order to<br />
<strong>in</strong>vestigate such complex plasmas, we have developed a<br />
three-dimensi<strong>on</strong>al particle based <strong>in</strong>tegrated code, EPIC3D,<br />
<strong>in</strong> which complex atomic and relaxati<strong>on</strong> processes are<br />
self-c<strong>on</strong>sistently <strong>in</strong>cluded [1].<br />
Based <strong>on</strong> the EPIC3D, we performed the simulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
laser-cluster <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> parameter regimes relevant to<br />
the experiment. We also performed the simulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
lightn<strong>in</strong>g/discharge process <str<strong>on</strong>g>of</str<strong>on</strong>g> a compressed ne<strong>on</strong> gas and<br />
successfully reproduced key physical processes such as<br />
streamer formati<strong>on</strong> and sprite events. Lightn<strong>in</strong>g and<br />
discharges are a well known process, but the details have<br />
not been fully clarified. The understand<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the process<br />
is crucially important not <strong>on</strong>ly from the academic<br />
viewpo<strong>in</strong>t but also for various <strong>in</strong>dustrial applicati<strong>on</strong>s.<br />
SIMULATION MODEL OF EPIC3D<br />
Here we describe EPIC3D, which was orig<strong>in</strong>ally based<br />
<strong>on</strong> a fully relativistic three-dimensi<strong>on</strong>al electromagnetic<br />
PIC technique. The phot<strong>on</strong> field is classically treated by<br />
the Maxwell equati<strong>on</strong>s <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> electric and magnetic<br />
field (E,B) or its potential form (A, φ). The charge density<br />
ρ and current density j are assigned <strong>on</strong> each spatial grid<br />
us<strong>in</strong>g the cloud-<strong>in</strong>-cell (CIC) method. For the (A, φ)<br />
versi<strong>on</strong>, the Poiss<strong>on</strong> equati<strong>on</strong> as well as the c<strong>on</strong>t<strong>in</strong>uity<br />
equati<strong>on</strong> are imposed, while for the (E,B) versi<strong>on</strong>, a local<br />
solver technique is employed [2]. We extended the model<br />
by <strong>in</strong>corporat<strong>in</strong>g the i<strong>on</strong>izati<strong>on</strong> and relaxati<strong>on</strong> processes<br />
described <strong>in</strong> the follow<strong>in</strong>g.<br />
2.1. Collisi<strong>on</strong> and relaxati<strong>on</strong> process<br />
We here <strong>in</strong>troduced a relativistic pair<strong>in</strong>g method by<br />
successive b<strong>in</strong>ary collisi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> particle pairs, which<br />
precisely c<strong>on</strong>serves the momentum and energy before and<br />
after the collisi<strong>on</strong> event <strong>in</strong> a relativistic regime. It has<br />
been certified <strong>in</strong> [3] that the method successfully resolves<br />
n<strong>on</strong>-local electr<strong>on</strong> heat transport where the temperature<br />
scale length becomes comparable to that <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong><br />
mean free path and the assumpti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Fick’s law (or<br />
diffusi<strong>on</strong> approximati<strong>on</strong>) breaks up.<br />
2.2. I<strong>on</strong>izati<strong>on</strong> process due to phot<strong>on</strong> field (Optical Field<br />
I<strong>on</strong>izati<strong>on</strong>-process)<br />
We <strong>in</strong>troduced an additi<strong>on</strong>al dimensi<strong>on</strong> <strong>in</strong> each atom<br />
that represents the <strong>in</strong>ternal electr<strong>on</strong>ic state <str<strong>on</strong>g>of</str<strong>on</strong>g> the atom [4].<br />
The i<strong>on</strong>izati<strong>on</strong> rate W (k) j (E), where the suffix denotes the<br />
k-th electr<strong>on</strong>ic state <str<strong>on</strong>g>of</str<strong>on</strong>g> j-th atom, is calculated from the<br />
local electric field E. Here we employed a cycle-averaged<br />
tunnel<strong>in</strong>g i<strong>on</strong>izati<strong>on</strong> rate <str<strong>on</strong>g>of</str<strong>on</strong>g> the Ammosov–Debre-Kra<strong>in</strong>ov<br />
(ADK) formula [5]. Then, the i<strong>on</strong>izati<strong>on</strong> <strong>in</strong>dex def<strong>in</strong>ed <strong>in</strong><br />
the range {0,1} is obta<strong>in</strong>ed as<br />
R (k) j (t) = 1 – exp[−W (k) j (E)Δt ]. (2.1)
log N (PIC particle)<br />
When the c<strong>on</strong>diti<strong>on</strong> Rj>R[0, 1] is satisfied, a free electr<strong>on</strong><br />
with the zero k<strong>in</strong>etic energy (Ee =0) is created and then<br />
the i<strong>on</strong> charge state is <strong>in</strong>creased by <strong>on</strong>e. Here, R[0,1]<br />
represents the random number def<strong>in</strong>ed <strong>in</strong> the range {0,1}.<br />
2.3. I<strong>on</strong>izati<strong>on</strong> process due to electr<strong>on</strong> impact<br />
Here we utilize the electr<strong>on</strong>–i<strong>on</strong> pairs <strong>in</strong>troduced <strong>in</strong> Sec.<br />
2.1 for the Coulomb collisi<strong>on</strong>s <strong>in</strong>side the computati<strong>on</strong>al<br />
mesh. After the Coulomb scatter<strong>in</strong>g takes place ( p (i) e →<br />
p (f) e), the i<strong>on</strong>izati<strong>on</strong> <strong>in</strong>dex is obta<strong>in</strong>ed from the cross<br />
secti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> electr<strong>on</strong> impact i<strong>on</strong>izati<strong>on</strong> σ (e) i (Ee). Thus, the<br />
i<strong>on</strong>izati<strong>on</strong> event is successively determ<strong>in</strong>ed for every<br />
electr<strong>on</strong>–i<strong>on</strong> pair. In the case where the i<strong>on</strong>izati<strong>on</strong> is<br />
switched <strong>on</strong>, a free electr<strong>on</strong> with Ee =0 is created from the<br />
i<strong>on</strong>. The i<strong>on</strong>izati<strong>on</strong> energy is subtracted from that <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
Coulomb-scattered electr<strong>on</strong> to keep the total momentum<br />
and energy c<strong>on</strong>servati<strong>on</strong>s.<br />
LASER-CLUSTER INTERACTION<br />
In Advance Phot<strong>on</strong> Research Center <str<strong>on</strong>g>of</str<strong>on</strong>g> JAEA,<br />
<strong>in</strong>teracti<strong>on</strong> experiments us<strong>in</strong>g Xe and Ar clusters and an<br />
ultra-short pulse laser <strong>in</strong> the range <str<strong>on</strong>g>of</str<strong>on</strong>g> 10 17 ~10 18 W/cm 2<br />
were performed and the energy distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> multiply<br />
charged i<strong>on</strong>s and c<strong>on</strong>trol <str<strong>on</strong>g>of</str<strong>on</strong>g> the energy distributi<strong>on</strong><br />
utiliz<strong>in</strong>g a pulse shap<strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser have been studied [2].<br />
In order to clarify the underly<strong>in</strong>g physical process <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<strong>in</strong>teracti<strong>on</strong>s, we performed simulati<strong>on</strong>s us<strong>in</strong>g the EPIC3D.<br />
The parameters are given as follows: cluster radius: a =<br />
24nm, electr<strong>on</strong> density: ne = 2.44 × 10 22 cm −3 , laser<br />
wavelength: λ1 = 820nm, pulse length: τ1 = 20fsec,<br />
maximum laser amplitude: IL =1.0×10 18 W/cm 2 .<br />
Figure 1 illustrates the i<strong>on</strong> charge state distributi<strong>on</strong> at<br />
t = 61fsec. The Ne-like charge state (Ar +8 ) is almost<br />
i<strong>on</strong>ized and the peak appears at Ar +9 . Figure 2 shows the<br />
i<strong>on</strong> energy distributi<strong>on</strong> for different charge state. The<br />
maximum i<strong>on</strong> energy <strong>in</strong>creases with the i<strong>on</strong> charge state<br />
and the energy distributi<strong>on</strong> with higher charge state (Ar +12<br />
− Ar +16 ) exhibits an <strong>in</strong>verted structure that is typically<br />
seen <strong>in</strong> cluster Coulomb explosi<strong>on</strong>. Figure 3 illustrates<br />
the density distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Ar +9 (a) and Ar +16 (b),<br />
respectively. The i<strong>on</strong>s with Ar +9 distributes dom<strong>in</strong>antly <strong>in</strong><br />
the core regi<strong>on</strong>, while those <str<strong>on</strong>g>of</str<strong>on</strong>g> Ar +16 distribute around the<br />
2.5 10 5<br />
2 10 5<br />
1.5 10 5<br />
1 10 5<br />
5 10 4<br />
Ar +9<br />
Ar +10<br />
Ar +11<br />
Ar +16<br />
0<br />
0 5 10<br />
charge stae<br />
15 20<br />
Fig. 1. I<strong>on</strong> charge<br />
state distributi<strong>on</strong> at<br />
t = 61fsec after the<br />
<strong>in</strong>teracti<strong>on</strong>. Field<br />
i<strong>on</strong>izati<strong>on</strong> and<br />
electr<strong>on</strong> impact<br />
i<strong>on</strong>izati<strong>on</strong> are taken<br />
<strong>in</strong>to account <strong>in</strong> the<br />
simulati<strong>on</strong>.<br />
log{f(E +n )}<br />
10 -1<br />
10 -2<br />
10 -3<br />
10 -4<br />
10 -5<br />
10 -6<br />
10 -1<br />
Ar +9 Ar +8<br />
10 0<br />
Ar +10 Ar +11<br />
10 1<br />
10 2<br />
log{E +n (keV)}<br />
Fig.2 (a)<br />
Ar +16<br />
(a) Z=9<br />
(b) Z=16<br />
Fig. 2 Energy<br />
distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong>s<br />
with different charge<br />
state at t = 61fsec.<br />
Different symbols,<br />
□ for Z=8, + for Z=9,<br />
△ for Z=15, ○ for<br />
Z=16, are used.<br />
Fig. 3. I<strong>on</strong> density distributi<strong>on</strong> <strong>in</strong> (x,y) doma<strong>in</strong> for different<br />
charge state, (a) Z = 9 and (b) Z = 16, at t = 61 fsec. The<br />
polarizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> laser electric field is x-directi<strong>on</strong>.<br />
expansi<strong>on</strong> fr<strong>on</strong>t. This is due to the fact that i<strong>on</strong>s with<br />
higher charge state are produced by str<strong>on</strong>g polarizati<strong>on</strong><br />
fields generated near the cluster surface [4]. Such<br />
polarizati<strong>on</strong> field is found to reach the value<br />
approximately 5 times larger than that <str<strong>on</strong>g>of</str<strong>on</strong>g> the orig<strong>in</strong>al laser<br />
field. I<strong>on</strong>s hav<strong>in</strong>g large accelerati<strong>on</strong> rate provide a<br />
maximum energy around MeV.<br />
DISCHARGE AND LIGHTNING PROCESS<br />
Recently, discharge/lightn<strong>in</strong>g phenomena, which are<br />
observed <strong>in</strong> the earth atmosphere and also <strong>in</strong> the<br />
i<strong>on</strong>osphere, have attracted c<strong>on</strong>siderable attenti<strong>on</strong> [5]. For<br />
example, high energy relativistic electr<strong>on</strong>s and even γrays<br />
were observed dur<strong>in</strong>g such lightn<strong>in</strong>g event. Here, we<br />
performed discharge simulati<strong>on</strong>s for high pressure ne<strong>on</strong><br />
gas with the density <str<strong>on</strong>g>of</str<strong>on</strong>g> 4.6×10 20 cm −3 (17 times the ideal<br />
gas), where high voltage electric field, E = 10 7 V/cm, is<br />
uniformly applied. A t<strong>in</strong>y i<strong>on</strong>izati<strong>on</strong> spot with Ne +2 is<br />
<strong>in</strong>itially set to trigger discharge.<br />
Figure 4 (a)-(c) show the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong> charge<br />
density and Fig. 5 shows the time history <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong> density<br />
with different charge state. The density <str<strong>on</strong>g>of</str<strong>on</strong>g> Ne +1 i<strong>on</strong> is<br />
found to slowly <strong>in</strong>crease, but suddenly exhibits<br />
exp<strong>on</strong>ential growth with fast time scale around t = 43psec.<br />
After the explosive growth <str<strong>on</strong>g>of</str<strong>on</strong>g> Ne +1 i<strong>on</strong>s, the i<strong>on</strong> density<br />
with charge state higher than Ne +1 , i.e. Ne +σ with σ ≥ 2 ,<br />
also explosively <strong>in</strong>creases with the growth rate larger than<br />
that <str<strong>on</strong>g>of</str<strong>on</strong>g> Ne +1 i<strong>on</strong> (Fig.5). Branch-like structures referred as<br />
10 3<br />
Ar +14<br />
Ar +15<br />
Ar +13<br />
Ar +12<br />
10 4
P(k x )<br />
(a) 44.3psec (b) 45.3pec (c) 45.9psec<br />
(d) i<strong>on</strong>izati<strong>on</strong> spots<br />
at t = 42psec<br />
Fig. 4. I<strong>on</strong> density distributi<strong>on</strong> at three different times, (a) 44.3,<br />
(b) 45.3, (c) 45.9psec, after prom<strong>in</strong>ent streamer formati<strong>on</strong> and<br />
avalanches takes place. Fig. 4(d) illustrates the i<strong>on</strong> spot<br />
distributi<strong>on</strong> prior to the avalanche.<br />
log(N)<br />
10 8<br />
10 7<br />
10 6<br />
10 5<br />
10 4<br />
1000<br />
100<br />
0 1 10 4<br />
2 10 4<br />
3 10 4<br />
4 10 4<br />
t(fsec)<br />
Fig. 5. Time history <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong> density for different charge state.<br />
Avalanche <str<strong>on</strong>g>of</str<strong>on</strong>g> Ne +1 i<strong>on</strong> is triggered around t = 43psec. The<br />
regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> t = 40-47psec is shown <strong>in</strong> order to see the details <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
