New results for the form factors of Pion, Kaon, and Proton at ... - Fisica

ECT Workshop on Form Factors Feb. 18–22, 2013
PRECISION FORM FACTORS OF
PIONS, KAONS & PROTONS
at the highest timelike momentum transfers
Kamal K. Seth
Northwestern University, Evanston IL 60208, USA
(kseth@northwestern.edu)
ECT Workshop on Form Factors
Trento, Feb. 18–22, 2013
Northwestern University 1 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
PREAMBLE
This workshop on form factors is largely devoted to nucleon form factors.
• And there is good reason for it.
Despite the obvious bias of DeBeers for diamonds,
ONLY PROTONS ARE FOREVER
And nucleons do make up nearly all of the Universe!
So, I will mention our results for protons, but the emphasis in my talk will be on
pions and kaons.
• And there are good reasons for it too.
Mesons, with only two quarks and no spin, are (or should be) hell–of–a lot simpler
to understand than baryons with three quarks!
And easier wins over harder, anytime.
Northwestern University 2 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
PRELIMINARIES
You have already heard about these preliminary considerations about form factors from
previous speakers, Diego Bettoni, Rinaldo Baldini, and Frank Maas.
1. Four momentum transfers defined as
Q 2 (4 mom.) 2 = Q 2 (3 mom.) 2 space −(energy)2 time
can be positive and spacelike, or negative and timelike.
e -
e -
p 1
γ *
p 2
h -
h -
p 1
h + h -
γ *
e + e -
2. Form factors are analytic functions of momentum transfer, and therefore, á la
Cauchy, for infinite momentum transfer
FF(spacelike, Q 2 = ∞) = FF(timelike, Q 2 = ∞)
All the latest measurements I will talk about are based on e + e − annihilation at the
collider at Cornell with data taken with the CLEO-c detector
e + e − → p¯p, π + π − , K + K − , K 0 SK 0 L
Northwestern University 3 K. K. Seth
p 2

ECT Workshop on Form Factors Feb. 18–22, 2013
PROTONS
I promised not to say much about protons. So, I will only present the latest results,
including our results (PRL 110, 022002 (2013)) published one month ago, and the
BaBar results (arXiv:1302.0055) listed two weeks ago.
(|Q 4 ||G M (Q 2 )|) / μ p (GeV 4 )
2.5
2
1.5
1
0.5
0
BABAR FNAL CLEO NU p
Timelike
Spacelike
α S 2
4 6 8 10 12 14 16 18 20
|Q 2 | (GeV 2 )
I want to draw your attention to three points.
1. Irrespective of some of the large error bars, it is clear that
|G E /G M |
FF(timelike) ≈ 2×FF(spacelike)
1.5
1
0.5
BABAR
PS170
2 2.25 2.5 2.75 3
M pp – (GeV/c 2 )
This feature, first observed in Fermilab p¯p → e + e − measurements, and confirmed
by CLEO measurements, is still unexplained. Diquark–quark structure of the proton
remains a contender.
Northwestern University 4 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
(|Q 4 ||G M (Q 2 )|) / μ p (GeV 4 )
2.5
2
1.5
1
0.5
0
BABAR FNAL CLEO NU p
Timelike
Spacelike
α S 2
4 6 8 10 12 14 16 18 20
|Q 2 | (GeV 2 )
|G E /G M |
1.5
1
0.5
BABAR
PS170
2 2.25 2.5 2.75 3
M pp – (GeV/c 2 )
2. The newest results, based on the CLEO-c data, have better than ±5% precision.
They reveal a new unexplained feature:
|Q 4 |GM(14.2 GeV 2 ) = (0.78±0.04)|Q 4 |GM(17.4 GeV 2 )
3. The newest results from BaBar show
|GE/GM| = 1.48±0.17 to 1.04±0.29 for |Q 2 | = 3.5 GeV 2 −9 GeV 2
The results for |Q 2 | < 4.2 GeV 2 differ completely from those of PS170 (NP B411,
3 (1994)).
4. BaBar has determined the asymmetry A(cosθp) = −0.025±0.014±0.003 for
|Q 2 | < 9 GeV 2 , showing no evidence from higher order contributions.
Northwestern University 5 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
Form Factors of Pions and Kaons
Mesons with two quarks are much simpler than baryons with three quarks. For pions
and kaons with spin 0, there is an additional simplification. There is no magnetic form
factor, and so, σ(e + e − → π + π − ) = (πα 2 /3s)βπ|Fπ(s)| 2 .
