t Hooft mechanism of confinement or dual Meissner effect
t Hooft mechanism of confinement or dual Meissner effect
t Hooft mechanism of confinement or dual Meissner effect
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Mandelstam – Polyakov – ’t <strong>Ho<strong>of</strong>t</strong> <strong>mechanism</strong><br />
<strong>of</strong> <strong>confinement</strong> <strong>or</strong> <strong>dual</strong> <strong>Meissner</strong> <strong>effect</strong><br />
The usual <strong>Meissner</strong> <strong>effect</strong>: magnetic field cannot penetrate into the superconduct<strong>or</strong> (except<br />
by burning out a narrow tube where superconductivity is destroyed = Abrikosov v<strong>or</strong>tex).<br />
Two infinitely thin and long solenoids are, at their endpoints, sources <strong>of</strong> the Coulomb-like<br />
magnetic field.<br />
Superconduct<strong>or</strong><br />
solenoid<br />
N<br />
Flux tube<br />
S<br />
solenoid<br />
Energy <strong>of</strong> the magnetic ‘monopole-antimonopole’ pair = E ⊥ · L =⇒ linear potential<br />
energy between monopoles.<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
Dual <strong>Meissner</strong> <strong>effect</strong>:<br />
• condensation <strong>of</strong> magnetic monopoles<br />
• quarks are sources <strong>of</strong> (col<strong>or</strong>ed) electric field<br />
• Along the tube connecting quarks the magnetic condensate is destroyed<br />
• electric field squeezed inside the tube<br />
= Abrikosov–Nielsen–Olesen v<strong>or</strong>tex<br />
Estimate <strong>of</strong> the string tension<br />
Landau–Ginsburg <strong>effect</strong>ive the<strong>or</strong>y <strong>of</strong> supersonductivity:<br />
E =<br />
∫<br />
d 3 r<br />
[<br />
B<br />
2<br />
2 + |(∂ i − ieA i )φ| 2 + λ 2 (φ 2 − v 2 ) 2 ]<br />
m W = ev, m H = λv, B = curl A.<br />
Dimensionless quantities: φ ′ = φ v , A′ i = A i<br />
m W<br />
,<br />
,<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
x = rm W , κ = λ e .<br />
E<br />
= Lv 2 ∫<br />
d 2 x<br />
[<br />
B<br />
′2<br />
2 + ∣ ∣(∂ i − ieA ′ i )φ′∣ ∣ 2 + κ 2 (φ ′ 2 − 1) 2 ]<br />
,<br />
v<strong>or</strong>tex transverse size ρ 0 ∼ 1 √ κ<br />
.<br />
String tension = σ = energy / length <strong>of</strong> the tube = v 2 [ O(1) + O<br />
(<br />
mH<br />
m W<br />
)].<br />
Londons’ limit: m H → ∞ =⇒ infinite-energy v<strong>or</strong>tex<br />
Bogomolny–Prasad–Sommerfeld limit: m H → 0.<br />
type-I superconduct<strong>or</strong>: (m H > m W ): no v<strong>or</strong>tices<br />
type-II superconduct<strong>or</strong>: (m H < m W ): yes<br />
One needs an analog <strong>of</strong> type II superconduct<strong>or</strong> with magnetic monopoles condensed.<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
Polyakov’s realization <strong>of</strong> <strong>confinement</strong><br />
d = 2 + 1 Ge<strong>or</strong>gi–Glashow model: Yang–Mills SU(2) fields interacting with the Higgs<br />
field in the triplet representation:<br />