avalanches.<br />
10 -3<br />
10 -4<br />
10 -5<br />
10 -6<br />
10 -7<br />
10 -2<br />
10 -1<br />
k x (wave number)<br />
+4<br />
Ne +1<br />
Ne +2<br />
Ne +3<br />
Ne Spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> B z Fig. 6.<br />
Wave number spectrum<br />
-1.86<br />
kx “streamer” develop from the <strong>in</strong>itial i<strong>on</strong>izati<strong>on</strong> spot<br />
[Fig.4(a)]. However, after the exp<strong>on</strong>ential growth,<br />
neighbor<strong>in</strong>g streamers c<strong>on</strong>nect each other (b) and develop<br />
to a complex net-like structure (c), which c<strong>on</strong>ta<strong>in</strong>s<br />
enormous branches with different spatial scales. This<br />
structure may corresp<strong>on</strong>d to the so-called “sprite”.<br />
10 0<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> self- <strong>in</strong>duced B z field<br />
dur<strong>in</strong>g the avalanche<br />
process at t = 45.3psec.<br />
The power law<br />
dependence with k x -1.86<br />
is obta<strong>in</strong>ed.<br />
Figure 4 (d) illustrates the spatial distributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> i<strong>on</strong><br />
charge state <strong>in</strong> early time before the explosive event takes<br />
place (t = 42psec). Many t<strong>in</strong>y i<strong>on</strong>izati<strong>on</strong> spots are found<br />
to appear <strong>in</strong> the entire system. When the number <str<strong>on</strong>g>of</str<strong>on</strong>g> spots<br />
(or equivalently “pack<strong>in</strong>g fracti<strong>on</strong>” <str<strong>on</strong>g>of</str<strong>on</strong>g> the spot) exceeds a<br />
certa<strong>in</strong> value, micro-scale discharges are triggered<br />
between neighbor<strong>in</strong>g i<strong>on</strong>izati<strong>on</strong> spots. Such a local event<br />
simultaneously propagates over the wide spatial regi<strong>on</strong>,<br />
lead<strong>in</strong>g to explosive “sprite” phenomen<strong>on</strong>. This process is<br />
similar to that <str<strong>on</strong>g>of</str<strong>on</strong>g> “forest burn<strong>in</strong>g” and/or “percolati<strong>on</strong>”<br />
dynamics. Furthermore, s<strong>in</strong>ce the electr<strong>on</strong> current is<br />
driven al<strong>on</strong>g i<strong>on</strong>izati<strong>on</strong> branches c<strong>on</strong>stitut<strong>in</strong>g sprites,<br />
electro-magnetic signals are emitted from the system.<br />
Figure 6 illustrates the wave number spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>duced<br />
magnetic fields obta<strong>in</strong>ed from the data at t = 45.9psec. A<br />
clear power low spectrum is found, suggest<strong>in</strong>g that the<br />
sprite shows a fractal nature that exhibits no special scales.<br />
It is <strong>in</strong>terest<strong>in</strong>g to note that similar spectrum was observed<br />
<strong>in</strong> low frequency electromagnetic signals dur<strong>in</strong>g lightn<strong>in</strong>g<br />
events <strong>in</strong> the atmosphere [5].<br />
CONCLUSION<br />
In order to <strong>in</strong>vestigate the complex plasmas dom<strong>in</strong>ated<br />
by atomic and relaxati<strong>on</strong> processes, we have developed a<br />
three-dimensi<strong>on</strong>al particle based <strong>in</strong>tegrated code, EPIC3D.<br />
Us<strong>in</strong>g the developed code, we performed the simulati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> laser-cluster <strong>in</strong>teracti<strong>on</strong> and also lightn<strong>in</strong>g process and<br />
found the prom<strong>in</strong>ent i<strong>on</strong>izati<strong>on</strong> dynamics. Complex<br />
i<strong>on</strong>izati<strong>on</strong> dynamics and i<strong>on</strong> accelerati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> different<br />
charge state were clarified. We found the complex<br />
structures dur<strong>in</strong>g lightn<strong>in</strong>g/discharge such as streamer and<br />
sprite. A mechanism lead<strong>in</strong>g to the sudden appearance <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the lightn<strong>in</strong>g/discharge was <strong>in</strong>vestigated. Through these<br />
simulati<strong>on</strong>s, we successfully reproduced prom<strong>in</strong>ent<br />
structure formati<strong>on</strong>, which lead to the key physical<br />
understand<strong>in</strong>gs <str<strong>on</strong>g>of</str<strong>on</strong>g> complex plasma state dom<strong>in</strong>ated by<br />
atomic and relaxati<strong>on</strong> processes.<br />
REFERENCES<br />
[1] Y. Kishimoto, Annual Report <str<strong>on</strong>g>of</str<strong>on</strong>g> the Earth Simulator<br />
Center, April 2002-Marcg 2003 (ISSN 1348-5822),<br />
Chapter 4 Epoch-Mak<strong>in</strong>g Simulati<strong>on</strong>, pp.201-205.<br />
[2] Y. Fukuda, K. Yamakawa, et al., Phys. Rev. A67,<br />
061201(R) (2003). Also, Y. Fukuda and K.<br />
Yamakawa, private communicati<strong>on</strong>, 2003.<br />
[3] Y. Kishimoto, T. Masaki, and T. Tajima, Phys.<br />
Plasmas 9, pp.589-601, February, 2002<br />
[4] V.P. Pasko, M.A. Stanley, J.D. Mathews, U.S. Inan,<br />
and T.W. Wood, Nature 416, pp.152-154, March,<br />
2002, H.T. Su, R.R. Hsu, A. BB. Chen et al., Nature<br />
423, pp.974-976, June, 2003<br />
[5] M.A. Uman, The lightn<strong>in</strong>g discharge, Academic, San<br />
Diego, 1987
Abstract<br />
X-RAY GENERATION VIA LASER COMPTON SCATTERING BY<br />
LASER-ACCELERATED ELECTRON BEAM ∗<br />
E. Miura † , R. Kuroda, H. Toyokawa, AIST, Tsukuba, Ibaraki 3058568, Japan<br />
S. Ishii, K. Tanaka, Tokyo University <str<strong>on</strong>g>of</str<strong>on</strong>g> Science, Noda, Chiba 2788510, Japan<br />
X-ray generati<strong>on</strong> by laser Compt<strong>on</strong> scatter<strong>in</strong>g us<strong>in</strong>g a<br />
quasi-m<strong>on</strong>oenergetic electr<strong>on</strong> beam with a narrow energy<br />
spread obta<strong>in</strong>ed by laser-driven plasma-based accelerati<strong>on</strong><br />
is reported. X-rays are produced by the collisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
femtosec<strong>on</strong>d laser pulse (140 mJ, 100 fs) with a quasim<strong>on</strong>oenergetic<br />
electr<strong>on</strong> beam with a peak energy <str<strong>on</strong>g>of</str<strong>on</strong>g> 50<br />
MeV and a charge <strong>in</strong> the m<strong>on</strong>oenergetic peak <str<strong>on</strong>g>of</str<strong>on</strong>g> 30 pC produced<br />
by focus<strong>in</strong>g an <strong>in</strong>tense laser pulse (700 mJ, 40 fs) <strong>on</strong><br />
a He gas jet. A well-collimated X-ray beam with a divergence<br />
angle <str<strong>on</strong>g>of</str<strong>on</strong>g> 5 mrad is obta<strong>in</strong>ed. The maximum phot<strong>on</strong><br />
energy and the yield <str<strong>on</strong>g>of</str<strong>on</strong>g> the X-rays are estimated to be 60<br />
keV and 10 5 phot<strong>on</strong>s/pulse.<br />
INTRODUCTION<br />
In laser-driven plasma-based accelerati<strong>on</strong>, electr<strong>on</strong>s are<br />
accelerated by the electric field <str<strong>on</strong>g>of</str<strong>on</strong>g> a plasma wave driven<br />
by an <strong>in</strong>tense laser pulse[1]. To realize next-generati<strong>on</strong><br />
electr<strong>on</strong> accelerators, the research has been <strong>in</strong>tensively<br />
c<strong>on</strong>ducted over a few decades. S<strong>in</strong>ce 2004, the generati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a well-collimated electr<strong>on</strong> beam with a narrow energy<br />
spread, that is quasi-m<strong>on</strong>oenergetic electr<strong>on</strong> (QME)<br />
beam, has been dem<strong>on</strong>strated by several groups[2, 3, 4, 5].<br />
The generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a QME beam with a peak energy <str<strong>on</strong>g>of</str<strong>on</strong>g> 1<br />
GeV [6], and a QME beam c<strong>on</strong>ta<strong>in</strong><strong>in</strong>g 0.5 nC electr<strong>on</strong>s <strong>in</strong><br />
the m<strong>on</strong>oenergetic peak [5], which is comparable to that<br />
achieved with radio-frequency (rf) accelerators, has been<br />
also dem<strong>on</strong>strated. The road toward the realizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
practical laser electr<strong>on</strong> accelerator is now open.<br />
In laser-driven plasma-based accelerati<strong>on</strong>, a high accelerati<strong>on</strong><br />
field more than 100 GV/m, which corresp<strong>on</strong>ds to<br />
thousands <str<strong>on</strong>g>of</str<strong>on</strong>g> those achieved by rf accelerators, is obta<strong>in</strong>ed.<br />
A compact electr<strong>on</strong> accelerator can be realized us<strong>in</strong>g such<br />
a high accelerati<strong>on</strong> field. Furthermore, the wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the accelerati<strong>on</strong> field, that is the wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g> the plasma<br />
wave, is short, <str<strong>on</strong>g>of</str<strong>on</strong>g> the order <str<strong>on</strong>g>of</str<strong>on</strong>g> tens <str<strong>on</strong>g>of</str<strong>on</strong>g> micrometers. Then,<br />
the electr<strong>on</strong> pulse durati<strong>on</strong> is extremely short, <str<strong>on</strong>g>of</str<strong>on</strong>g> the order<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a few tens <str<strong>on</strong>g>of</str<strong>on</strong>g> femtosec<strong>on</strong>ds. The set <str<strong>on</strong>g>of</str<strong>on</strong>g> such unique characteristics<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> laser-driven plasma-based accelerati<strong>on</strong> enables<br />
us to realize a compact, all-optical, ultrashort X-ray<br />
source based <strong>on</strong> laser Compt<strong>on</strong> scatter<strong>in</strong>g, that is scatter<strong>in</strong>g<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s by energetic electr<strong>on</strong>s.<br />
∗ A part <str<strong>on</strong>g>of</str<strong>on</strong>g> this study was f<strong>in</strong>ancially supported by the Budget for Nuclear<br />
Research <str<strong>on</strong>g>of</str<strong>on</strong>g> the M<strong>in</strong>istry <str<strong>on</strong>g>of</str<strong>on</strong>g> Educati<strong>on</strong>, Culture, Sports, Science, and<br />
Technology, Japan, based <strong>on</strong> the screen<strong>in</strong>g and counsel<strong>in</strong>g by the Atomic<br />
Energy Commissi<strong>on</strong>.<br />
† e-miura@aist.go.jp<br />
So far, generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an ultrashort X-ray pulse by laser<br />
Compt<strong>on</strong> scatter<strong>in</strong>g has been dem<strong>on</strong>strated by us<strong>in</strong>g a femtosec<strong>on</strong>d<br />
laser pulse and a picosec<strong>on</strong>d electr<strong>on</strong> pulse from<br />
rf accelerators[7, 8]. The X-ray pulse durati<strong>on</strong> is determ<strong>in</strong>ed<br />
by the <strong>in</strong>teracti<strong>on</strong> time between the laser and electr<strong>on</strong><br />
pulses. To obta<strong>in</strong> a femtosec<strong>on</strong>d X-ray pulse, 90 ◦<br />
scatter<strong>in</strong>g geometry should be adopted for a picosec<strong>on</strong>d<br />
electr<strong>on</strong> pulse. In c<strong>on</strong>trast, 180 ◦ scatter<strong>in</strong>g (head-<strong>on</strong> collisi<strong>on</strong>)<br />
geometry is available for a femtosec<strong>on</strong>d electr<strong>on</strong><br />
pulse. There are some advantages <str<strong>on</strong>g>of</str<strong>on</strong>g> us<strong>in</strong>g 180 ◦ scatter<strong>in</strong>g<br />
geometry. Even though the electr<strong>on</strong> energy and the charge<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong> beam are the same, the phot<strong>on</strong> energy and the<br />
yield <str<strong>on</strong>g>of</str<strong>on</strong>g> X-rays are higher than those for 90 ◦ scatter<strong>in</strong>g geometry.<br />
Recently, X-ray generati<strong>on</strong> has been dem<strong>on</strong>strated<br />
us<strong>in</strong>g a laser-accelerated electr<strong>on</strong> beam[9]. However, the<br />
X-ray energy was around 1 keV and the yield was not so<br />
high, because an electr<strong>on</strong> beam with Maxwell-like energy<br />
distributi<strong>on</strong> was used. To obta<strong>in</strong> a bright X-ray source with<br />
higher phot<strong>on</strong> energy, a QME beam with a higher energy<br />
and a larger charge is necessary.<br />
In this paper, we report X-ray generati<strong>on</strong> by laser Compt<strong>on</strong><br />
scatter<strong>in</strong>g us<strong>in</strong>g a QME beam obta<strong>in</strong>ed by laser-driven<br />
plasma-based accelerati<strong>on</strong>.<br />
EXPERIMENTAL CONDITIONS<br />
Figure 1 shows the experimental setup. In the follow<strong>in</strong>g<br />
part, laser pulses for electr<strong>on</strong> accelerati<strong>on</strong> and laser Compt<strong>on</strong><br />
scatter<strong>in</strong>g are called ”ma<strong>in</strong> laser pulse” and ”collid<strong>in</strong>g<br />
laser pulse”, respectively. A p-polarized ma<strong>in</strong> laser pulse<br />
(700 mJ, 40 fs, 800 nm) was focused <strong>on</strong> the edge <str<strong>on</strong>g>of</str<strong>on</strong>g> a He<br />
gas jet us<strong>in</strong>g an <str<strong>on</strong>g>of</str<strong>on</strong>g>f-axis parabolic mirror with a focal length<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> 720 mm. The laser spot diameter <strong>in</strong> vacuum was 13 µm<br />