• The classic debate about when |Q2 | is large enough for the validity of pQCD
took place in the 1980s between Brodsky et al. and Isgur et al. Unfortunately, the
then existing experimental data for pions and kaons was too poor to be of much
use in settling the debate.
|Q 2 | |F π (Q 2 )| (GeV 2 )
Q 2 F π (Q 2 ) (GeV 2 )
3
2.5
2
1.5
1
0.5
Timelike
π
|Q 2 | (GeV 2 0
0 2 4 6 8 10 12 14
)
1.4
1.2
1
0.8
0.6
0.4
0.2
0.225
0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025
0
0 0.05 0.1 0.15 0.2 0.25 0.3
Spacelike
Q 2 (GeV 2 0
0 2 4 6 8 10 12 14
)
π
|Q 2 | |F K (Q 2 )| (GeV 2 )
Q 2 F K (Q 2 ) (GeV 2 )
3
2.5
2
1.5
1
0.5
Timelike
K
|Q 2 | (GeV 2 0
0 2 4 6 8 10 12 14
)
1.4
1.2
1
0.8
0.6
0.4
0.2
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.02 0.04 0.06 0.08 0.1 0.12
Spacelike
K
Q 2 (GeV 2 0
0 2 4 6 8 10 12 14
)
For |Q 2 | > 5 GeV 2
Up to ±100% errors
Data limited to
Q 2 < 2.5 GeV 2 for π
Q 2 < 0.12GeV 2 forK
Northwestern University 6 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
Spacelike Form Factors of Pions and Kaons
Spacelike form factors of mesons are very difficult to measure, because meson targets
do not exist. Two different methods have been used.
1. Fπ and FK from Elastic Scattering of pions/kaons off atomic electrons,
π(K)e − → π(K)e − . Unfortunately, in this approach the momentum transfer is
very small. At CERN for 200 GeV pions, Q 2 (π) ≤ 0.25 GeV 2 and
Q 2 (K) ≤ 0.11 GeV 2 were realized.
2. Fπ from Electroproduction of pions,
e − p → e − π − p (e − π + n), has serious theoretical problems
and uncertainties. The good precision data are confined
to Q 2 < 2.45 GeV 2 (JLab).
FK from Electroproduction of kaons — No data exist.
• As you will see, excellent timelike form factor data for π
and K at large Q 2 now exist. It is a pity that the
corresponding spacelike data do not exist to allow us to
determine if the ratio of timelike/spacelike form factors
for mesons is also ≈ 2, as it is for protons.
Q 2 F π (GeV/c) 2
e
γ *
F π
e /
π + (π - )
p (n) n (p)
0.8
0.6
0.4
0.2
QCD Sum Rules (Nesterenko, 1982)
pQCD (Bakulev et al, 2004)
BSE-DSE (Maris and Tandy, 2000)
Disp. Rel. (Geshkenbein, 2000)
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Northwestern University 7 K. K. Seth
CERN π-e scattering
DESY (Ackermann)
DESY (Brauel)
JLab (Tadevosyan)
this work
Q 2 (GeV/c) 2

ECT Workshop on Form Factors Feb. 18–22, 2013
Measurements of Pion and Kaon Timelike Form Factors
Timelike form factors are determined by measuring the cross sections
e + e − → h + h − , h = π,K,p, etc.
They have the great advantage of being able to determine form factors of any
hadron, but they have the great problem of detecting the desired hadron pair in
presence of several orders of magnitude larger background of QED–produced e + e − and
µ + µ − pairs.
• The MC–generated figure shows that for √ s = 3.77 GeV, the expected peak
counts for π, K, and p range from 500 to 100 counts, whereas the peak counts for
electrons and muons are 10 7 and ∼ 10 6 , i.e., lepton rejection at 3 to 4 orders of
magnitude is required.
Northwestern University 8 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
Measurements of Pion and Kaon Timelike Form Factors (cont’d)
I will not bore you with the nitty-gritty of how using all the detector components of
CLEO-c, the drift chambers, the central calorimeter, the RICH detector, and muon
detector, we were able to identify π, K, and p cleanly, in presence of the monstrous
backgrounds of electrons and muons. Here is how clean!