⎧(<br />
) 2<br />
⎫<br />
∫ ⎪⎨ F a (<br />
S = d 3 ij<br />
x + 1<br />
⎪⎩ 4g 2 2<br />
D ab<br />
i φb) 2 [ 2 + λ (φ a ) 2 − v 2] 2<br />
⎪⎬<br />
⎪⎭ .<br />
’t <strong>Ho<strong>of</strong>t</strong>–Polyakov monopole is a local minimum <strong>of</strong> this action:<br />
{<br />
φ a = ∓n a vΦ(r), n a = xa 0, r → 0<br />
r , Φ(r) → 1, r → ∞<br />
A a i<br />
1 − R(r)<br />
= ǫ aij n j , R(r) →<br />
r<br />
{ 1, r → 0<br />
0, r → ∞<br />
Magnetic field strength, B i<br />
r→∞<br />
∼<br />
n i<br />
r2 , is that <strong>of</strong> the magnetic monopole !<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
Action = v g w (<br />
mH<br />
m W<br />
)<br />
≫ 1,<br />
w(0) = 4π (in the BPS limit)<br />
To add up hedgehogs, one has first to comb their hair!<br />
‘Stringy’ <strong>or</strong> singular gauge<br />
The the<strong>or</strong>y is invariant under gauge transf<strong>or</strong>mations:<br />
φ = φ a τ a<br />
2<br />
A i = A a τ a<br />
i<br />
2<br />
→<br />
S(x)φS † (x),<br />
→ S(x)A i S † (x) + iS∂ i S † .<br />
Choose the unitary gauge-transf<strong>or</strong>mation matrix S(x) such that<br />
S(n · τ)S † = τ 3 =⇒ S(θ, φ) = e −iφ 2 τ 3<br />
e iθ 2 τ 2<br />
e iφ 2 τ 3<br />
.<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
φ ′ = S φ S † = τ 3<br />
2 v Φ(r) → τ 3<br />
2 v,<br />
A ′ i<br />
=<br />
⎧<br />
⎨<br />
⎩<br />
A ′ r<br />
= 0<br />
A ′ θ<br />
= only inside c<strong>or</strong>e<br />
A ′ φ<br />
= inside c<strong>or</strong>e ∓ τ 3 1−cos θ<br />
2r sin θ .<br />
Magnetic field strength outside the monopole c<strong>or</strong>e at r ≫ 1/m W :<br />
B ′ r = ( curl A ′) r = 1<br />
r sin θ<br />
∂<br />
∂θ<br />
( )<br />
sin θ A ′ φ<br />
= ± τ 3<br />
2r 2.<br />
string singularity<br />
(gauge artifact)<br />
magnetic field<br />
<strong>of</strong> a monopole<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
Monopole interactions<br />
One can add up (anti)monopoles only in the singular (‘stringy’) gauge where the Higgs<br />
field φ a r→∞<br />
−→ δ a3 v.<br />
The interaction <strong>of</strong> two (anti)monopoles is Coulomb-like at large separations:<br />
U int = 4π 1 (<br />
)<br />
±1 − e −r 12 m H .<br />
g 2 r 12<br />
Two gluons (out <strong>of</strong> three) W ± = A1 √±iA 2<br />
2<br />
have large masses m W = gv and decouple.<br />
The third gluon (the ‘photon’) is massless, but there are monopoles around.<br />
Monopole ensemble<br />
Monopole ‘weight’ <strong>or</strong> ‘fugacity’<br />
ζ = (pre−exponent) · exp(−Action) = controllably small.<br />
At small momenta, the the<strong>or</strong>y becomes the the<strong>or</strong>y <strong>of</strong> plasma <strong>of</strong> magnetic charges <strong>of</strong><br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
opposite sign. The grand canonical partition function is<br />
Z = ∑<br />
ζ N + ζ N −<br />
N<br />
N + ,N + ! N − !<br />
−<br />
∏<br />
∫<br />
m<br />
d 3 z m exp<br />
(<br />
− 4π<br />
g 2 ∑<br />
m