at full width at half maximum (FWHM). The peak <strong>in</strong>tensity<br />
was 4.7 × 10 18 W/cm 2 . The gas jet was ejected from<br />
a supers<strong>on</strong>ic nozzle with a c<strong>on</strong>ical shape. The diameter <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the nozzle exit was 1.6 mm. The focal positi<strong>on</strong> was set at<br />
1 mm above the nozzle exit. A p-polarized collid<strong>in</strong>g pulse<br />
(140 mJ, 100 fs, 800 nm) was focused around the exit <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
ma<strong>in</strong> laser pulse from the gas jet us<strong>in</strong>g an <str<strong>on</strong>g>of</str<strong>on</strong>g>f-axis parabolic<br />
mirror with a focal length <str<strong>on</strong>g>of</str<strong>on</strong>g> 300 mm. The laser spot diameter<br />
<strong>in</strong> vacuum was 9 µm at FWHM. The <strong>in</strong>cident angle <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the collid<strong>in</strong>g laser pulse was 20 ◦ to the propagati<strong>on</strong> axis <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the ma<strong>in</strong> laser pulse.<br />
X-rays produced by laser Compt<strong>on</strong> scatter<strong>in</strong>g were emitted<br />
<strong>on</strong> the coaxial directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong> beam. The<br />
electr<strong>on</strong> beam was bended by a magnetic field and spatially<br />
separated from the X-rays. Both X-rays and electr<strong>on</strong>s<br />
were <strong>in</strong>cident <strong>on</strong> a phosphor screen (DRZ-HGH, Mit-
the positi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the nozzle exit with 1.6-mm diameter. In<br />
Fig. 3(d), the ma<strong>in</strong> pulse propagated from top to bottom<br />
and the collid<strong>in</strong>g pulse propagated from lower right to upper<br />
left <strong>in</strong> the directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> 20 ◦ to the ma<strong>in</strong> laser propagati<strong>on</strong><br />
axis. As shown by the arrow, a bright spot was observed,<br />
<strong>on</strong>ly the synchr<strong>on</strong>ized collisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the two laser pulses was<br />
achieved. It is supposed that the bright spot <strong>in</strong>dicates the<br />
collisi<strong>on</strong> po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> the two laser pulses. The collisi<strong>on</strong> po<strong>in</strong>t<br />
was set near the edge <str<strong>on</strong>g>of</str<strong>on</strong>g> the nozzle exit, which was near the<br />
extracti<strong>on</strong> positi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong> beam from a plasma.<br />
(a)<br />
(b)<br />
(c)<br />
Collid<strong>in</strong>g Ma<strong>in</strong> Ma<strong>in</strong><br />
(d)<br />
Collisi<strong>on</strong><br />
po<strong>in</strong>t<br />
1 mm<br />
Collid<strong>in</strong>g<br />
Collisi<strong>on</strong><br />
po<strong>in</strong>t<br />
Figure 3: (a)-(c) Shadowgraph images observed for different<br />
delay times <str<strong>on</strong>g>of</str<strong>on</strong>g> a probe pulse to a ma<strong>in</strong> pulse: (a) -1.33<br />
ps, (b) -0.67 ps, and (c) 0 ps. (d) Thoms<strong>on</strong>-scattered light<br />
image when the synchr<strong>on</strong>ized collisi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the ma<strong>in</strong> and collid<strong>in</strong>g<br />
laser pulses is achieved. The bright spot <strong>in</strong>dicates the<br />
collisi<strong>on</strong> po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> the two laser pulse.<br />
X-ray generati<strong>on</strong><br />
X-rays produced by laser Compt<strong>on</strong> scatter<strong>in</strong>g were observed,<br />
when a QME beam with a c<strong>on</strong>siderably high charge<br />
was obta<strong>in</strong>ed. Figure 4(a) shows a image <str<strong>on</strong>g>of</str<strong>on</strong>g> X-rays. The<br />
image was obta<strong>in</strong>ed <strong>in</strong> a s<strong>in</strong>gle shot. From the energyresolved<br />
electr<strong>on</strong> image simultaneously observed with the<br />
image shown <strong>in</strong> Fig. 4(a), the peak energy and the charge <strong>in</strong><br />
the m<strong>on</strong>oenergetic peak <str<strong>on</strong>g>of</str<strong>on</strong>g> the QME beam was 50 MeV and<br />
30 pC, respectively. Figure 4(b) shows the <strong>in</strong>tensity pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the image <strong>in</strong> the vertical directi<strong>on</strong>. The divergence angle<br />
<strong>in</strong> vertical directi<strong>on</strong> was 5 mrad at FWHM. The divergence<br />
angle <strong>in</strong> horiz<strong>on</strong>tal directi<strong>on</strong> was 7 mrad at FWHM. In laser<br />
Compt<strong>on</strong> scatter<strong>in</strong>g, a collimated X-ray beam can be obta<strong>in</strong>ed.<br />
The divergence angle <str<strong>on</strong>g>of</str<strong>on</strong>g> the X-ray beam is given by<br />
∼ 1/γ. Here, γ is the Lorentz factor <str<strong>on</strong>g>of</str<strong>on</strong>g> an electr<strong>on</strong> energy.<br />
The divergence <str<strong>on</strong>g>of</str<strong>on</strong>g> an X-ray beam is estimated to be approximately<br />
10 mrad from the observed peak energy <str<strong>on</strong>g>of</str<strong>on</strong>g> 50 MeV.<br />
The observed divergence angle was close to the predicted<br />
value from the electr<strong>on</strong> energy. The maximum phot<strong>on</strong> energy<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the X-rays was estimated to be 60 keV from the<br />
peak energy <str<strong>on</strong>g>of</str<strong>on</strong>g> the QME beam and the <strong>in</strong>teracti<strong>on</strong> angle.<br />
The X-ray yield was also estimated to be approximately<br />
10 5 phot<strong>on</strong>s/pulse from the charge <str<strong>on</strong>g>of</str<strong>on</strong>g> the QME beam and<br />
the irradiati<strong>on</strong> c<strong>on</strong>diti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the collid<strong>in</strong>g laser pulse by <strong>in</strong>clud<strong>in</strong>g<br />
the dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> the differential cross secti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
scatter<strong>in</strong>g <strong>on</strong> the scattered angle.<br />
The allowance range <str<strong>on</strong>g>of</str<strong>on</strong>g> the delay between the ma<strong>in</strong> and<br />
collid<strong>in</strong>g laser pulses for X-ray generati<strong>on</strong> was <strong>in</strong>vestigated.<br />
The allowance range was approximately 100 fs, corresp<strong>on</strong>d<strong>in</strong>g<br />
to the durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the collid<strong>in</strong>g pulse. This result<br />
suggests that a pulse durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a QME beam is nearly<br />
equal to or less than 100 fs. The generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an ultrashort<br />
electr<strong>on</strong> pulse by laser-drive plasma-based accelerati<strong>on</strong> has<br />
been dem<strong>on</strong>strated at the same time.<br />
70<br />
(a) (b)<br />
Vertical positi<strong>on</strong> [pixel]<br />
60<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
0 200 400 600 800<br />
Intensity [arb. unit]<br />
Figure 4: (a) Image <str<strong>on</strong>g>of</str<strong>on</strong>g> X-rays produced by laser Compt<strong>on</strong><br />
scatter<strong>in</strong>g and (b) the vertical pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile <str<strong>on</strong>g>of</str<strong>on</strong>g> the image. A wellcollimated<br />
X-ray beam with a divergence angle <str<strong>on</strong>g>of</str<strong>on</strong>g> 5 mrad<br />
at FWHM is observed.<br />
SUMMARY<br />
X-ray generati<strong>on</strong> by laser Compt<strong>on</strong> scatter<strong>in</strong>g has been<br />
dem<strong>on</strong>strated us<strong>in</strong>g a QME beam with a peak energy <str<strong>on</strong>g>of</str<strong>on</strong>g> 50<br />
MeV and a charge <strong>in</strong> the m<strong>on</strong>oenergetic peak <str<strong>on</strong>g>of</str<strong>on</strong>g> 30 pC obta<strong>in</strong>ed<br />
by laser-driven plasma-based accelerati<strong>on</strong>. A wellcollimated<br />
X-ray beam with a divergence angle <str<strong>on</strong>g>of</str<strong>on</strong>g> 5 mrad<br />
at FWHM was obta<strong>in</strong>ed. The maximum phot<strong>on</strong> energy and<br />
the yield <str<strong>on</strong>g>of</str<strong>on</strong>g> the X-rays were estimated to be 60 keV and<br />
10 5 phot<strong>on</strong>s/pulse from the peak energy and the charge <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the QME beam.<br />
REFERENCES<br />
[1] T. Tajima and J. M. Daws<strong>on</strong>, Phys. Rev. Lett. 43 (1979) 267.<br />
[2] E. Miura et al., Appl. Phys. Lett. 86 (2005) 251501.<br />
[3] S. P. D. Mangles et al., Nature 431 (2004) 535.<br />
[4] C. G. R. Geddes et al., Nature 431 (2004) 538.<br />
[5] J. Faure et al., Nature 431 (2004) 541.<br />
[6] W. P. Leemans et al., Nature <strong>Physics</strong> 2 (2006) 696.<br />
[7] R. Schoenle<strong>in</strong> et al., Science 274 (1996) 236.<br />
[8] M. Yorozu et al., Appl. Phys. B 74 (2002) 327.<br />
[9] H. Schwoerer et al., Phys. Rev. Lett. 96 (2006) 014802.<br />
[10] S. Masuda et al., Rev. Sci. Instrum. 79 (2008) 083301.<br />
[11] S. Masuda and E. Miura, Appl. Phys. Express 1 (2008)<br />
086002.<br />
[12] E. Miura and S. Masuda, Appl. Phys. Express 2 (2009)<br />
126003.
The electr<strong>on</strong> beam is bent to the dump prevent<strong>in</strong>g it from<br />
strik<strong>in</strong>g the detector.<br />
Upgrade po<strong>in</strong>ts from FFTB<br />
We upgraded this m<strong>on</strong>itor to measure the even smaller<br />
beam sizes to be available at ATF2. The laser wavelength<br />
has been modified from 1064 nm to 532 nm us<strong>in</strong>g a sec<strong>on</strong>d<br />
harm<strong>on</strong>ics generator. The laser optics was newly designed<br />
and c<strong>on</strong>structed by implement<strong>in</strong>g a laser wire scheme to<br />
measure a larger horiz<strong>on</strong>tal beam size, and by enabl<strong>in</strong>g different<br />
cross<strong>in</strong>g angles <str<strong>on</strong>g>of</str<strong>on</strong>g> split laser beams to measure a<br />
wide (diverse) range <str<strong>on</strong>g>of</str<strong>on</strong>g> vertical beam sizes. The gamma<br />
detector for Sh<strong>in</strong>take M<strong>on</strong>itor has also been newly developed.<br />
MEASUREMENT SCHEME<br />
The Sh<strong>in</strong>take M<strong>on</strong>itor employs the <strong>in</strong>terference pattern<br />
created by splitt<strong>in</strong>g laser beams and cross<strong>in</strong>g them at the<br />
focal po<strong>in</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> beam. Laser <strong>in</strong>terference plays<br />
the role <str<strong>on</strong>g>of</str<strong>on</strong>g> chang<strong>in</strong>g the electr<strong>on</strong> beam size <strong>in</strong> corresp<strong>on</strong>dence<br />
to the gamma signal modulati<strong>on</strong>. In their <strong>in</strong>tersect<strong>in</strong>g<br />
regi<strong>on</strong>, the electromagnetic fields <str<strong>on</strong>g>of</str<strong>on</strong>g> the two laser beams<br />
form a stand<strong>in</strong>g wave (<strong>in</strong>terference fr<strong>in</strong>ge). The number <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Compt<strong>on</strong> scattered phot<strong>on</strong>s varies accord<strong>in</strong>g to the phase<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the stand<strong>in</strong>g wave where the electr<strong>on</strong>s pass through.<br />
With a smaller beam size, the number <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s variates<br />
more significantly compared to a large beam size (Fig.3).<br />
Figure 3: Different modulati<strong>on</strong> for different beam size<br />
Therefore the parameter called “Modulati<strong>on</strong> Depth” can<br />
be def<strong>in</strong>ed as<br />
M ≡ N+ − N−<br />
= | cos θ|exp(−2(kyσy)<br />
N+ + N−<br />
2 ) (1)<br />
us<strong>in</strong>g the maximum number <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s N+ and the m<strong>in</strong>imum<br />
number <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s N− are<br />
The beam size can be obta<strong>in</strong>ed from modulati<strong>on</strong> depth.<br />
Calculati<strong>on</strong> formula is<br />
σy = d<br />
√<br />
2π<br />
| cos θ|<br />
2ln( )<br />
M<br />
(2)<br />
In eq.2, d is a distance between peak to peak <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
strength <str<strong>on</strong>g>of</str<strong>on</strong>g> the magnetic field. d = π<br />
ky<br />
ϕ is the half <str<strong>on</strong>g>of</str<strong>on</strong>g> θ.<br />
= λ<br />
2 s<strong>in</strong>(ϕ) .<br />
SYSTEMATIC ERROR SOURCES<br />
There are 2 types <str<strong>on</strong>g>of</str<strong>on</strong>g> errors related to the measurement<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the Sh<strong>in</strong>take M<strong>on</strong>itor. The first type is related to the<br />
strength <str<strong>on</strong>g>of</str<strong>on</strong>g> gamma signal an example <str<strong>on</strong>g>of</str<strong>on</strong>g> which is detector<br />
resoluti<strong>on</strong>, Beam current jitter also bel<strong>on</strong>gs to this category.<br />
This type <str<strong>on</strong>g>of</str<strong>on</strong>g> errors c<strong>on</strong>sequently lead to the error bars<br />