Counts / 0.02
Counts / 0.02
Counts / 0.02
180
160
140
120
s = 3772 MeV
MC
(a) PIONS
π
350
300
250
s = 3772 MeV
MC
(b) KAONS
K
70
60
50
s = 3772 MeV
MC
(c) PROTONS
p
100
200
40
80
150
30
60
40
100
20
20
0
120
0.96
μ
0.98 1
s = 3772 MeV
1.02
(d) PIONS
50
0
1.04300
0.96 0.98 1
s = 3772 MeV
10
0
1.02 1.04140
0.96
(e) KAONS
0.98 1
s = 3772 MeV
1.02 1.04
(f) PROTONS
100
250
120
80
200
100
60
150
80
60
40
100
40
20
50
20
0
0.96
50
0.98 1
s = 4170 MeV
0
1.02 1.04
(g) PIONS 120 0.96 0.98 1
s = 4170 MeV
0
1.02 1.04 0.96
(h) KAONS 25
0.98 1
s = 4170 MeV
1.02 1.04
(i) PROTONS
40
30
20
100
80
60
40
20
15
10
10
20
5
0
0.96 0.98 1 1.02 1.04
Xπ
0
40.96 0.98 1 1.02
0
1.044
0.96 0.98 1 1.02 1.04 4
X
X
Xh ≡ [E(h + )+E(h − )]/ √ s
K
p
3.77 GeV—MC
3.77 GeV—Data
4.17 GeV—Data
Northwestern University 9 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
|Q 2 | |F π (Q 2 )| (GeV 2 )
3
2.5
2
1.5
1
0.5
Latest Pion and Kaon Timelike Form Factors
(PRL 95, 261803 (2005)) (PRL 110, 022002 (2013)))
BABAR CLEO Present π
QCDSR
J/ψ
pQCD
α S
AdS/QCD
0
0 2.5 5 7.5 10 12.5 15 17.5 20
|Q 2 | (GeV 2 )
|Q 2 | |F K (Q 2 )| (GeV 2 )
3
2.5
2
1.5
1
0.5
REF. 18 CLEO Present K
J/ψ
α S
0
0 2.5 5 7.5 10 12.5 15 17.5 20
|Q 2 | (GeV 2 )
Q 2 (GeV 2 ) N(π + π − ) σ(Born), pb. Fπ(|Q 2 |)×10 Q 2 Fπ(|Q 2 |) (GeV 2 )
13.5 26±5 9.0±1.8 0.75±0.09 1.02±0.13
14.2 661±26 6.36±0.25 0.65±0.01 0.92±0.04
17.4 213±12 2.89±0.16 0.48±0.01 0.84±0.03
Q 2 (GeV 2 ) N(K + K − ) σ(Born), pb. FK(|Q 2 |)×10 Q 2 FK(|Q 2 |) (GeV 2 )
13.5 71±9 5.7±0.7 0.63±0.04 0.91±0.14
14.2 1564±40 3.95±0.10 0.54±0.01 0.76±0.02
17.4 644±25 2.23±0.09 0.44±0.01 0.77±0.03
Northwestern University 10 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
|Q 2 | |F π (Q 2 )| (GeV 2 )
3
2.5
2
1.5
1
0.5
QCDSR
J/ψ
0
0 2.5 5 7.5 10 12.5 15 17.5 20
|Q 2 | (GeV 2 )
Pion and Kaon Form Factors
BABAR CLEO Present π
pQCD
α S
AdS/QCD
The important experimental results are:
|Q 2 | |F K (Q 2 )| (GeV 2 )
3
2.5
2
1.5
1
0.5
REF. 18 CLEO Present K
J/ψ
α S
0
0 2.5 5 7.5 10 12.5 15 17.5 20
|Q 2 | (GeV 2 )
1. There is a remarkable agreement of the form factors for both pions and kaons with
the dimensional counting rule prediction of QCD that |Q 2 |Fπ,K are nearly
constant, varying with |Q 2 | only weakly as αS(|Q 2 |).
2. The existing theoretical predictions underpredict the magnitude of Fπ(|Q 2 |) at
large |Q 2 | by large factors, ≥ 2. More about this later.
3. Fπ(14.2 GeV 2 )/FK(14.2 GeV 2 ) = 1.21±0.03,
Fπ(17.4 GeV 2 )/FK(17.4 GeV 2 ) = 1.09±0.04 = (fπ/fK) 2 = 0.67±0.01.
More about this later.
Northwestern University 11 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
Theoretical Implications
Lattice lives in Euclidean time, and is not capable of addressing timelike form factors.
So we have to live with QCD–based models for timelike form factor predictions.
• The starting point of the existing calculations is factorization, with
F(Q 2 ) = ψin ×TH ×ψ ∗ out
The meson wave functions ψin,out represent
soft components, not calculable
perturbatively. TH represents the hard
interaction, “hopefully calculable in
perturbative QCD.”
ψ π T H ψ π *
• Since ab initio the wave functions are not known, various empirical ψ functions
0.5
have been used.
• Two examples→
(x)
ψ
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
Chernyak
2
(1-2x) x(1-x)
asymptotic
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Northwestern University 12 K. K. Seth
x(1-x)
x

ECT Workshop on Form Factors Feb. 18–22, 2013
Theoretical Implications (cont’d)
Lepage and Brodsky used φ(x) ∝ fπx(1−x) which is true for asymptotically
large momentum transfers when the q and ¯q share momenta equally, and obtained
|Q 2 |Fπ(|Q 2 |) = 16παS|Q 2 |×f 2 π = 0.21 GeV 2 , |Q 2 | = 17.4 GeV 2 ,
with fπ = 130.41 MeV, and assuming αS = 0.25.