= exp − 1 2<br />
4π<br />
g 2 ∫ ∫<br />
= exp − 4π<br />
g 2 ∑<br />
m
Hence<br />
∫ { ∫<br />
Z = Dw exp −<br />
d 3 x<br />
[<br />
( )]}<br />
1<br />
4π<br />
2 (∂ iw) 2 − 2ζ cos<br />
g w .<br />
( ) 4π<br />
−2ζ cos<br />
g w<br />
≃ −2ζ + 1 2 µ2 w 2 ,<br />
µ 2 = 2ζ<br />
( 4π<br />
g<br />
) 2<br />
Debye mass!<br />
The plasma partition function is mathematically equivalent to the ‘Sine-G<strong>or</strong>don’ field<br />
the<strong>or</strong>y. The field w has the meaning <strong>of</strong> the <strong>dual</strong> potential: it gets a nonzero mass owing<br />
to the Debye screening in the monopole plasma.<br />
Z = exp<br />
⎧<br />
⎪⎨<br />
⎪⎩ 2ζV + 1<br />
12π<br />
[<br />
2ζ<br />
( 4π<br />
g<br />
⎫<br />
)<br />
]3<br />
2 2 ⎪⎬<br />
V + . . .<br />
⎪⎭<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
Average monopole density:<br />
¯N<br />
V = 1 V<br />
∂ ln Z<br />
∂ ln ζ<br />
= 2ζ + Coulomb c<strong>or</strong>rections<br />
In the weak-coupling regime g ≪ v monopoles are heavy, their density is small, everything<br />
is under control, this the<strong>or</strong>y becomes exact!<br />
Other ways to present plasma<br />
Z = ∑ N ±<br />
ζ N + +N −<br />
N + !N − !<br />
×<br />
∫<br />
Dρ(x) δ<br />
∏<br />
∫<br />
m<br />
d 3 z m e −4π ∑<br />
g 2<br />
qmqn<br />
|zm−zn|<br />
(<br />
ρ − ∑ )<br />
q m δ(x − z m )<br />
δ(...) =<br />
∫<br />
Dµ(x) exp<br />
[ ∫<br />
i<br />
d 3 x µρ − i ∑ ]<br />
q m µ(z m )<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
∫<br />
∫ [<br />
( ) ]<br />
= DρDµDw exp − d 3 1 4π<br />
x<br />
2 (∂ iw) 2 −2ζ cos<br />
g w − µ −iµρ .<br />
New variables: 4π g w − µ = ψ;<br />
4π<br />
g<br />
w + µ = φ ←−integrate over<br />
Z =<br />
∫ ∫<br />
DρDψ exp<br />
d 3 x<br />
[<br />
1<br />
2<br />
( 4π<br />
g<br />
) 2<br />
ρ 1 △ ρ−iψρ+2ζ cos ψ ]<br />
=<br />
∫<br />
∫<br />
Dρ exp<br />
d 3 x<br />
[<br />
1<br />
2<br />
( 4π<br />
g<br />
) 2<br />
ρ 1 △ ρ − V (ρ) ]<br />
V (ρ) = ρ ln<br />
(<br />
ρ<br />
2ζ + √<br />
1 + ρ2<br />
4ζ 2 )<br />
+ 2ζ<br />
√<br />
1 + ρ2<br />
4ζ 2<br />
ρ≪ζ<br />
−→<br />
−2ζ + ρ2<br />
4ζ<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
Unusual: div B 3 = −4πρ, curl B 3 = 0 (!)<br />
The ‘kinetic energy’ <strong>of</strong> monopole density is nothing but the magnetic energy:<br />
Hence<br />
∫<br />
− d 3 x 1 2<br />
( 4π<br />
g<br />
) 2<br />
ρ 1 △ ρ = 1 ∫<br />
∫<br />
d 3 1<br />
xB<br />
2g 2 i ∂ i<br />
△ ∂ jB j = d 3 x B iB i<br />
2g . 2<br />
∫<br />
∫<br />
Z = DB i δ(curlB) Dψ exp d 3 x<br />
[<br />
]<br />
− B2<br />
2g + idivB 2 4π ψ + 2ζ cos ψ .<br />
Confinement (= Area law f<strong>or</strong> large Wilson loops<br />
Wilson loop<br />
∮<br />
W = Tr P exp i<br />
dx i A a i<br />
τ a<br />
2 .<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
At large distances from monopoles only colour A 3 i<br />
Stokes the<strong>or</strong>em:<br />
∫<br />
W = Tr exp i d 2 S i B 3 i<br />
component survives =⇒ can use the<br />