at each gamma signal po<strong>in</strong>t and relate to the statistical error<br />
and the fitt<strong>in</strong>g accuracy <str<strong>on</strong>g>of</str<strong>on</strong>g> the modulati<strong>on</strong>. The sec<strong>on</strong>d<br />
type is related to the reducti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> modulati<strong>on</strong>. For example,<br />
c<strong>on</strong>trast <str<strong>on</strong>g>of</str<strong>on</strong>g> laser fr<strong>in</strong>ge, beam positi<strong>on</strong> jitter bel<strong>on</strong>gs to<br />
this type <str<strong>on</strong>g>of</str<strong>on</strong>g> errors. These c<strong>on</strong>sequently become systematic<br />
beam size measurement errors.<br />
Us<strong>in</strong>g the parameters Cα, Cβ, which represents the reducti<strong>on</strong><br />
factor <str<strong>on</strong>g>of</str<strong>on</strong>g> each sources, the measured modulati<strong>on</strong><br />
Mmeas can be written as<br />
Mmeas = CαCβ...Mideal<br />
Table 1: Error sources<br />
Source Value<br />
Error bar Detector resoluti<strong>on</strong> 8 %<br />
at each po<strong>in</strong>t Beam current jitter 1%(us<strong>in</strong>g ICT)<br />
Laser power jitter 3 %<br />
Total 9 %<br />
Reducti<strong>on</strong> BPM jitter 99.5 %<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> modulati<strong>on</strong> Spherical wavefr<strong>on</strong>t 99.4%<br />
Laser temporalcoherence<br />
99.7%<br />
Laser alighnmentaccuracy<br />
98.1%<br />
Beamsize growth <strong>in</strong>fr<strong>in</strong>ge<br />
99.9%<br />
Tilt <str<strong>on</strong>g>of</str<strong>on</strong>g> fr<strong>in</strong>ge 97.3%<br />
Total 94 %<br />
6 % <str<strong>on</strong>g>of</str<strong>on</strong>g> the reducti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> modulait<strong>on</strong> corresp<strong>on</strong>ds to 2.9<br />
nm systematic error <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> 37 nm measurement.<br />
CROSSING ANGLE MODE<br />
The Sh<strong>in</strong>take M<strong>on</strong>itor for ATF2 has 3 cross<strong>in</strong>g angle<br />
modes for vertical beam size measurement and laser wire<br />
mode for horiz<strong>on</strong>tal beam size measurement.<br />
For the different cross<strong>in</strong>g angle θ, the value <str<strong>on</strong>g>of</str<strong>on</strong>g> d and the<br />
range <str<strong>on</strong>g>of</str<strong>on</strong>g> measurable beam size is shown <strong>in</strong> table 2<br />
The c<strong>on</strong>t<strong>in</strong>uous changable 2-8 degree mode was implemented<br />
for the measurement down to several micr<strong>on</strong>s,<br />
which overlaps with traditi<strong>on</strong>al wire scanner ranges.<br />
(3)
Table 2: cross<strong>in</strong>g angle and the range <str<strong>on</strong>g>of</str<strong>on</strong>g> measurable beam<br />
size<br />
θ d range beam size<br />
2 deg 15.2µ m 1.4 - 6.0 µ m<br />
8 deg 3.81µ m 0.36 -1.4 µ m<br />
30 deg 1.28µ m 100 - 360 nm<br />
174 deg 266nm 25 - 100 nm<br />
A s<strong>in</strong>gle laser path mode, without cross<strong>in</strong>g is also implemented<br />
for functi<strong>on</strong><strong>in</strong>g as a laser wire. It enables to<br />
measure σx beam size.<br />
BEAM TIME RESULT<br />
The measurement with 7.96 deg. mode was performed<br />
until now. Fig.4 is the measurement <strong>in</strong> May <str<strong>on</strong>g>of</str<strong>on</strong>g> this year.<br />
The size was 313 ± 31(stat.) +0<br />
−40 (sys.)nm Corresp<strong>on</strong>d<strong>in</strong>g<br />
modulati<strong>on</strong> depth was 0.85.<br />
Signal Energy / ICT Charge [arb. units]<br />
100<br />
80<br />
60<br />
40<br />
20<br />
0<br />
0 2 4 6 8 10 12 14<br />
phase [rad]<br />
Figure 4: Measurement <str<strong>on</strong>g>of</str<strong>on</strong>g> beam size<br />
At that time, beam c<strong>on</strong>diti<strong>on</strong> was as follows. The background<br />
amount was 15 GeV. S/N was about 10, signal<br />
amount was much more than background. The beta functi<strong>on</strong>(vertical)<br />
at IP was 1mm, 10 larger times than the nom<strong>in</strong>al<br />
value. This was from the beam tun<strong>in</strong>g issue. When<br />
the beam is focused to 37nm, background is estimated to<br />
<strong>in</strong>crease because <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g focus<strong>in</strong>g. But if the background<br />
become 100 times larger, the resoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the detetctor is<br />
less than 10 %.<br />
UPGRADE POINTS IN THE SUMMER<br />
SHUTDOWN 2010<br />
Laser positi<strong>on</strong> stabilizati<strong>on</strong><br />
In order to stabilize the laser positi<strong>on</strong> at IP, an actuator<br />
followd by a PSD have been newly <strong>in</strong>stalled <strong>in</strong> fr<strong>on</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
20 m transport l<strong>in</strong>e<br />
Focal po<strong>in</strong>t scan<br />
In the case <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g>fset <str<strong>on</strong>g>of</str<strong>on</strong>g> the beam from the laser focus<br />
po<strong>in</strong>t, spherical wavefr<strong>on</strong>t effect would cause systematic<br />
errors. To prevent this we added a system for scann<strong>in</strong>g the<br />
laser focus positi<strong>on</strong>.<br />
Tilt m<strong>on</strong>itor<br />
Tilt <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terfrence fr<strong>in</strong>ge relative to the beam axis<br />
would also br<strong>in</strong>g about systematic errors. To counter this<br />
effect, 2 PSDs have been <strong>in</strong>stalled to m<strong>on</strong>itor the tilt <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
beam.<br />
SUMMARY AND NEAR FUTURE PLAN<br />
The study reported <strong>in</strong> this paper can be c<strong>on</strong>cluded based<br />
<strong>on</strong> the follow<strong>in</strong>g three major po<strong>in</strong>ts.<br />
1. Sh<strong>in</strong>take m<strong>on</strong>itor at ATF2 is desg<strong>in</strong>ed to measure vertical<br />
beam sizes from several micr<strong>on</strong> down to 25 nm<br />
2. About 300 nm beam size measurements were performed<br />
with 2 degree laser cross<strong>in</strong>g angle mode<br />
3. 37 nm beam size measurement is planned to be<br />
achieved and verified dur<strong>in</strong>g the next run period(Nov<br />
2010 May 2011)<br />
ACKNOWLEDGMENT<br />
The authors are very grateful for the researchers who<br />
took part <strong>in</strong> the comissi<strong>on</strong><strong>in</strong>g <str<strong>on</strong>g>of</str<strong>on</strong>g> the electr<strong>on</strong> beam at ATF2.<br />
We would like to express our thanks for the support<strong>in</strong>g fund<br />
by <strong>KEK</strong>.<br />
REFERENCES<br />
[1] C. Petit-Jean-Genaz and J. Poole, “JACoW, A service to<br />
the Accelerator Community”, EPAC’04, Lucerne, July 2004,<br />
THZCH03, p. 249, http://www.JACoW.org.<br />
[2] A. Name and D. Pers<strong>on</strong>, Phys. Rev. Lett. 25 (1997) 56.<br />
[3] A.N. Other, “A Very Interest<strong>in</strong>g Paper”, EPAC’96,<br />
Sitges, June 1996, MOPCH31, p. 7984 (1996),<br />
http://www.JACoW.org.<br />
[4] T.Suehara, et al. Nucl. Instrum. Methods Phys. Res., Sect. A<br />
616, 1 (2010)
INVESTIGATING THE ONE-PHOTON ANNIHILATION CHANNEL IN AN<br />
e − e + PLASMA CREATED FROM VACUUM IN STRONG LASER FIELDS<br />
D.B. Blaschke, University <str<strong>on</strong>g>of</str<strong>on</strong>g> Wroclaw, 50-204 Wroclaw, Poland; JINR, 141980 Dubna, Russia<br />
G. Röpke, Institut für Physik, Universität Rostock, D-18051 Rostock, Germany<br />
V.V. Dmitriev, S.A. Smolyansky ∗ , A.V. Tarakanov, Saratov State University, 410026 Saratov, Russia<br />
Abstract<br />
It is well known that <strong>in</strong> the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g external<br />
electromagnetic fields many processes forbidden <strong>in</strong> standard<br />
QED become possible. One example is the <strong>on</strong>ephot<strong>on</strong><br />
annihilati<strong>on</strong> process c<strong>on</strong>sidered recently by the<br />
present authors <strong>in</strong> the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> a k<strong>in</strong>etic approach to<br />
the quasiparticle e − e + γ plasma created from vacuum <strong>in</strong><br />
the focal spot <str<strong>on</strong>g>of</str<strong>on</strong>g> two counter-propagat<strong>in</strong>g laser beams. In<br />
these works the doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> large values <str<strong>on</strong>g>of</str<strong>on</strong>g> the adiabaticity<br />
parameter γ ≫ 1 (corresp<strong>on</strong>d<strong>in</strong>g to multiphot<strong>on</strong> processes)<br />
was c<strong>on</strong>sidered. In the present work we estimate the <strong>in</strong>tensity<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> stemm<strong>in</strong>g from phot<strong>on</strong> annihilati<strong>on</strong> <strong>in</strong><br />
the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> the effective mass model where γ 1,<br />
corresp<strong>on</strong>d<strong>in</strong>g to large electric fields E Ec = m 2 /e and<br />
high ”laser” field frequencies ν m (the doma<strong>in</strong> characteristic<br />
for X-ray lasers <str<strong>on</strong>g>of</str<strong>on</strong>g> the next generati<strong>on</strong>). Under such<br />
limit<strong>in</strong>g c<strong>on</strong>diti<strong>on</strong>s the result<strong>in</strong>g effect is sufficiently large<br />
to be accessible to experimental observati<strong>on</strong>.<br />
INTRODUCTION<br />
The planned experiments [1] for the observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> an<br />
e − e + plasma created from the vacuum <strong>in</strong> the focal spot <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
two counter-propagat<strong>in</strong>g optical laser beams with the <strong>in</strong>tensity<br />
I 10 21 W/cm 2 raises the problem <str<strong>on</strong>g>of</str<strong>on</strong>g> an accurate theoretical<br />
descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the experimental manifestati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the dynamical Schw<strong>in</strong>ger effect [2], see also Refs. [3, 4, 5].<br />
The exist<strong>in</strong>g predicti<strong>on</strong> [6] <strong>in</strong> the doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>gly subcritical<br />
fields E ≪ Ec = m 2 /e <str<strong>on</strong>g>of</str<strong>on</strong>g> a c<strong>on</strong>siderable number<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> sec<strong>on</strong>dary annihilati<strong>on</strong> phot<strong>on</strong>s is not rather c<strong>on</strong>v<strong>in</strong>c<strong>in</strong>g<br />
because it is based <strong>on</strong> the S-matrix approach for<br />
the descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> quasiparticle excitati<strong>on</strong>s <strong>in</strong> the presence<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a str<strong>on</strong>g external electric field. In particular, this approach<br />
does not take <strong>in</strong>to account vacuum polarizati<strong>on</strong> effects.<br />
Apparently, an adequate approach for the descripti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum excitati<strong>on</strong>s <strong>in</strong> str<strong>on</strong>g electromagnetic fields<br />
is a k<strong>in</strong>etic theory <strong>in</strong> the quasiparticle representati<strong>on</strong>. The<br />
simplest k<strong>in</strong>etic equati<strong>on</strong> (KE) <str<strong>on</strong>g>of</str<strong>on</strong>g> such type for the e − e +<br />
subsystem has been obta<strong>in</strong>ed for the case <str<strong>on</strong>g>of</str<strong>on</strong>g> l<strong>in</strong>early polarized,<br />
time dependent and spatially homogeneous electric<br />
fields [2]. Some generalizati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the KE <strong>in</strong> the fermi<strong>on</strong><br />
sector have been worked out <strong>in</strong> Refs. [7, 8, 9]. It can be expected,<br />
that electromagnetic field fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the e − e +<br />
plasma are accompanied by the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> real phot<strong>on</strong>s<br />
∗ smol@sgu.ru<br />
which can be registered far from the focal spot. The first<br />
two equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the BBGKY cha<strong>in</strong> for the phot<strong>on</strong> sector<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the e − e + plasma were obta<strong>in</strong>ed <strong>in</strong> [10, 11]. This level is<br />
sufficient for the k<strong>in</strong>etic descripti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>on</strong>e-phot<strong>on</strong> annihilati<strong>on</strong>.<br />
In the presence <str<strong>on</strong>g>of</str<strong>on</strong>g> an external field such process<br />
is not forbidden [12]. In the works [10, 11] it was shown<br />
that the spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> the sec<strong>on</strong>dary phot<strong>on</strong>s <strong>in</strong> the low frequency<br />
doma<strong>in</strong> k ≪ m has the character <str<strong>on</strong>g>of</str<strong>on</strong>g> the flicker<br />
noise. In the work [13] the <strong>in</strong>clusi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> vacuum polarizati<strong>on</strong><br />
effects <strong>in</strong> the <strong>on</strong>e-phot<strong>on</strong> radiati<strong>on</strong> spectrum led to an<br />
