1. QED Radiative Corrections for Precision Measurements of Nucleon Form Factors
Andrei Afanasev
Jefferson Lab
Nucleon 2005
INFN, Frascati, October 12, 2005

2. Main problem
Uncertainties in QED radiative corrections limit interpretability of precision experiments on electron-hadron scattering

3. Plan of talk Radiative corrections for electron scattering
Model-independent and model-dependent; soft and hard photons
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

4. Elastic Nucleon Form Factors
Based on one-photon exchange approximation
Two techniques to measure
Latter due to: Akhiezer, Rekalo; Arnold, Carlson, Gross

5. Do the techniques agree?
SLAC/Rosenbluth
~5% difference in cross-section
x5 difference in polarization
JLab/Polarization
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

6. Basics of QED radiative corrections
(First) Born approximation
Initial-state radiation
Final-state radiation
Cross section ~ dω/ω => integral diverges logarithmically: IR catastrophe
Vertex correction => cancels divergent terms; Schwinger (1949)
Multiple soft-photon emission: solved by exponentiation,
Yennie-Frautschi-Suura (YFS), 1961

7. Complete radiative correction in O(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
Mo&Tsai; Meister&Yennie formalism
Further work by Bardin&Shumeiko; Maximon&Tjon; AA, Akushevich, Merenkov;
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, Melnitchouk,Tjon, Phys.Rev.Lett.91:142304,2003
Chen, AA, Brodsky, Carlson, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004

8. Separating soft 2-photon exchange
Tsai; Maximon & Tjon (k→0)
Grammer &Yennie prescription PRD 8, 4332 (1973) (also applied in QCD calculations)
Shown is the resulting (soft) QED correction to cross section
Already included in experimental data analysis
NB: Corresponding effect to polarization transfer and/or asymmetry is zero ε
δSoft
Q2= 6 GeV2

9. 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 kinematics of elastic ep-scattering measurements, cross sections are more sensitive to RC

10. Lorentz Structure of 2-γ amplitude
Generalized form factors are functions of two Mandelstam invariants;
Specific dependence is determined by nucleon structure

11. New Expressions for Observables
We can formally define ep-scattering observables in terms of the new form factors:
For the target asymmetries: Ax=Px, Ay=Py, Az=-Pz

12. 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/

13. 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
Note the additional effective (axial-vector)2 interaction; absence of mass terms

14. `Hard’ contributions to generalized form factors
GPD integrals
Two-photon-exchange form factors from GPDs

15. Two-Photon Effect for Rosenbluth Cross Sections
Data shown are from Andivahis et al, PRD 50, 5491 (1994)
Included GPD calculation of two-photon-exchange effect
Qualitative agreement with data:
Discrepancy likely reconciled

16. Updated Ge/Gm plot AA, Brodsky, Carlson, Chen, Vanderhaeghen,
Phys.Rev.Lett.93:122301,2004; hep-ph/

17. Full Calculation of Bethe-Heitler Contribution
Additional work by AA et 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)
Additional effect of full soft+hard brem → +1.2% correction to ε-slope
Resolves additional ~25% of Rosenbluth/polarization discrepancy!

18. Polarization transfer
Also corrected by two-photon exchange, but with little impact on Gep/Gmp extracted ratio

19. Charge asymmetry
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 , Spokepersons W. Brooks et al.

20. Single-Spin Asymmetries in Elastic Scattering
Parity-conserving
Observed spin-momentum correlation of the type:
where k1,2 are initial and final electron momenta, s is a polarization vector
of a target OR beam
For elastic scattering asymmetries are due to absorptive part of 2-photon exchange amplitude
Parity-Violating

21. Quark+Nucleon Contributions to An
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
Only minor role of quark mass

22. Radiative Corrections for Weak Processes
Semi-Leptonic processes involving nucleons
Neutrino-nucleon scattering
Per cent level reached by NuTeV. Radiative corrections for DIS calculated at a partonic level (D. Bardin et al.)
Neutron beta-decay: Important for Vud measurements; axial-vector coupling gA
Marciano, Sirlin, PRL 56, 22 (1986); Ando et al., Phys.Lett.B595: ,2004; Hardy, Towner, PRL94:092502,2005
Parity-violating DIS: Bardin, Fedorenko, Shumeiko, Sov.J.Nucl.Phys.32:403,1980; J.Phys.G7:1331,1981, up to 10% effect from rad.corrections
Parity-violating elastic ep (strange quark effects, weak mixing angle)

