Two-Photon Exchange for Elastic Electron-Nucleon
Scattering
Andrei Afanasev
Jefferson Lab
JLab Users Workshop
June 22, 2005
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Plan of talk
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Two-photon exchange effects in the process e+p→e+p
Models for two-photon exchange
Cross sections
Polarization transfer
Single-spin asymmetries
Parity-violating asymmetry
Refined bremsstrahlung calculations
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Elastic Nucleon Form Factors
•Based on one-photon exchange approximation
M fi  M 1fi
1
fi
M  e ue  ueu p ( F1 (t )  
•Two techniques to measure
2
   q
2m
F2 (t ))u p
   0 (GM 2    GE 2 ) : Rosenbluth technique
GE  2 (1   )
Px
Ax


Pz
Az
GM  1   2
: Polarizati on technique
GE  F1  F2 , GM  F1  F2
( Py  0)
Latter due to: Akhiezer, Rekalo; Arnold, Carlson, Gross
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Do the techniques agree?
SLAC/Rosenbluth
~5% difference in cross-section
JLab/Polarization
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Both early SLAC and Recent JLab experiments on (super)Rosenbluth separations followed
Ge/Gm~const
JLab measurements using polarization transfer technique give different results (Jones’00,
Gayou’02)
Radiative corrections, in particular, a short-range part of 2-photon
exchange is a likely origin of the discrepancy
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Electron Scattering: LO and NLO in αem
Radiative Corrections:
• Electron vertex correction (a)
• Vacuum polarization (b)
• Electron bremsstrahlung (c,d)
• Two-photon exchange (e,f)
• Proton vertex and VCS (g,h)
• Corrections (e-h) depend on the
nucleon structure
•Guichon&Vanderhaeghen’03:
Can (e-f) account for the Rosenbluth
vs. polarization experimental
discrepancy? Look for ~3% ...
Main issue: Corrections dependent on nucleon structure
Model calculations:
•Blunden, Melnitchuk,Tjon, Phys.Rev.Lett.91:142304,2003
•Chen, AA, Brodsky, Carlson, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Separating soft photon exchange
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Tsai; Maximon & Tjon
We used Grammer &Yennie prescription PRD 8, 4332 (1973) (also applied in
QCD calculations)
Shown is the resulting (soft) QED correction to cross section
NB: Corresponding effect to polarization transfer and/or asymmetry is zero
ε
δSoft
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Q2= 6 GeV2
Andrei Afanasev, JLab Users Workshop, 6/22/05
Lorentz Structure of 2-γ amplitude
Generalized form factors are functions of two Mandelstam invariants;
Specific dependence is determined by nucleon structure
M fi  M 1fi  M 2fi
M
1
fi
 ue  ueu p ( F1 (t )  
   q
2m
F2 (t ))u p
M 2fi  Ve  V p  Ae  Ap
Ve  ue  ue , V p  u p ( F1 ' ( s, u )  
   q
2m
F2 ' ( s, u ))u p
Ae  ue   5ue , Ap  u pGA ( s, u )   5u p
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Andrei Afanasev, JLab Users Workshop, 6/22/05
New Expressions for Observables
We can formally define ep-scattering observables in terms of
the new form factors:
   0 (| GM |2     | GE |2 2  (1   )(1   2 ) Re GM* G A )
Re( GE* GM )  2 (1   )  Re( GE* G A ) 1   2 (1   )
Px

Pz
| GM |2  1   2  Re( GM* G A )2  (1   )
Py 
Im( GE* GM )  2 (1   )  Im( GE* G A ) 1   2 (1   )
| GM |2     | GE |2 2  (1   )(1   2 ) Re GM* G A
For the target asymmetries: Ax=Px, Ay=Py, Az=-Pz
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Calculations using Generalized
Parton Distributions
Model schematics:
• Hard eq-interaction
•GPDs describe quark
emission/absorption
•Soft/hard separation
•Use Grammer-Yennie
prescription
Hard interaction with
a quark
AA, Brodsky, Carlson, Chen, Vanderhaeghen,
Phys.Rev.Lett.93:122301,2004; hep-ph/0502013
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Short-range effects; on-mass-shell quark
(AA, Brodsky, Carlson, Chen,Vanderhaeghen)
Two-photon probe directly interacts with a (massless) quark
Emission/reabsorption of the quark is described by GPDs
eq2  em e
2
Aeq  eq 
(V  Vq  fV  Ae  Aq  f A ),
t 2
Ve , q  ue , q  ue , q , Ae , q  ue , q   5ue, q
u
t
t 1
u
1
s
fV  2[log(  )  i ] log(  2 )  [ (log( )  i )  log(  )] 
s