essential change <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> KE which was <strong>in</strong>vestigated<br />
<strong>in</strong> a broad spectral band <strong>in</strong>clud<strong>in</strong>g the annihilati<strong>on</strong> doma<strong>in</strong><br />
ν ∼ 2m. First we have c<strong>on</strong>sidered the doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> large<br />
adiabaticity parameters γ ≫ 1, where the phot<strong>on</strong> radiati<strong>on</strong><br />
from the focal spot turns out to be very small. However,<br />
the tendency <str<strong>on</strong>g>of</str<strong>on</strong>g> the effect to grow for γ → 1 has been discovered.<br />
This is just the doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> practical <strong>in</strong>terest for<br />
parameters <str<strong>on</strong>g>of</str<strong>on</strong>g> modern lasers.<br />
In the present work the effective mass model is c<strong>on</strong>sidered<br />
which allows to <strong>in</strong>vestigate the phot<strong>on</strong> radiati<strong>on</strong> <strong>in</strong> the<br />
doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> rather str<strong>on</strong>g fields not restricted to specific values<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the adiabacity parameter. Some crude estimati<strong>on</strong>s <strong>in</strong><br />
the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> this model [12] lead to an unexpectedly<br />
large total phot<strong>on</strong> producti<strong>on</strong> <strong>in</strong>tensity.<br />
EFFECTIVE MASS MODEL<br />
We will proceed from the phot<strong>on</strong> k<strong>in</strong>etic equati<strong>on</strong><br />
F ˙ ( e<br />
k, t) =<br />
2<br />
4k(2π) 3<br />
t<br />
dt ′<br />
<br />
d 3 pe −iθ(p,p+ k, k;t ′ ,t) ×<br />
×K(p, p + k, k; t ′ , t)f(p, t ′ )f(p + k, t ′ ) + c.c. (1)<br />
for the <strong>on</strong>e-phot<strong>on</strong> annihilati<strong>on</strong> mechanism tak<strong>in</strong>g <strong>in</strong>to account<br />
vacuum polarizati<strong>on</strong> effects <strong>in</strong> the low density approximati<strong>on</strong><br />
[13]. In Eq. (1) F ( k, t) and f(p, t) are the<br />
phot<strong>on</strong> and electr<strong>on</strong> (positr<strong>on</strong>) distributi<strong>on</strong> functi<strong>on</strong>s, respectively,<br />
k is the wave vector <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiated phot<strong>on</strong> and<br />
θ(p1, p2, k; t ′ , t) =<br />
t<br />
t ′<br />
dτ [ω(p1, τ) + ω(p2, τ) − k] (2)<br />
is the high frequency phase.<br />
The two-time c<strong>on</strong>voluti<strong>on</strong> K(p, p+ k, k; t ′ , t) <str<strong>on</strong>g>of</str<strong>on</strong>g> the foursp<strong>in</strong>ors<br />
is a slowly vary<strong>in</strong>g functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> its variables and can
e replaced by its average K → K0 ∼ 1, which is sufficient<br />
for coarse estimati<strong>on</strong>s.<br />
The effective mass model [12] is based <strong>on</strong> the approximati<strong>on</strong><br />
ω(p, t) =<br />
<br />
m 2 +<br />
<br />
p − e 2 A(t) →<br />
→ ω∗(p) = m 2 ∗ + p 2 , (3)<br />
with the effective mass def<strong>in</strong>ed by the relati<strong>on</strong><br />
m 2 ∗ = m 2 + e 2 <br />
A 2<br />
(t) = m 2 + e 2 E 2 0/2ν 2 =<br />
= m 2 (1 + 1/2γ 2 ), (4)<br />
where 〈...〉 denotes the time averag<strong>in</strong>g operati<strong>on</strong>, ν is the<br />
frequency <str<strong>on</strong>g>of</str<strong>on</strong>g> the periodic ”laser” field and E0 is its field<br />
strength amplitude, γ = (Ec/E0)(ν/m) is the adiabaticity<br />
parameter.<br />
In this approximati<strong>on</strong> the phase (2) becomes m<strong>on</strong>ochromatic<br />
θ(p1, p2, k; t ′ , t) = Ω∗(p1, p2, k)(t − t ′ ), (5)<br />
Ω∗(p1, p2, k) = ω∗(p1) + ω∗(p2) − k, (6)<br />
i.e. the approximati<strong>on</strong> (3) leads to a suppressi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> multiphot<strong>on</strong><br />
processes (it corresp<strong>on</strong>ds to the large values <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
adiabacity parameter γ ≫ 1) and the mismatch (6) can be<br />
compensated by the harm<strong>on</strong>ics <str<strong>on</strong>g>of</str<strong>on</strong>g> the fermi<strong>on</strong> distributi<strong>on</strong><br />
functi<strong>on</strong>s <strong>in</strong> Eq. (1) <strong>on</strong>ly.<br />
The <strong>in</strong>specti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the fermi<strong>on</strong> distributi<strong>on</strong> functi<strong>on</strong><br />
shows, <strong>in</strong> particular, that it oscillates basically with twice<br />
the laser frequency<br />
f(p, t) = 1<br />
2 ¯ f(p) [1 − cos(2νt)] . (7)<br />
The substituti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Eqs. (5) and (7) <strong>in</strong>to the k<strong>in</strong>etic equati<strong>on</strong><br />
(1) allows to perform the time <strong>in</strong>tegrati<strong>on</strong>, lead<strong>in</strong>g to<br />
the appearance <str<strong>on</strong>g>of</str<strong>on</strong>g> two harm<strong>on</strong>ics <strong>in</strong> the radiati<strong>on</strong> spectrum<br />
<strong>on</strong>ly (the 2 nd and the 4 th ),<br />
˙<br />
F ( k, t) = −A (2) ( k) cos(2νt) + A (4) ( k) cos(4νt), (8)<br />
A (2) ( k) = π2 K0α<br />
2k<br />
A (4) ( k) = π2 K0α<br />
8k<br />
<br />
<br />
d 3 p<br />
(2π) 3 ¯ f(p) ¯ f(p + k)δ (2ν − Ω∗) , (9)<br />
d 3 p<br />
(2π) 3 ¯ f(p) ¯ f(p + k)δ (4ν − Ω∗) . (10)<br />
It is important that a c<strong>on</strong>stant comp<strong>on</strong>ent is absent here,<br />
because the mismatch (6) could not be compensated <strong>in</strong> this<br />
case by other sources <str<strong>on</strong>g>of</str<strong>on</strong>g> the time dependence <strong>on</strong> the r.h.s.<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (1). 1<br />
Thus, <strong>in</strong> the case <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>f<strong>in</strong>ite system the soluti<strong>on</strong> (8)<br />
can be <strong>in</strong>terpreted as ”breath<strong>in</strong>g” <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> subsystem.<br />
However, the situati<strong>on</strong> is changed, when the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the e − e + γ plasma is c<strong>on</strong>sidered <strong>in</strong> a small spatial doma<strong>in</strong><br />
1 This is <strong>in</strong> c<strong>on</strong>trast to the case γ ≫ 1, where account<strong>in</strong>g for multiphot<strong>on</strong><br />
processes <strong>in</strong> the phase (2) leads to a c<strong>on</strong>stant comp<strong>on</strong>ent [13].<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the focal spot with volume ∼ λ 3 due to the vacuum<br />
c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the absence <str<strong>on</strong>g>of</str<strong>on</strong>g> the e − e + γ plasma <strong>in</strong> the <strong>in</strong>itial<br />
moment <str<strong>on</strong>g>of</str<strong>on</strong>g> switch<strong>in</strong>g <strong>on</strong> the laser field. In this case<br />
<strong>on</strong>e can expect, that all annihilati<strong>on</strong> phot<strong>on</strong>s generated <strong>in</strong><br />
the first half-period <str<strong>on</strong>g>of</str<strong>on</strong>g> the field will leave the volume <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the system and therefore <strong>in</strong> the next half-period the reverse<br />
process (phot<strong>on</strong> transformati<strong>on</strong> to e − e + plasma) will be<br />
impossible. Such a mechanism leads to a pulsati<strong>on</strong> pattern<br />
for the phot<strong>on</strong> radiati<strong>on</strong> from the focal spot. It corresp<strong>on</strong>ds<br />
to <strong>in</strong>troduc<strong>in</strong>g the c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a positive def<strong>in</strong>ite phot<strong>on</strong><br />
producti<strong>on</strong> rate <strong>on</strong> the r.h.s <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (8).<br />
For estimati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the amplitudes (9), (10) let us <strong>in</strong>troduce<br />
the additi<strong>on</strong>al model approximati<strong>on</strong> <strong>in</strong> the spirit <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />
model (3)<br />
ω∗(p + k) → ω∗(p, k) = ω 2 ∗(p) + k 2 (11)<br />
and the isotropisati<strong>on</strong> c<strong>on</strong>diti<strong>on</strong> ¯ f(p + k) → ¯ f(p + k). The<br />
<strong>in</strong>tegrals <strong>on</strong> the r.h.s <str<strong>on</strong>g>of</str<strong>on</strong>g> Eqs. (9), (10) can then be calculated.<br />
For example,<br />
A (2) ( k) = K0α<br />
4k ¯ f(p0) ¯ f(p0 + k) ω∗(p0)ω∗(p0, k)<br />
ω∗(p0) + ω∗(p0, k) p0,<br />
where<br />
p0 =<br />
(12)<br />
<br />
4ν 2 (k + ν) 2<br />
(k + 2ν) 2 − m2 ∗ (13)<br />
is the root <str<strong>on</strong>g>of</str<strong>on</strong>g> the equati<strong>on</strong> Ω∗ − 2ν = 0. From Eq. (13) it<br />
is follows the threshold c<strong>on</strong>diti<strong>on</strong> 2<br />
2ν(k + ν)<br />
k + 2ν m∗ . (14)<br />
This c<strong>on</strong>diti<strong>on</strong> is rather n<strong>on</strong>trivial because the effective<br />
mass (4) depends also <strong>on</strong> ν. The m<strong>in</strong>imal permissible value<br />
ν = 2m∗ corresp<strong>on</strong>ds to k = 0. For the 4 th harm<strong>on</strong>ic the<br />
threshold value falls to ν = m∗, which is close to the parameters<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the XFEL [15].<br />
The 1/k - dependence <strong>on</strong> the r.h.s. <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (12) corresp<strong>on</strong>ds<br />
to the flicker noise. This feature <strong>in</strong> the spectrum<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> radiated annihilati<strong>on</strong> phot<strong>on</strong>s has been found first <strong>in</strong><br />
Ref. [10].<br />
The number <str<strong>on</strong>g>of</str<strong>on</strong>g> phot<strong>on</strong>s with the frequency k ly<strong>in</strong>g <strong>in</strong> the<br />
<strong>in</strong>terval [k, k + dk] and radiated from the focal spot with<br />
the volume λ 3 = ν −3 per time <strong>in</strong>terval is<br />
d 2 N<br />
dtdk<br />
= 8πk2<br />
ν 3<br />
F ˙ ( k, t). (15)<br />
The fracti<strong>on</strong> <strong>on</strong> the r.h.s. <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (12) is a slow functi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the frequencies k and ν and for the sake <str<strong>on</strong>g>of</str<strong>on</strong>g> a prelim<strong>in</strong>ary<br />
estimati<strong>on</strong> it can be replaced by m∗/2. For the 2 nd<br />
harm<strong>on</strong>ic we then obta<strong>in</strong> from Eqs. (12) and (15)<br />
d 2 N (2)<br />
dtdk<br />
= 2παK0km∗<br />
ν 3<br />
¯f(p0) ¯ f(p0 + k)p0 . (16)<br />
2 A similar effect was found first <strong>in</strong> the theory describ<strong>in</strong>g the absorpti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> a weak signal by the e − e + plasma created from vacuum [14].
As a representative characteristics <str<strong>on</strong>g>of</str<strong>on</strong>g> the effectiveness <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the radiati<strong>on</strong> from the focal spot doma<strong>in</strong> we will c<strong>on</strong>sider<br />
the total phot<strong>on</strong> number per time <strong>in</strong>terval,<br />
˙N (2) = 2παK0m∗<br />
ν 3<br />
<br />
kmax<br />
0<br />
dk k ¯ f(p0) ¯ f(p0 + k)p0 . (17)<br />
The electr<strong>on</strong> and positr<strong>on</strong> distributi<strong>on</strong> functi<strong>on</strong>s enter<strong>in</strong>g<br />
here are def<strong>in</strong>ed as the soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the corresp<strong>on</strong>d<strong>in</strong>g n<strong>on</strong>perturbative<br />
k<strong>in</strong>etic equati<strong>on</strong> [2, 7] describ<strong>in</strong>g vacuum creati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> e − e + pairs under the acti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a str<strong>on</strong>g, time dependent<br />
electric field <str<strong>on</strong>g>of</str<strong>on</strong>g> a stand<strong>in</strong>g wave <str<strong>on</strong>g>of</str<strong>on</strong>g> two counter<br />
propagat<strong>in</strong>g laser beams. The cut<str<strong>on</strong>g>of</str<strong>on</strong>g>f parameter kmax =<br />
2m∗ is <strong>in</strong>troduced <strong>in</strong> order to take <strong>in</strong>to account the annihilati<strong>on</strong><br />
phot<strong>on</strong>s <strong>in</strong> the radiati<strong>on</strong> spectrum.<br />
The fermi<strong>on</strong> distributi<strong>on</strong> functi<strong>on</strong> f(p, t) is a rapidly decreas<strong>in</strong>g<br />
functi<strong>on</strong> with its maximum <strong>in</strong> the po<strong>in</strong>t p = 0 [3].<br />
On this basis for a rough estimati<strong>on</strong> <strong>on</strong>e can put p0 = 0 <strong>in</strong><br />
the arguments <str<strong>on</strong>g>of</str<strong>on</strong>g> these functi<strong>on</strong>s <strong>on</strong> the r.h.s. <str<strong>on</strong>g>of</str<strong>on</strong>g> Eq. (17),<br />
˙N (2) = 2παK0m∗<br />
ν 3<br />
¯f(0)<br />
<br />
kmax<br />
0<br />
dk k ¯ f(k)p0 , (18)<br />
where accord<strong>in</strong>g to Eq. (13)<br />
m<br />
p0(k) =<br />
2 <br />
∗<br />
48 + 56<br />
k + 4m∗<br />
k<br />
+ 15<br />
m∗<br />
k2<br />
m2 <br />
∗<br />
m<br />
<br />
48 + 56<br />
4<br />
k<br />
, (19)<br />
m∗<br />