This is a factor 4 smaller than what we measure.
Further, by replacing fπ by fK = 156.1 MeV, assuming identical wave function for
kaons, we obtain
|Q 2 |FK(|Q 2 |) = 16παS|Q 2 |×f 2 K = 0.31 GeV2 , |Q 2 | = 17.4 GeV 2 .
This is a factor 2.6 smaller than what we measure.
Further, the ratio, which removes the dependence on the assumed wave functions,
is predicted to be
Fπ(17.4 GeV 2 )/FK(17.4 GeV 2 ) = f 2 π/f 2 K = 0.70±0.01.
This is (36±1)% smaller than our measured value of 1.09±0.04.
With the precision measurements now available, it is quite obvious that
something is very wrong.
Could it be the wave functions?
Northwestern University 13 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
Since the relation
Theoretical Implications (cont’d)
Fπ(|Q 2 |)/FK(|Q 2 |) = f 2 π/f 2 K
is based on assuming identical wave functions for pions and kaons, Lepage and
Brodsky (1980) conjectured that because the s-quark in the kaon is ∼ 27 times
heavier than the 〈u,d〉 quarks in the pion, and the SU(3) flavor symmetry is
broken, the kaon wave function may differ from the pion wave function by acquiring
an asymmetric component, and account for the observed violation of the above
relation.
As a matter of fact, Chernyak and
Zhitnitsky (1984) were able to explain the
experimental observation in terms of their
two–humped wave function, and a rather large
effect of SU(3) breaking in the kaon.
Unfortunately, the CZ wave functions, and the conclusions drawn from them, “have
largely been discredited” (Brown et al. 2006), and we have to look elsewhere for the
explanation.
Northwestern University 14 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
Recently some quenched lattice calculations
(Braun et al., PRD 74, 074501 (2006)), and
AdS/CFT light–front QCD model calculations
(Brodsky and de Teremond,
arXiv:0802.0514[hep-ph] (2008)), have
addressed the question of pion and kaon wave
functions. Both obtain a much smaller
asymmetric component in the kaon wave
function, and a much smaller effect of
SU(3) breaking than CZ proposed.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
π K
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
What is needed to resolve theoretical differences is an experimental measure of the
effect of SU(3) breaking in the kaon.
We have now made the first such measurement.
Northwestern University 15 K. K. Seth
(x)
ψ
x

ECT Workshop on Form Factors Feb. 18–22, 2013
Estimating SU(3) Breaking in the Kaon
Lepage and Brodsky (1980) suggested that a large violation of the
Fπ/FK = f2 π/f2 K identity can arise if there is a substantial SU(3) breaking effect in the
kaon wave function. They predicted that a large SU(3) breaking effect would lead to a
large form factor for the neutral kaon, and FKSKL /FK + K− of the order one, and
suggested that FKSKL should be measured.
Following this suggestion, we have made the first ever measurement of the form factor
FKSKL (|Q2 |) at |Q 2 | = 17.4 GeV 2 .
Since the cross section for e + e − → KSKL is expected to be small, and KL are not
detected, careful criteria for event identification had to be developed, and their efficacy
tested. We have done so by measuring ψ(2S) → KSKL at √ s = 3.686 GeV using the
same event selection criteria, and confirmed that we obtain B(ψ(2S) → KSKL) in
agreement with its known value.
Northwestern University 16 K. K. Seth

ECT Workshop on Form Factors Feb. 18–22, 2013
Counts / 0.002
8
7
6
5
4
3
s = 4.17 GeV
2
1
0
0.9 0.95 1 1.05 1.1
E(K
S
) / E
beam
Counts / 0.002
60
50
40
30
20
10
s = 3.686 GeV
0
0.9 0.95 1 1.05 1.1
For e + e − → KSKL at √ s = 4.17 GeV, we obtain 4 events in the signal region, and a
background estimate of 2 counts, which leads to, for |Q 2 | = 17.4 GeV 2 :
E(K
S
) / E
beam
FKSKL (|Q2 |) = 3.9×10 −3 , or 90% CL of (0−7.0)×10 −3 ,
FKSKL (|Q2 |)/F K + K −(|Q 2 |) = 0.09, or 90% CL of 0−0.16.
In other words, the SU(3) breaking effect on the ratio is found to be very
small, certainly much less than of “the order of one”.
To come back to the original problem of Fπ/FK(expt.) = f2 π/f2 K , it is now apparent
that it can not be attributed to SU(3) breaking alone. The problem remains
unresolved.
• Here is a challenge worthy of the best theoretical attempts.
Northwestern University 17 K. K. Seth