2 s
t
u
t
(u 2  s 2 ) 1
1
s
s
u2  s2
2 u
2

[ 2 (log ( )   )  2 log(  )(log(  )  i 2 )]  i
4
s
t
u
t
t
2su
t 1
u
1
s
f A   [ (log( )  i )  log(  )] 
2 s
t
u
t
(u 2  s 2 ) 1
1
s
s
t2
2 u
2

[ 2 (log ( )   )  2 log(  )(log(  )  i 2 )]  i
4
s
t
u
t
t
2 su
Note the additional effective (axial-vector)2 interaction; absence of mass terms
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Andrei Afanasev, JLab Users Workshop, 6/22/05
`Hard’ contributions to generalized
form factors
GPD integrals
Two-photon-exchange form factors from GPDs
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Two-Photon Effect for Rosenbluth Cross Sections
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Data shown are from Andivahis et al,
PRD 50, 5491 (1994)
Included GPD calculation of twophoton-exchange effect
Qualitative agreement with data:
Discrepancy likely reconciled
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Two-Photon Effect for Rosenbluth Cross Sections
.
Blunden, Melnitchouk, Tjon, Phys.Rev.Lett.91:142304,2003; nuclth/0506039
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Updated Ge/Gm plot
AA, Brodsky, Carlson, Chen, Vanderhaeghen,
Phys.Rev.Lett.93:122301,2004; hep-ph/0502013
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Andrei Afanasev, JLab Users Workshop, 6/22/05
Polarization transfer
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Also corrected by two-photon
exchange, but with little impact on
Gep/Gmp extracted ratio
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Charge asymmetry
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Cross sections of electron-proton
scattering and positron-proton
scattering are equal in one-photon
exchange approximation
. Different for two- or more
photon exchange
To be measured in JLab Experiment 04-116, Spokepersons W. Brooks et al.
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Quark+Nucleon Contributions to An
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Single-spin asymmetry or polarization normal to the scattering plane
Handbag mechanism prediction for single-spin asymmetry/polarization of elastic
ep-scattering on a polarized proton target

2 (1   ) 1 
1 
G
Im(
A
)

G
Im(
B
)
 E

M

R 
2

~
No dependence on GPD H
An 
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Only minor role of quark mass
Andrei Afanasev, JLab Users Workshop, 6/22/05
Parity Violating elastic e-N scattering
Longitudinally polarized electrons,
unpolarized target

 R   L   GF Q 2  AE  AM  AA
A

 R   L  4 2  2 unpol
AE   ( ) GEZ GE

AM   G GM
Z
M
AA  (1  4 sin 2 W )  G Ae GM
e

p e
p
2

 = Q2/4M2
 = [1+2(1+)tan2( /2)]-1
’= [(+1)(1-2)]1/2
Neutral weak form factors contain explicit contributions from strange sea
GEZ, M (Q 2 )  (1  4 sin 2 W )(1  RAp )GEp, M  (1  RAn )GEn , M  GEs , M
G Ae (Q 2 )  G AZ  (FA  R e )  s
GZA(0) = 1.2673 ± 0.0035 (from b decay)
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
8 2
 3.45
1  4 sin 2 W
2γ-exchange Correction to Parity-Violating
Electron Scattering
γ
Electromagnetic
.
γ
Z0
x
Neutral Weak
New parity violating terms due to (2gamma)x(Z0) interference should
be added:

Z
Re AAA
AAV
 ( 12 )  (1   )(1   2 ) Re( GA GMZ )e

Z
Re AAA
AVA
 12 (1  4 sin 2 W )(1   ) Re( GA GAZ )e
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GPD Calculation of 2γ×Z-interference
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Can be used at higher Q2, but points at a
problem of additional systematic corrections
for parity-violating electron scattering. The
effect evaluated in GPD formalism is the
largest for backward angles:
AA & Carlson, hep-ph/0502128, Phys. Rev.
Lett. 94, 212301 (2005):
measurements of strange-quark content of the
nucleon are affected, Δs may shift by ~10%
Important note: (nonsoft) 2γ-exchange amplitude has no 1/Q2 singularity;
1γ-exchange is 1/Q2 singular => At low Q2, 2γ-corrections is suppressed as Q2
P. Blunden used this formalism and evaluated correction of 0.16% for Qweak
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Proton Mott Asymmetry at Higher Energies
Spin-orbit interaction of electron moving in a Coulomb field
N.F. Mott, Proc. Roy. Soc. London, Set. A 135, 429 (1932).
Transverse beam SSA, note (α me/Ebeam) suppression, units are
parts per million calculation by AA et al, hep-ph/0208260
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Due to absorptive part of two-photon exchange amplitude (elastic
contribution: dotted, elastic+inelastic: solid curve for target case)
Nonzero effect observed by SAMPLE Collab for beam asymmetry
(S.Wells et al., PRC63:064001,2001) for 200 MeV electrons
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MAMI data on Mott Asymmetry
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F. Maas et al.,
Phys.Rev.Lett.94:082001, 2005
Pasquini, Vanderhaeghen:
Surprising result: Dominance of
inelastic intermediate excitations
Elastic intermediate
state
Inelastic excitations
dominate
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Beam Normal Asymmetry
(AA, Merenkov)
Q 2
d 3k Lb H b
A  2
Im

 D ( s, Q 2 )  2k0
Q12Q22
e
n
1
Lb  Tr (kˆ2  me )  (kˆ1  me )(1   5ˆe ) b (kˆ  me ) 
4
1
H b  Tr ( pˆ 2  M ) ( pˆ1  M )(1   5ˆp )Tb
4
aˆ  a  
Lb q  Lb q2  Lb q1b  H b q  H b q2  H b q1b  0
Gauge invariance essential in cancellation of infra-red singularity:
Lb H b  0 if
Q12 and / or Q22  0
Feature of the normal beam asymmetry: After me is factored out, the remaining
expression is singular when virtuality of the photons reach zero in the loop integral!
But why are the expressions regular for the target SSA?!
Also available calculations by Gorshtein, Guichon, Vanderhaeghen, Pasquini;
Confirm quasi-real photon exchange enhancement in the nucleon resonance region
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Phase Space Contributing to the absorptive
part of 2γ-exchange amplitude
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2-dimensional integration (Q12, Q22) for the elastic intermediate state
3-dimensional integration (Q12, Q22,W2) for inelastic excitations
`Soft’ intermediate electron;
Both photons are hard collinear
Example: MAMI A4
E= 855 MeV
Θcm= 57 deg
One photon is
Hard collinear
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Special property of normal beam asymmetry
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AA, Merenkov, Phys.Lett.B599:48,2004, Phys.Rev.D70:073002,2004;
+Erratum (2005)
Reason for the unexpected behavior: hard collinear quasi-real photons
. Intermediate photon is collinear to the parent electron
. It generates a dynamical pole and logarithmic enhancement of inelastic
excitations of the intermediate hadronic state
.
For s>>-t and above the resonance region, the asymmetry is given by:
2
(

m
)
Q
F1  F2
Q2
e
e
An   p
 2
(log( 2 )  2)
2
2
8
F1  F2
me
Also suppressed by a standard diffractive factor exp(-BQ2), where B=3.5-4 GeV-2
Compare with no-structure asymmetry at small θ:
me 3
A 