because the small kmax ≪ m∗ is effective <strong>in</strong> the <strong>in</strong>tegral<br />
(18). As the result, we obta<strong>in</strong> the follow<strong>in</strong>g order <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude<br />
estimate<br />
˙N (2) ∼ αm∗ ¯ f 2 (0) . (20)<br />
For the XFEL parameters E0 = 0.24Ec and λ = 15 nm<br />
[15] we have accord<strong>in</strong>g to the k<strong>in</strong>etic theory <strong>in</strong> the e − e +<br />
sector ¯ f(0) ∼ 10 −2 . From Eq. (20) then follows<br />
˙N (2) ∼ 10 17 s −1 . (21)<br />
For the 4 th harm<strong>on</strong>ic with the oscillati<strong>on</strong> amplitude (10)<br />
the threshold for the generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> annihilati<strong>on</strong> phot<strong>on</strong>s is<br />
lowered (see discussi<strong>on</strong> after Eq. (14)) but the <strong>in</strong>tensity <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the phot<strong>on</strong> radiati<strong>on</strong> is also lowered so that the order <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g> (21) rema<strong>in</strong>s unchanged.<br />
SUMMARY<br />
We have c<strong>on</strong>sidered the effective mass model [12] which<br />
allows a rather simple soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the k<strong>in</strong>etic equati<strong>on</strong> describ<strong>in</strong>g<br />
(<strong>in</strong> the framework <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>on</strong>e-phot<strong>on</strong> annihilati<strong>on</strong><br />
mechanism) the phot<strong>on</strong> radiati<strong>on</strong> from the focal spot <str<strong>on</strong>g>of</str<strong>on</strong>g> two<br />
counter-propagat<strong>in</strong>g laser beams. This simple model leads<br />
to c<strong>on</strong>siderable depleti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the spectrum <str<strong>on</strong>g>of</str<strong>on</strong>g> parametric oscillati<strong>on</strong>s<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the e − e + γ plasma: <strong>on</strong>ly the 2 nd and 4 th harm<strong>on</strong>ics<br />
rema<strong>in</strong> due to the c<strong>on</strong>diti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the absence <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
c<strong>on</strong>stant comp<strong>on</strong>ent <strong>in</strong> the phot<strong>on</strong> producti<strong>on</strong> rate. Thus<br />
a compensati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the mismatch (6) is possible by means<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> these two harm<strong>on</strong>ics <strong>on</strong>ly. The doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> applicability<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> this model is limited to the X-ray doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the laser<br />
radiati<strong>on</strong>. The model suggests a high <strong>in</strong>tegral lum<strong>in</strong>osity <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
∼ 10 15 phot<strong>on</strong>s per sec from the focal spot. Other features<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the model are: the 1/k-behavior <strong>in</strong> the <strong>in</strong>frared doma<strong>in</strong><br />
k ≪ m (the flicker noise <str<strong>on</strong>g>of</str<strong>on</strong>g> electrodynamic nature) and the<br />
threshold effect. These results are encourag<strong>in</strong>g for a further<br />
detailed study <str<strong>on</strong>g>of</str<strong>on</strong>g> the phot<strong>on</strong> k<strong>in</strong>etics <strong>on</strong> the basis <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the <strong>on</strong>e-phot<strong>on</strong> annihilati<strong>on</strong> mechanism <strong>in</strong> the doma<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
small adiabacity parameters γ 1.<br />
ACKNOWLEDGEMENTS: We thank our colleagues<br />
G. Gregori, C.D. Murphy, A.V. Prozorkevich, C.D. Roberts<br />
and S. Schmidt for their collaborati<strong>on</strong>. A.M. Fedotov,<br />
D. Habs, B. Kämpfer, H. Ruhl and R. Sauerbrey are acknowledged<br />
for their c<strong>on</strong>t<strong>in</strong>ued <strong>in</strong>terest <strong>in</strong> our work and<br />
valuable discussi<strong>on</strong>s. A.V.T would like to thank the Organiz<strong>in</strong>g<br />
Committee for the warm and stimulat<strong>in</strong>g atmosphere<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the PIF2010.<br />
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Abstract<br />
Unruh radiati<strong>on</strong> and Interference effect ∗<br />
Satoshi Iso † , Yasuhiro Yamamoto ‡ and Sen Zhang § , <strong>KEK</strong>, Tsukuba, Japan<br />
A uniformly accelerated charged particle feels the vacuum<br />
as thermally excited and fluctuates around the classical<br />
trajectory. Then we may expect additi<strong>on</strong>al radiati<strong>on</strong> besides<br />
the Larmor radiati<strong>on</strong>. It is called Unruh radiati<strong>on</strong>. In<br />
this report, we review the calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong><br />
with an emphasis <strong>on</strong> the <strong>in</strong>terference effect between the<br />
vacuum fluctuati<strong>on</strong> and the radiati<strong>on</strong> from the fluctuat<strong>in</strong>g<br />
moti<strong>on</strong>. Our calculati<strong>on</strong> is based <strong>on</strong> a stochastic treatment<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> the particle under a uniform accelerati<strong>on</strong>. The basics <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the stochastic equati<strong>on</strong> are reviewed <strong>in</strong> another report <strong>in</strong> the<br />
same proceed<strong>in</strong>g [2]. In this report, we ma<strong>in</strong>ly discuss the<br />
radiati<strong>on</strong> and the <strong>in</strong>terference effect.<br />
STOCHASTIC ALD EQUATION<br />
The Unruh radiati<strong>on</strong> is the additi<strong>on</strong>al radiati<strong>on</strong> expected<br />
to be emanated by a uniformly accelerated charged particle<br />
[3]. A uniformly accelerated observer feels the quantum<br />
vacuum as thermally excited with the Unruh temperature<br />
TU = a/2πckB. Hence as the ord<strong>in</strong>ary Unruh-de Wit detector,<br />
a charged particle <strong>in</strong>teract<strong>in</strong>g with the radiati<strong>on</strong> field<br />
can be expected to fluctuate around the classical trajectory.<br />
Is there additi<strong>on</strong>al radiati<strong>on</strong> associated with this fluctuat<strong>in</strong>g<br />
moti<strong>on</strong>? It is the issue <str<strong>on</strong>g>of</str<strong>on</strong>g> the present report.<br />
In order to formulate the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> such fluctuat<strong>in</strong>g<br />
moti<strong>on</strong>, we make use <str<strong>on</strong>g>of</str<strong>on</strong>g> the stochastic technique. Namely,<br />
we solve a set <str<strong>on</strong>g>of</str<strong>on</strong>g> equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the accelerated particle and<br />
the radiati<strong>on</strong> field <strong>in</strong> a semiclassical approximati<strong>on</strong>. By<br />
semiclassical, we mean that the radiati<strong>on</strong> field is treated<br />
as a quantum field while the particle is treated classically.<br />
S<strong>in</strong>ce the accelerated particle dissipates its energy<br />
through the Larmor radiati<strong>on</strong>, the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong> c<strong>on</strong>ta<strong>in</strong>s<br />
the radiati<strong>on</strong> damp<strong>in</strong>g term. This is the Abraham-<br />
Lorentz-Dirac (ALD) equati<strong>on</strong>. Furthermore, s<strong>in</strong>ce the accelerated<br />
particle feels the M<strong>in</strong>kowski vacuum as thermally<br />
excited, a noise term is also <strong>in</strong>duced <strong>in</strong> the equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> moti<strong>on</strong>.<br />
The stochastic equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the accelerated charged<br />
particle is called the stochastic ALD equati<strong>on</strong> and derived<br />
by [4].<br />
We c<strong>on</strong>sider the scalar QED whose acti<strong>on</strong> is given by<br />
∫<br />
S[z, ϕ, h] = − m<br />
∫<br />
+<br />
dτ √ ˙z µ ∫<br />
˙zµ +<br />
d 4 x 1 2<br />
(∂µϕ)<br />
2<br />
d 4 x j(x; z)ϕ(x). (1)<br />
∗ Based <strong>on</strong> a poster presentati<strong>on</strong> by Y.Yamamoto and [1].<br />
† satoshi.iso@kek.jp<br />
‡ yamayasu@post.kek.jp<br />
§ zhangsen@post.kek.jp<br />
where<br />
∫<br />
j(x; z) = e<br />
dτ √ ˙z µ ˙zµδ 4 (x − z(τ)), (2)<br />
We choose the parametrizati<strong>on</strong> τ to satisfy ˙z 2 = 1.<br />
By solv<strong>in</strong>g the Heisenberg equati<strong>on</strong> for ϕ, we get the<br />
stochastic ALD equati<strong>on</strong> for the charged particle:<br />
m ˙v µ − F µ − e2<br />
12π (vµ ˙v 2 + ¨v µ ) = −e −→ ω µ ϕh(z) (3)<br />
where v µ = ˙z µ . The dissipative term corresp<strong>on</strong>ds to loss<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> energy through the radiati<strong>on</strong> and it is called the radiati<strong>on</strong><br />
damp<strong>in</strong>g term. On the other hand, the noise term comes<br />
from the Unruh effect, namely, <strong>in</strong>teracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a uniformly<br />
accelerated particle with the thermal bath <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong><br />
field.<br />
We can easily solve the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> small fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the transverse velocities v i = v i 0+δv i <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> the quantum<br />
fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> the field ϕh (or its Fourier tranformed<br />
field φ) as<br />
where<br />
δ˜v i (ω) = eh(ω)∂iφ(ω), (4)<br />
δv i ∫<br />
dω<br />
(τ) =<br />
2π δ˜vi (ω)e −iωτ ,<br />
∫<br />
dω<br />
−iωτ<br />
∂iϕh(τ) = ∂iφ(ω)e<br />
2π<br />
(5)<br />
(6)<br />
h(ω) =<br />
1<br />
. (7)<br />
−imω + e2<br />
12π (ω2 + a 2 )<br />
In the follow<strong>in</strong>g, as an ideal case we c<strong>on</strong>sider a uniformly<br />
accelerated charged particle <strong>in</strong> the scalar QED, and <strong>in</strong>vestigate<br />
the radiati<strong>on</strong> from such a particle. The ma<strong>in</strong> issue is<br />
the effect <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>in</strong>terference.<br />
RADIATION AND INTERFERENCE<br />
Now we calculate the radiati<strong>on</strong> emanated from the uniformly<br />
accelerated charged particle. First let’s c<strong>on</strong>sider the<br />
2-po<strong>in</strong>t functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong> field. S<strong>in</strong>ce the field is<br />
written as a sum <str<strong>on</strong>g>of</str<strong>on</strong>g> the quantum fluctuati<strong>on</strong> (a homogeneous<br />
soluti<strong>on</strong>) ϕh and the <strong>in</strong>homogeneous soluti<strong>on</strong> <strong>in</strong> the<br />
presence <str<strong>on</strong>g>of</str<strong>on</strong>g> the charged particle ϕI, the 2-po<strong>in</strong>t functi<strong>on</strong> is<br />
given by<br />
⟨ϕ(x)ϕ(y)⟩ − ⟨ϕh(x)ϕh(y)⟩ (8)<br />
= ⟨ϕI(x)ϕh(y)⟩ + ⟨ϕh(x)ϕI(y)⟩ + ⟨ϕI(x)ϕI(y)⟩.<br />
The Unruh radiati<strong>on</strong> estimated <strong>in</strong> [3] is c<strong>on</strong>ta<strong>in</strong>ed <strong>in</strong><br />
⟨ϕIϕI⟩, which <strong>in</strong>clude the Larmor radiati<strong>on</strong>. We need special<br />
care <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference terms. As discussed <strong>in</strong> [5],
the <strong>in</strong>terference terms ⟨ϕIϕh⟩ + ⟨ϕhϕI⟩ may possibly cancel<br />
the Unruh radiati<strong>on</strong> <strong>in</strong> ⟨ϕIϕI⟩ after the thermalizati<strong>on</strong><br />
occurs. The cancellati<strong>on</strong> is explicitly shown for an <strong>in</strong>ternal<br />
detector, but it is not obvious whether the same cancellati<strong>on</strong><br />
occurs for the case <str<strong>on</strong>g>of</str<strong>on</strong>g> a charged particle we are c<strong>on</strong>sider<strong>in</strong>g.<br />
The <strong>in</strong>homogeneous soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the scalar field is written<br />
as<br />
∫<br />
ϕI(x) = e<br />
dτGR(x − z(τ)) =<br />
e<br />
. (9)<br />
4πρ(x)<br />
ρ(x) = ˙z(τ x −) · (x − z(τ x −)), (10)<br />
where τ x − satisfies (x − z(τ x −)) 2 = 0, x0 > z0 (τ x −),<br />
which is the proper time <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle whose radiati<strong>on</strong><br />
travels to the space-time po<strong>in</strong>t x. Hence, z(τ x −) lies <strong>on</strong> an<br />
<strong>in</strong>tersecti<strong>on</strong> between the particle’s world l<strong>in</strong>e and the light<br />
c<strong>on</strong>e extend<strong>in</strong>g from the observer’s positi<strong>on</strong> x (See Fig 1).<br />
We write the superscript x to make the x dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> τ<br />
explicitly.<br />
The particle’s trajectory is fluctuat<strong>in</strong>g and expressed as<br />
z = z0 + δz + δ2z + · · · where we have expanded the<br />
tragectory with respect to the <strong>in</strong>teracti<strong>on</strong> with the radiati<strong>on</strong><br />
field (i.e. e). Then ρ is also expanded as ρ = ρ0 + δρ +<br />
δ2ρ + · · · and (9) becomes<br />
(<br />
) )<br />
2<br />
ϕI = e<br />
4πρ0<br />
1 − δρ<br />
+<br />