τ 3<br />
2 = e i 2 Φ + e − i 2 Φ ,<br />
where Φ = ∫ d 2 S i B 3 i<br />
is the flux <strong>of</strong> the magnetic field created by monopoles and<br />
antimonopoles in the plasma, through the surface spanned over the Wilson loop.<br />
Wilson ( loop averaged over the ensemble <strong>of</strong> monopoles: have to plunge the source term<br />
∫ )<br />
i<br />
exp<br />
2 d<br />
2−→ S·−→ B into the partition function, and integrate over all possible magnetic<br />
fields.<br />
F<strong>or</strong> large loops (say, lying in the z =0 plane), one can use the saddle-point method and<br />
find the ‘best’ fields −→ B, ψ minimizing the energy together with the surface source:<br />
B i (x, y, z) = δ iz B(z), ψ = ψ(z),<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
1<br />
g 2B(z) + i<br />
4π<br />
i dB<br />
dz<br />
f<strong>or</strong> x, y inside the contour. The solution:<br />
dψ<br />
dz = i 2 δ(z),<br />
− 8πζ sin ψ = 0,<br />
e −µ|z|<br />
B = i g2 µ<br />
π 1 + e −2µ|z|,<br />
(<br />
ψ = 4 sign(z) atan e −µ|z|) .<br />
The solution c<strong>or</strong>responds to a purely imaginary double layer <strong>of</strong> monopoles around the<br />
surface.<br />
String tension<br />
< W > = exp (−σ Area) ,<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
σ = g2 µ<br />
= 2g √<br />
2ζ,<br />
2π 2 π<br />
prop<strong>or</strong>tional to the square root <strong>of</strong> the mean monopole density N/V = 2ζ.<br />
No massless states are left in the the<strong>or</strong>y. F<strong>or</strong> example, consider the c<strong>or</strong>relation function <strong>of</strong><br />
the magnetic field (in momentum space):<br />
< B 3 i (p)B3 j (−p) ><br />
(<br />
= g 2 δ ij − p )<br />
ip j<br />
+ (4π) 2 N p i p j<br />
p 2 V p − N<br />
4 (4π)2 V<br />
µ 2 p i p j<br />
p 2 + µ 2 p 4<br />
= g 2 (<br />
δ ij −<br />
p )<br />
(<br />
ip j<br />
4π<br />
, Debye mass : µ 2 = 2ζ<br />
p 2 + µ 2 g<br />
) 2<br />
.<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12
Best dreams fulfilled! [A. Polyakov, Nucl. Phys. B (1977)]. However,<br />
i) d = 2 + 1 and ii) the gauge group SU(2) is explicitly broken by the Higgs field down<br />
to U(1) where the <strong>dual</strong> photon gets the mass.<br />
M<strong>or</strong>e recent achievements:<br />
• N = 2, d = 3 + 1 supersymmetric the<strong>or</strong>y, s<strong>of</strong>tly broken to N = 1 supersymmetric<br />
the<strong>or</strong>y. It is also shown to possess <strong>confinement</strong> and mass gap – m<strong>or</strong>e <strong>or</strong> less due to<br />
the same <strong>mechanism</strong> (<strong>of</strong> monopole condensation) [Seiberg and Witten (1994), Douglas<br />
and Shenker (1995)]<br />
• pure Yang–Mills d = 3 + 1 non-supersymmetric the<strong>or</strong>y [D.D. and Petrov (2007)] is<br />
also shown to possess <strong>confinement</strong>, at least at the semiclassical level!<br />
Confinement in the 3d Ge<strong>or</strong>gi–Glashow model D. Diakonov, L-12