s
e
n
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Input parameters
.
Used σγp from parameterization by N. Bianchi at al., Phys.Rev.C54 (1996)1688
(resonance region) and Block&Halzen, Phys.Rev. D70 (2004) 091901
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Predictions for normal beam asymmetry
Use fit to experimental data on σγp and exact 3-dimensional integration
over phase space of intermediate 2 photons
Data expected from
HAPPEX, G0
HAPPEX
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No suppression for beam asymmetry with energy
at fixed Q2
x10-6
x10-9
SLAC E158 kinematics
Parts-per-million vs. parts-per
billion scales: a consequence of
soft Pomeron exchange, and hard
collinear photon exchange
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Beam SSA in the resonance region
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Trouble with `Handbag’ for beam normal asymmetry
Model schematics:
• Hard eq-interaction
•GPDs describe quark
emission/absorption
•Soft/hard photon separation
•Use Grammer-Yennie
prescription
•Hard interaction with a quark
•Applied for BSSA by Gorshtein,
Guichon, Vanderhaeghen, NP A741,
234 (2004)
•Exchange of hard collinear photons is kinematically forbidden if one assumes
a handbag approximation (placing quarks on mass shell) , BUT…
•Collinear-photon-exchange enhancement (up to two orders of magnitude) is
allowed for off-mass-shell quarks (higher twists) and Regge-like contributions
=> If the handbag approximation is violated at ≈ 0.5% level,
It would result in (0.5%)log2(Q2/me2) ≈100% level correction to beam asymmetry
But target asymmetry, TPE corrections to Rosenbluth and polarization transfer
predictions will be violated at the same 0.5% level
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Lessons from SSA in Elastic ep-Scattering
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Collinear photon exchange present in (light particle) beam SSA
(Electromagnetic) gauge invariance of is essential for cancellation of
collinear-photon exchange contribution for a (heavy) target SSA
Unitarity is very helpful in reducing model dependence of calculations
at small scattering angles, in particular for beam asymmetry
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Mott Asymmetry Promoted to High Energies
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.
.
Excitation of inelastic hadronic intermediate states by the consecutive
exchange of two photons leads to logarithmic and double-logarithmic
enhancement due to contributions of hard collinear quasi-real photons for the
beam normal asymmetry
The strongest enhancement has a form: log2(Q2/me2) => two orders of
magnitude+unsuppressed angular dependence
. Can be generalized to transverse asymmetries in light spin-1/2 particle
scattering via massless gauge boson exchange
. Beam asymmetry at high energies is strongly affected by effects beyond
pure Coulomb distortion
. Supports Qiu-Sterman twist-3 picture of SSA in QCD
What else can we learn from elastic beam SSA?
. Check implications of elastic hadron scattering in QCD
(Can large An,t in pp elastic be due to onset of collinear multi-gluon
exchange + inelastic excitations?)
. Large SSA in deep-inelastic collisions due to hard collinear gluon
exchange; in pQCD need NNLO to obtain unsupressed collinear gluons
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Full Calculation of Bethe-Heitler Contribution
Additional work by AA at al., using MASCARAD (Phys.Rev.D64:113009,2001)
Full calculation including soft and hard bremsstrahlung
Radiative leptonic tensor in full form
AA et al, PLB 514, 269 (2001)
1
Lr    Tr (kˆ2  m) (1   5ˆe )( kˆ1  m) 
2
  kˆ    kˆ 
 k1
k 2 
  
  


k

k
k

k
2
k

k
2k  k 2
1
2 
1

 k1
k 2 
  kˆ    kˆ 
  
  


2k  k1 2k  k 2
 k  k1 k  k 2 
Additional effect of full soft+hard brem → +1.2% correction to ε-slope
Resolves additional ~25% of Rosenbluth/polarization discrepancy!
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Bethe-Heitler corrections to
polarization transfer and cross sections
AA, Akushevich, Merenkov Phys.Rev.D64:113009,2001;
AA, Akushevich, Ilychev, Merenkov, PL B514, 269 (2001)
Pion threshold: um=mπ2
In kinematic of elastic ep-scattering measurements, cross sections are more sensitive to RC
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Two-photon exchange for electron-proton scattering
.
.
.
.
.
Quark-level short-range contributions are substantial (3-4%) ; correspond to J=0 fixed
pole (Brodsky-Close-Gunion, PRD 5, 1384 (1972)).
Structure-dependent radiative corrections calculated using GPDs bring into agreement
the results of polarization transfer and Rosenbluth techniques for Gep measurements
Full treatment of brem corrections removed ~25% of R/P discrepancy in addition to 2γ
Collinear photon exchange + inelastic excitations dominance for beam asymmetry
Experimental tests of multi-photon exchange
. Comparison between electron and positron elastic scattering (JLab E04-116)
. Measurement of nonlinearity of Rosenbluth plot (JLab E05-017)
. Search for deviation of angular2 dependence of polarization and/or asymmetries
from Born behavior at fixed Q (JLab E04-019)
. Elastic single-spin asymmetry or induced polarization (JLab E05-015)
. 2γ add-ons for parity-violating measurements (HAPPEX, G0)
Through active theoretical support emerged JLab research program of
Testing precision of the electromagnetic probe
Double-virtual VCS studies with two spacelike photons
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