ρ0<br />
( δρ<br />
ρ0<br />
− δ2 ρ<br />
ρ0<br />
+ · · ·<br />
. (11)<br />
The first term is the classical potential, but s<strong>in</strong>ce the particle’s<br />
trajectory deviates from the classical <strong>on</strong>e, the potential<br />
also receives correcti<strong>on</strong>s.<br />
Inhomogeneous part<br />
By <strong>in</strong>sert<strong>in</strong>g the expansi<strong>on</strong> (11), the correlator <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>homogeneous<br />
soluti<strong>on</strong> ϕI becomes<br />
⟨ϕI(x)ϕI(y)⟩ (12)<br />
(<br />
e<br />
) 2 1<br />
=<br />
4π ρ0(x)ρ0(y)<br />
)<br />
×<br />
.<br />
(<br />
1 + ⟨δρ(x)δρ(y)⟩<br />
ρ0(x)ρ0(y) + ⟨(δρ(x))2 ⟩<br />
ρ2 0 (x) + ⟨(δρ(y))2 ⟩<br />
ρ2 0 (y)<br />
The first term gives the Larmor radiati<strong>on</strong>. The other terms<br />
corresp<strong>on</strong>d to the radiati<strong>on</strong> <strong>in</strong>duced by the fluctuati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the particle’s moti<strong>on</strong>.<br />
The calculati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> these terms are easy, because <strong>on</strong>e can<br />
write ⟨δρδρ⟩ <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> ⟨δ ˙z i δ ˙z i ⟩ = ⟨δv i δv i ⟩, which can<br />
be obta<strong>in</strong>ed by solv<strong>in</strong>g the dynamics <str<strong>on</strong>g>of</str<strong>on</strong>g> fluctuati<strong>on</strong>s <strong>in</strong> the<br />
stochastic ALD equati<strong>on</strong> [1, 2]. They become<br />
⟨ϕI(x)ϕI(y)⟩ (13)<br />
(<br />
e<br />
) [<br />
2 1<br />
=<br />
1 + e<br />
4π ρ0(x)ρ0(y)<br />
2<br />
∫<br />
dω<br />
2π |h(ω)|2IS(ω) )<br />
×<br />
]<br />
.<br />
( i i −iω(τ<br />
x y e x y<br />
−−τ− )<br />
ρ0(x)ρ0(y) + xixi ρ2 0 (x) + yiyi ρ2 0 (y)<br />
S<strong>in</strong>ce we are c<strong>on</strong>sider<strong>in</strong>g the fluctuat<strong>in</strong>g moti<strong>on</strong> whose frequency<br />
is smaller than the accelerati<strong>on</strong>, IS can be approximately<br />
given by a 3 /12π 2 .<br />
Figure 1: The hyperbolic l<strong>in</strong>e <strong>in</strong> the right wedge denotes<br />
the world l<strong>in</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle. The po<strong>in</strong>ts OF and OR are<br />
observers <strong>in</strong> the future and right wedges, respectively. For<br />
an observer <strong>in</strong> the right wedge, the light-c<strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the observer<br />
has two <strong>in</strong>tersecti<strong>on</strong>s with the world l<strong>in</strong>e, and the<br />
proper time <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>tersecti<strong>on</strong>s are given by τ R ± . For an<br />
observer <strong>in</strong> the future wedge, there is <strong>on</strong>ly <strong>on</strong>e <strong>in</strong>tersecti<strong>on</strong><br />
<strong>on</strong> the particle’s real trajectory which corresp<strong>on</strong>ds to τ F − .<br />
The other soluti<strong>on</strong> T F + = τ F + + iπ/a is complex. One may<br />
<strong>in</strong>terpret this complex proper time as the <strong>in</strong>tersecti<strong>on</strong> between<br />
the light-c<strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the observer and the world l<strong>in</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> a<br />
virtual particle with a real proper time τ F + <strong>in</strong> the left wedge.<br />
The superscript letters R or F are used to dist<strong>in</strong>guish two<br />
different observers, but we do not use them <strong>in</strong> the body <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
the paper to leave the space for the observer’s positi<strong>on</strong> x.<br />
Interference Term<br />
The calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference terms is a bit more<br />
<strong>in</strong>volved. The <strong>in</strong>homogeneous soluti<strong>on</strong> ϕI is expanded as<br />
(11). S<strong>in</strong>ce the lead<strong>in</strong>g term which has a n<strong>on</strong>vanish<strong>in</strong>g correlati<strong>on</strong><br />
with the quantum fluctuati<strong>on</strong> ϕh is the sec<strong>on</strong>d term,<br />
we have<br />
⟨ϕI(x)ϕh(y)⟩ + ⟨ϕh(x)ϕI(y)⟩<br />
= − e<br />
(<br />
⟨δρ(x)ϕh(y)⟩<br />
4π<br />
ρ 2 0 (x)<br />
+ ⟨ϕh(x)δρ(y)⟩<br />
ρ 2 0 (y)<br />
)<br />
. (14)<br />
The fluctuati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the distance δρ is written <strong>in</strong> terms <str<strong>on</strong>g>of</str<strong>on</strong>g> δv i<br />
which is the soluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the stochastic ALD equati<strong>on</strong> (4),<br />
and we obta<strong>in</strong><br />
⟨ϕh(x)δρ(y)⟩ = −ey i<br />
∫<br />
dω<br />
y<br />
e−iωτ−h(ω)⟨ϕh(x)∂iφ(ω)⟩. 2π
The <strong>in</strong>tegrand can be written as<br />
∫<br />
⟨ϕh(x)∂iφ(ω)⟩ = dτe iωτ<br />
( )<br />
∂<br />
⟨ϕh(x)ϕh(y)⟩<br />
∂yi y=z(τ)<br />
∫<br />
= − dτe iωτ<br />
(<br />
∂P (x, ω)<br />
∂xi )<br />
, (15)<br />
where<br />
∫<br />
P (x, ω) = dτ<br />
e iωτ<br />
(x 0 − z 0 (τ) − iϵ) 2 − (x 1 − z 1 (τ)) 2 − x 2 ⊥<br />
x 2 ⊥ = (x2 ) 2 + (x 3 ) 2 is the transverse distance. The τ <strong>in</strong>tegral<br />
can be calculated by the c<strong>on</strong>tour <strong>in</strong>tegral. The residues<br />
are located where the <strong>in</strong>variant length between the observed<br />
po<strong>in</strong>t x and a po<strong>in</strong>t <strong>on</strong> the particle’s trajectory vanishes.<br />
The c<strong>on</strong>diti<strong>on</strong> is noth<strong>in</strong>g but the c<strong>on</strong>diti<strong>on</strong> that the radiati<strong>on</strong><br />
field propagates <strong>on</strong> the light c<strong>on</strong>e. Fig.1 shows such<br />
a situati<strong>on</strong>. It is <strong>in</strong>terest<strong>in</strong>g that the c<strong>on</strong>diti<strong>on</strong> for residues<br />
has a soluti<strong>on</strong> <strong>on</strong> an <strong>in</strong>tersecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the light-c<strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> the observer<br />
and the virtual path <str<strong>on</strong>g>of</str<strong>on</strong>g> a particle (dotted l<strong>in</strong>e <strong>in</strong> the<br />
left wedge). We skip the calculati<strong>on</strong>s and show the f<strong>in</strong>al<br />
results <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference terms;<br />
⟨ϕI(x)ϕh(y)⟩ + ⟨ϕh(x)ϕI(y)⟩ (16)<br />
=<br />
×<br />
where<br />
−iae2xiy i<br />
(4π) 2ρ0(x) 2ρ0(y) 2<br />
[<br />
e<br />
x y<br />
−iω(τ−−τ e<br />
+ e<br />
− e<br />
∫<br />
dω 1<br />
2π 1 − e−2πω/a − ) h(−ω) ( aL2 x<br />
2ρ0(x)<br />
x y<br />
−iω(τ−−τ− ) − h(ω) ( aL2y 2ρ0(y)<br />
x<br />
−iω(τ+ −τ y<br />
x<br />
−iω(τ− − iω<br />
a<br />
)<br />
iω )<br />
+<br />
a<br />
− ) h(−ω) ( − aL2x iω )<br />
− Zx(−ω)<br />
2ρ0(x) a<br />
y<br />
−τ+ ) h(ω) ( − aL2y iω ) ]<br />
+ Zy(−ω)<br />
2ρ0(y) a<br />
Zx(ω) = e πω/a θ(x 0 − x 1 ) + θ(x 1 − x 0 ) (17)<br />
L 2 x = −x µ xµ + 1<br />
a2 , L2y = −y µ yµ + 1<br />
. (18)<br />
a2 Partial Cancellati<strong>on</strong><br />
The correlati<strong>on</strong> functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>homogeneous terms<br />
(13) depends <strong>on</strong>ly <strong>on</strong> τ−. The <strong>in</strong>terference terms c<strong>on</strong>ta<strong>in</strong><br />
both <str<strong>on</strong>g>of</str<strong>on</strong>g> terms depend<strong>in</strong>g <strong>on</strong> τ− and τ+; the first term <strong>in</strong> the<br />
parenthesis <str<strong>on</strong>g>of</str<strong>on</strong>g> (17) depends <strong>on</strong>ly <strong>on</strong> τ−, so it is the term<br />
that may cancel the <strong>in</strong>homogeneous terms (i.e. the Unruh<br />
radiati<strong>on</strong>). Us<strong>in</strong>g the relati<strong>on</strong><br />
h(ω) + h(−ω) = e2<br />
6π (ω2 + a 2 )|h(ω)| 2 , (19)<br />
<strong>on</strong>e can show that a part <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>terference terms<br />
iae2xiy i<br />
(4π) 2ρ0(x) 2ρ0(y) 2<br />
×<br />
∫ x y<br />
−iω(τ<br />
dω e −−τ− 2π<br />
)<br />
1 − e−2πω/a ( iω<br />
h(−ω)<br />
a<br />
)<br />
+ h(ω)iω<br />
a<br />
(20)<br />
.<br />
cancels the first correcti<strong>on</strong> term <str<strong>on</strong>g>of</str<strong>on</strong>g> the <strong>in</strong>homogeneous part<br />
<strong>in</strong> (13). This term was obta<strong>in</strong>ed by tak<strong>in</strong>g a derivative <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
x<br />
iωτ e − <strong>in</strong> P (x, ω). But note that the cancellati<strong>on</strong> occurs <strong>on</strong>ly<br />
partially. Furthermore, the τ+-dependent terms <strong>in</strong> the <strong>in</strong>terference<br />
terms cannot be canceled with the Unruh radiati<strong>on</strong>.<br />
The Energy Momentum Tensor<br />
Given the 2-po<strong>in</strong>t functi<strong>on</strong>, we can calculate the energy<br />
momentum tensor <str<strong>on</strong>g>of</str<strong>on</strong>g> the radiati<strong>on</strong><br />
⟨Tµν(x)⟩ = ⟨: ∂µϕ∂νϕ − 1<br />
2 gµν∂ α ϕ∂αϕ :⟩. (21)<br />
It is a sum <str<strong>on</strong>g>of</str<strong>on</strong>g> the classical and the fluctuati<strong>on</strong> parts; Tµν =<br />
Tcl,µν + Tfluc,µν. The classical part is given by<br />
Tcl,µν ∼ e2∂µρ0∂νρ0 (4π) 2ρ4 . (22)<br />
0<br />
It corresp<strong>on</strong>ds to the energy momentum tensor <str<strong>on</strong>g>of</str<strong>on</strong>g> the Larmor<br />
radiati<strong>on</strong>. From (10) it can be seen to be proporti<strong>on</strong>al<br />
to a 2 /r 2 where a is the accelerati<strong>on</strong> and r is the spacial<br />
distance from the particle to the observer. Tfluc,µν is the<br />
energy momentum <str<strong>on</strong>g>of</str<strong>on</strong>g> the additi<strong>on</strong>al radiati<strong>on</strong><br />
Tfluc,µν = (xi ) 2<br />
− e2 a 2 L 2 x<br />
(4π) 2 ρ 3 0<br />
(<br />
ρ 2 0<br />
[ (e 2<br />
π Im − 6ma2 I1L 2 x<br />
ρ0<br />
)<br />
Tcl,µν<br />
mI3 ∂µτ x −∂ντ x − + 2mI1<br />
ρ0L2 (xµ∂νρ0 + xν∂µρ0)<br />
x<br />
+ e2Im 12πL2 (xµ∂ντ<br />
x<br />
x − + xν∂µτ x −)<br />
− e2Im (∂µτ<br />
24πρ0<br />
x −∂νρ0 + ∂ντ x −∂µρ0)<br />
) ]<br />
(23)<br />
where I1 = 3<br />
2mae2 , I3 ∼ Ω2 −I1 ≪ a2I1, Im = I3+a2 I1 ∼<br />
a2I1. Hence, these terms orig<strong>in</strong>at<strong>in</strong>g from the fluctuat<strong>in</strong>g<br />
moti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the particle is proporti<strong>on</strong>al to a3 , and smaller<br />
by a factor <str<strong>on</strong>g>of</str<strong>on</strong>g> a compared to the above Larmor radiati<strong>on</strong>.<br />
Though they have different angular distributi<strong>on</strong>, there is an<br />
overall factor (x2 i ) <strong>in</strong> fr<strong>on</strong>t and they vanish at the forward<br />
directi<strong>on</strong>. Together with the l<strong>on</strong>g relaxati<strong>on</strong> time discussed<br />
<strong>in</strong> [2], the detecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Unruh radiati<strong>on</strong> seems to be very<br />
difficult experimentally.<br />
REFERENCES<br />
[1] S. Iso, Y. Yamamoto and S. Zhang, arXiv:1011.4191 [hep-th].<br />
[2] S. Iso, Y. Yamamoto and S. Zhang, <strong>in</strong> the same proceed<strong>in</strong>g,<br />
“Can we detect ”Unruh radiati<strong>on</strong>” <strong>in</strong> the high <strong>in</strong>tensity laser?”<br />
[3] P. Chen and T. Tajima, Phys. Rev. Lett. 83 (1999) 256.<br />
[4] P. R. Johns<strong>on</strong> and B. L. Hu, arXiv:quant-ph/0012137.<br />
Phys. Rev. D 65 (2002) 065015 [arXiv:quant-ph/0101001].<br />
P. R. Johns<strong>on</strong> and B. L. Hu, Found. Phys. 35, 1117 (2005)<br />
[arXiv:gr-qc/0501029].<br />
[5] D. J. Ra<strong>in</strong>e, D. W. Sciama and P. G. Grove, Proc. R. Soc.<br />
L<strong>on</strong>d. A (1991) 435, 205-215
P r o g r a m <str<strong>on</strong>g>of</str<strong>on</strong>g> PIF2010<br />
th<br />
N o v (Wednesday) . 2 4<br />
08:30 Registrati<strong>on</strong> starts @ foyer <strong>in</strong> fr<strong>on</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> Kobayashi Hall<br />
[Open<strong>in</strong>g and tutorial-1 : chaired by T shiki o Tajima (LMU, Munich)]<br />
09:00 Open<strong>in</strong>g address by Satoshi Iso (<strong>KEK</strong>, the c<strong>on</strong>ference chair) [10]<br />
09:10 Welcome speech by Fumihiko Takasaki (<strong>KEK</strong>) [10]<br />
09:20 Gerard Mourou (Ecole Polytechnique) [60]<br />
“Extreme Light for High Energy <strong>Physics</strong>”<br />
10:20 break [15]<br />
[N<strong>on</strong>l<strong>in</strong>ear QED : chaired by Toshiaki Tauchi (<strong>KEK</strong>)]<br />
10:35 Thomas He<strong>in</strong>zl (Plymouth U.) [45]<br />
“QED <strong>in</strong> ultra-<strong>in</strong>tense laser fields”<br />
11:20 Kaoru Yokoya (<strong>KEK</strong>) [30]<br />
“Beam-beam <strong>in</strong>teracti<strong>on</strong> <strong>in</strong> <str<strong>on</strong>g>Internati<strong>on</strong>al</str<strong>on</strong>g> L<strong>in</strong>ear Collider”<br />
11:50 Anth<strong>on</strong>y Hart<strong>in</strong> (DESY) [25]<br />
“Sec<strong>on</strong>d Order QED processes <strong>in</strong> the Furry Picture and their Radiative Correcti<strong>on</strong>s”<br />
12:15 lunch [75]<br />
[Heavy-i<strong>on</strong> collisi<strong>on</strong>s and Quark-Glu<strong>on</strong> Plasma : chaired by Tetsuo Hatsuda (U. T o )] k y o<br />
13:30 Kazunori Itakura (<strong>KEK</strong>) [30]<br />
“Str<strong>on</strong>g Field Dynamics <strong>in</strong> Heavy I<strong>on</strong> Collisi<strong>on</strong>s”<br />
14:00 Andreas Ipp (TU Vienna) [25]<br />
“Yoctosec<strong>on</strong>d phot<strong>on</strong> pulse generati<strong>on</strong> <strong>in</strong> heavy i<strong>on</strong> collisi<strong>on</strong>s”<br />
14:25 Kenji Fukushima (Keio U.) [25]<br />
“Fields, <strong>in</strong>stant<strong>on</strong>s, and currents”<br />
14:50 Kenji Morita (GSI) [25]<br />
“Critical behavior <str<strong>on</strong>g>of</str<strong>on</strong>g> charm<strong>on</strong>ium : QCD sec<strong>on</strong>d order Stark effect”
15:15 break [15]<br />
[Unruh radiati<strong>on</strong> : chaired by Kensuke Homma (Hiroshima U.)]<br />
15:30 Ralf Schutzhold (Essen U.) [45]<br />
“Fundamental Quantum Effects <strong>in</strong> Str<strong>on</strong>g Lasers”<br />
16:15 Sen Zhang (<strong>KEK</strong>) [25]<br />
“Does an uniformly accelerated electr<strong>on</strong> radiate Unruh radiati<strong>on</strong>? ”<br />
16:40 Frieder Lenz (Erlangen-Nuernberg & RIKEN) [25]<br />
“Quantum fields <strong>in</strong> accelerated frames”<br />
17:05 break [15]<br />
[Axi<strong>on</strong>-like particle searches : chaired by Hiroshi Azechi (ILE, Osaka U.)]<br />
17:20 Axel L<strong>in</strong>dner (DESY) [45]<br />
“Sh<strong>in</strong><strong>in</strong>g light through walls: enroute towards a new particle physics fr<strong>on</strong>tier”<br />
18:05 Kensuke Homma (Hiroshima U.) [25]<br />
“Prob<strong>in</strong>g extremely light fields via res<strong>on</strong>ance scatter<strong>in</strong>g by laser focus<strong>in</strong>g”<br />
th<br />
N o v (Thursday) . 2 5<br />
[Tutorial-2 : chaired by Dietrich Habs (LMU, Munich)]<br />
09:00 Gerald Dunne (C<strong>on</strong>necticut U.) [60]<br />
“The Heisenberg-Schw<strong>in</strong>ger Effect: N<strong>on</strong>perturbative Pair Producti<strong>on</strong> from Vacuum”<br />
10:00 break [15]<br />
[Schw<strong>in</strong>ger mechanism <strong>in</strong> QGP and c<strong>on</strong>densed matter : chaired by Masayuki Asakawa (Osaka<br />
U.)]<br />
10:15 Naoto Tanji (Tokyo U. Komaba) [25]<br />
“Dynamical view <str<strong>on</strong>g>of</str<strong>on</strong>g> pair creati<strong>on</strong> via the Schw<strong>in</strong>ger mechanism”<br />
10:40 Aiichi Iwazaki (Nishogakusha U.) [25]<br />
“Exact soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> pair producti<strong>on</strong>s <strong>in</strong> str<strong>on</strong>g electric fields with f<strong>in</strong>ite sizes”<br />
11:05 Takashi Oka (Tokyo U.) [25]<br />
“Str<strong>on</strong>g field physics <strong>in</strong> c<strong>on</strong>densed matter”<br />
11:30 Sh<strong>in</strong> Nakamura (Kyoto) [25]<br />
“N<strong>on</strong>-l<strong>in</strong>ear charge transport <strong>in</strong> plasma under str<strong>on</strong>g field”
11:55 lunch & group photo [85]<br />
[Recent developments <strong>in</strong> Schw<strong>in</strong>ger mechanism 1: chaired by Philip Bambade (LAL, Orsay)]<br />
13:20 Dietrich Habs (LMU,Munich) [45]<br />
“Vacuum Pair Creati<strong>on</strong>”<br />
14:05 Hartmut Ruhl (LMU,Munich) [25]<br />
“QED cascad<strong>in</strong>g: A particle <strong>in</strong> cell model”<br />
14:30 N<strong>in</strong>a Elk<strong>in</strong>a (LMU, Munich) [25]<br />
“Numerical simulati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> QED cascades <strong>in</strong> circularly polarized laser fields”<br />
14:55 break [15]<br />
[Recent developments <strong>in</strong> Schw<strong>in</strong>ger mechanism 2: chaired by Gerald Dunne (C<strong>on</strong>necticut U.)]<br />
15:10 Sergei Bulanov (Kansai Phot<strong>on</strong> Science Institute, JAEA) [45]<br />
“On the Schw<strong>in</strong>ger limit atta<strong>in</strong>ability with extreme power lasers”<br />
15:55 Natalia Naumova (Ecole Polytechnique) [25]<br />
“Pair Creati<strong>on</strong> <strong>in</strong> QED-Str<strong>on</strong>g Pulsed Laser Fields Interact<strong>in</strong>g with Electr<strong>on</strong> Beams”<br />
16:20 He<strong>in</strong>rich Hora (New South Wales) [25]<br />
“Accelerati<strong>on</strong> up to black hole c<strong>on</strong>diti<strong>on</strong>s and B-mes<strong>on</strong> decay”<br />
16:45 POSTER SESSION at the foyer <strong>in</strong> fr<strong>on</strong>t <str<strong>on</strong>g>of</str<strong>on</strong>g> Kobayashi Hall [90] (~ 18:15)<br />
18:30 Buses leave for banquet<br />
19:00 Banquet at Okura Fr<strong>on</strong>tier Hotel Tsukuba (~21:00)<br />
th<br />
N o v ( F r i . d a y ) 2 6<br />
[Recent progress <str<strong>on</strong>g>of</str<strong>on</strong>g> ultra-<strong>in</strong>tense lasers : chaired by Alex Borisov (U. Ill<strong>in</strong>ois at Chicago)]<br />
09:00 Hideaki Takabe (ILE, Osaka U.) [45]<br />
“Present Status <str<strong>on</strong>g>of</str<strong>on</strong>g> Ultra-<strong>in</strong>tense Lasers and High-Field <strong>Physics</strong> <strong>in</strong> the World”<br />
09:45 Baifei Shen (Shanghai Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Optics and F<strong>in</strong>e Mechanics) [25]<br />
“Generati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> ultra <strong>in</strong>tense X ray approach<strong>in</strong>g the Schw<strong>in</strong>ger limit”<br />
10:10 Charles Rhodes (U. Ill<strong>in</strong>ois at Chicago) [25]<br />
“Reach<strong>in</strong>g the Schw<strong>in</strong>ger Limit with X-Rays”
10:35 break (20)<br />
[Magnetars : chaired by David Salzmann (Weizmann Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Science)]<br />
10:55 Kazunori Kohri (<strong>KEK</strong>) [30]<br />
“N<strong>on</strong>l<strong>in</strong>ear QED effects by str<strong>on</strong>g magnetic field <strong>in</strong> astrophysics”<br />
11:25 Kazuo Makishima (Tokyo U.) [45]<br />
“W i -Band d e X-ray Observati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> Magnetars”<br />
12:10 Toshitaka Tatsumi (Kyoto U.) [25]<br />
“QCD orig<strong>in</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> str<strong>on</strong>g magnetic fields <strong>in</strong> compact stars”<br />
12:35 lunch @ foyer [105]<br />
[New technologies : chaired by Kim<strong>in</strong>ori K<strong>on</strong>do (JAEA)]<br />
14:20 Kazuhisa Nakajima (<strong>KEK</strong>) [45]<br />
“Recent progress and prospects <strong>on</strong> laser-plasma accelerati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> charged particles”<br />
15:05 Masaki Kando (JAEA) [25]<br />
“Fly<strong>in</strong>g Mirror as a tool to access ultra-high field”<br />
15:30 break [15]<br />
[Overview : chaired by Tohru Takahashi (Hiroshima U.)]<br />
15:45 Toshiki Tajima (LMU, Munich) [60]<br />
“High field science”<br />
16:45 Clos<strong>in</strong>g remarks by Tohru Takahashi (Hiroshima U., the c<strong>on</strong>ference chair) [10]
List <str<strong>on</strong>g>of</str<strong>on</strong>g> Participants<br />
Name Instituti<strong>on</strong> Name Instituti<strong>on</strong><br />
AKAGI, Tomoya Hiroshima University NAKAMURA, Sh<strong>in</strong> Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Kyoto University<br />
ARAKI, Sakae <strong>KEK</strong> NAKAMURA, Gen Hiroshima University<br />
ASAKAWA, Masayuki Osaka University NARA, Yasushi Akita <str<strong>on</strong>g>Internati<strong>on</strong>al</str<strong>on</strong>g> University<br />
AZECHI, Hiroshi Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Laser Eng<strong>in</strong>eer<strong>in</strong>g Osaka University NAUMOVA, Natalia Ecole Polytechnique<br />
BAIOTTI, Luca Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Laser Eng<strong>in</strong>eer<strong>in</strong>g, Osaka University NISHIMURA, Hiroaki Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Laser Eng<strong>in</strong>eer<strong>in</strong>g, Osaka University<br />
BAMBADE, S.Philip LAL-Orsay NISHIMURA, Jun <strong>KEK</strong><br />
BORISOV, B.Alex University <str<strong>on</strong>g>of</str<strong>on</strong>g> Ill<strong>in</strong>ois at Chicago NODA, Akira Institute for Chemical Research, Kyoto University<br />
BULANOV, V.Sergei Kansai Phot<strong>on</strong> Science Institute, JAEA NOZAKI, Mitsuaki <strong>KEK</strong><br />
DELERUE, Nicolas LAL, Orsay OHTA, Masahiro <strong>KEK</strong><br />
DUNNE, V.Gerald University <str<strong>on</strong>g>of</str<strong>on</strong>g> C<strong>on</strong>necticut OKA, Takashi University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo<br />
ELKINA, N<strong>in</strong>a Ludwig-Maximilian University, Munich OKADA, Yasuhiro <strong>KEK</strong><br />
ENOTO, Teruaki KIPAC / Stanford University OKAZAWA, Susumu SOKENDAI, <strong>KEK</strong><br />
FUJII, Keisuke <strong>KEK</strong> OMORI, Tsunehiko <strong>KEK</strong><br />
FUJII, Hirotsugu University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo OROKU, Masahiro University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo<br />
FUJII, Yasunori Waseda University POSCH, Paul <strong>KEK</strong><br />
FUJITSUKA, Masashi SOKENDAI, <strong>KEK</strong> RHODES, K. Charles University <str<strong>on</strong>g>of</str<strong>on</strong>g> Ill<strong>in</strong>ois at Chicago<br />
FUJIWARA, Mamoru Research Center for Nuclear <strong>Physics</strong>, Osaka Univ. RUHL, Hartmut Ludwig-Maximilians-University Munich<br />
FUKUSHIMA, Kenji Keio University SAEKI, Takayuki <strong>KEK</strong><br />
GOTO, Hajime Graduate University for Advanced Studies SALZMANN, DAVID Weizmann Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Science<br />
HABS, Dietrich Faculty <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, LMU Munich SASAKI, Toshihiko University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo, <strong>KEK</strong><br />
HARTIN, Anth<strong>on</strong>y DESY SCHUETZHOLD, Ralf Fakultät für Physik der Universität Duisburg-Essen<br />
HATSUDA, Tetsuo Phys. Dep., Univ. Tokyo SEKINO, Yasuhiro Okayama Institute for Quantum <strong>Physics</strong><br />
HATTORI, Koichi <strong>KEK</strong> SHEN, Baifei Shanghai Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Optics and F<strong>in</strong>e Mechanics<br />
HEINZL, Thomas University <str<strong>on</strong>g>of</str<strong>on</strong>g> Plymouth, Comput<strong>in</strong>g and Mathematics SHIMADA, Kengo SOKENDAI, <strong>KEK</strong><br />
HIDAKA, Yoshimasa Department <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>Physics</strong>, Kyoto University SHIMIZU, Katsuya KYOKUGEN, Osaka University<br />
HOMMA, Kensuke Hiroshima University / LMU SHIMIZU, Hirotaka <strong>KEK</strong><br />
HONDA, Yosuke <strong>KEK</strong> SOUDA, Hikaru Institute for Chemical Research, Kyoto University<br />
HORA, He<strong>in</strong>rich University <str<strong>on</strong>g>of</str<strong>on</strong>g> New South Wales SUNAHARA, Atsushi Institute for Laser Technology<br />
IPP, Andreas Vienna University <str<strong>on</strong>g>of</str<strong>on</strong>g> Technology SUZUKI, Atsuto <strong>KEK</strong><br />
ISO, Satoshi <strong>KEK</strong> TAJIMA, Toshiki LMU<br />
ITAKURA, Kazunori <strong>KEK</strong> TAKABE, Hideaki ILE and School <str<strong>on</strong>g>of</str<strong>on</strong>g> Science, Osaka University<br />
IWATA, Natsumi Kyoto University TAKAHASHI, Tohru Hiroshima University<br />
IWAZAKI, Aiichi Nishogakusha Universeity TAKASAKI, Fumihiko <strong>KEK</strong><br />
KANDO, Masaki Japan Atomic Energy Agency TANJI, Naoto University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo<br />
KISHIMOTO, Yausaki Kyoto University TARAKANOV, Alexander Saratov State University, <strong>Physics</strong> Department<br />
KITAMOTO, Hiroyuki SOKENDAI, <strong>KEK</strong> TATSUMI, Toshitaka Kyoto University<br />
KITAZAWA, Yoshihisa <strong>KEK</strong> TATSUMI, Daisuke Nati<strong>on</strong>al Astr<strong>on</strong>omical Observatory <str<strong>on</strong>g>of</str<strong>on</strong>g> Japan<br />
KOHRI, Kazunori <strong>KEK</strong> TAUCHI, Toshiaki <strong>KEK</strong><br />
KOYAMA, Kazuyoshi University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo TERUNUMA, Nobuhiro <strong>KEK</strong><br />
KUBO, Kiyoshi <strong>KEK</strong> TOMARU, TAKAYUKI <strong>KEK</strong><br />
KUBOTA, Hirohisa SOKENDAI, <strong>KEK</strong> UEDA, Ken-ichi University <str<strong>on</strong>g>of</str<strong>on</strong>g> Electro-Communicati<strong>on</strong>s<br />
KUMADA, Masayuki NIRS URAKAWA, Junji <strong>KEK</strong><br />
KUMITA, Tetsuro Tokyo Metropolitan University YABANA, Kazuhiro University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tsukuba<br />
LENZ, Frieder University <str<strong>on</strong>g>of</str<strong>on</strong>g> Erlangen-Nuernberg YAMAGUCHI, Yohei University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo<br />
LINDNER, Axel DESY YAMAMOTO, Yasuhiro SOKENDAI/<strong>KEK</strong><br />
MAKISHIMA, Kazuo University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo YAN, Jacquel<strong>in</strong>e University <str<strong>on</strong>g>of</str<strong>on</strong>g> Tokyo<br />
MATSUBA, Shunya SOKENDAI, <strong>KEK</strong> YAZAKI, Koichi Hashimoto Math. Phys. Lab., Nish<strong>in</strong>a Center, RIKEN<br />
MINAKATA, Hisakazu Tokyo Metropolitan University YOKOYA, Kaoru <strong>KEK</strong><br />
MIURA, Eisuke AIST ZHANG, Sen <strong>KEK</strong><br />
MORITA, Kenji GSI<br />
MOUROU, A. Gerard Institut Lumière Extrême/Ecole Polytechnique ENSTA<br />
MURAKAMI, Masakatsu Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Laser Eng<strong>in</strong>eer<strong>in</strong>g, Osaka University Secretaries<br />
MUROYA, Sh<strong>in</strong> Matsumoto University IKEDA, Kimiyo <strong>KEK</strong><br />
NAKAI, Mitsuo Institute <str<strong>on</strong>g>of</str<strong>on</strong>g> Laser Eng<strong>in</strong>eer<strong>in</strong>g, Osaka University KUSAMA, Hitomi <strong>KEK</strong><br />
NAKAJIMA, Kazuhisa <strong>KEK</strong> SHISHIDO, Tamao <strong>KEK</strong>