23. Parity Violating elastic e-N scattering
Longitudinally polarized electrons, unpolarized target
t = Q2/4M2
e = [1+2(1+t)tan2(q /2)]-1
e’= [t(t+1)(1-e2)]1/2
Neutral weak form factors contain explicit contributions from strange sea
GZA(0) = ± (from b decay)

24. Born and Box diagrams for elastic ep-scattering
(d) Computed by Marciano,Sirlin, Phys.Rev.D29:75,1984, Erratum-ibid.D31:213,1985 for atomic PV
(c) Presumed small, e.g., M. Ramsey-Musolf, Phys.Rev. C60 (1999)

25. New Expressions for PV asymmetry
PV-asymmetry, Born Approximation
PV asymmetry in terms of generalized form factors including multi-photon exchange

26. 2γ-exchange Correction to Parity-Violating Electron ScatteringZ0 γ γ
Electromagnetic
Neutral Weak
New parity violating terms due to (2gamma)x(Z0) interference should be added:

27. GPD Calculation of 2γ×Z-interference
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/ , Phys. Rev. Lett. 94, (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

28. Two-photon exchange for PV electron-proton scattering
Quark-level short-range contributions are substantial (2-3%)
2γ-correction to parity-violating asymmetry does not cancel. May reach a few per cent for GeV momentum transfers
Corrections are angular-dependent, not reducible to re-definition of coupling constants
Revision of γZ-box contribution and extension of model calculations to lower Q2 is necessary
Experimentally measurable directly by comparing electrons vs positrons on a spin-0 target- it is difficult => in the meantime need to rely on the studies of 2γ-effect for parity-conserving observables

29. Parity-Conserving Single-Spin Asymmetries in Scattering Processes (early history)
N. F. Mott, Proc. R. Soc. (London), A124, 425 (1929), noticed that polarization and/or asymmetry is due to spin-orbit coupling in the Coulomb scattering of electrons (Extended to high energy ep-scattering by AA et al., 2002).
Julian Schwinger, Phys. Rev. 69, 681 (1946); ibid., 73, 407 (1948), suggested a method to polarize fast neutrons via spin-orbit interaction in the scattering off nuclei
Lincoln Wolfeinstein, Phys. Rev. 75, 1664 (1949); A. Simon, T.A.Welton, Phys. Rev. 90, 1036 (1953), formalism of polarization effects in nuclear reactions

30. 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);
BNSA for electron-muon scattering: Barut, Fronsdal, Phys.Rev.120, 1871 (1960);
BNSA for electron-proton scattering: Afanasev, Akushevich, Merenkov, hep-ph/
Transverse beam SSA, units are parts per million
Figures from AA et al, hep-ph/
Due to absorptive part of two-photon exchange amplitude; shown is elastic contribution
Nonzero effect observed by SAMPLE Collaboration (S.Wells et al., PRC63:064001,2001) for 200 MeV electrons
Calculations of Diaconescu, Ramsey-Musolf (2004): low-energy expansion version of hep-ph/

31. Phase Space Contributing to the absorptive part of 2γ-exchange amplitude
2-dimensional integration (Q12, Q22) for the elastic intermediate state
3-dimensional integration (Q12, Q22,W2) for inelastic excitations
Examples: MAMI A4
E= 855 MeV
Θcm= 57 deg;
SAMPLE, E=200 MeV
`Soft’ intermediate electron;
Both photons are hard collinear
One photon is
Hard collinear

32. MAMI data on Mott Asymmetry
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

33. Beam Normal Asymmetry (AA, Merenkov)
Gauge invariance essential in cancellation of infra-red singularity for target asymmetry
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 calculations by Vanderhaeghen, Pasquini (2004); Gorschtein, hep-ph/
Confirm quasi-real photon exchange enhancement

34. Peaking Approximation
Dominance of collinear-photon exchange =>
Can replace 3-dimensional integral over (Q12,Q22,W) with one-dimensional integral along the line (Q12≈0; Q22=Q2(s-W2)/(s-M2))
Save computing time
Avoid uncertainties associated with (unknown) double-virtual Compton amplitude
Provides more direct connection to VCS and RCS observables

35. Special property of normal beam asymmetry
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:
Also suppressed by a standard diffractive factor exp(-BQ2), where B=3.5-4 GeV-2 Compare with no-structure asymmetry at small θ:

36. 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)

37. 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 from HAPPEX
More to come from G0
2γ-exchange is in
diffractive regime
HAPPEX

38. No suppression for beam asymmetry with energy at fixed Q2
SLAC E158 kinematics
Parts-per-million vs. parts-per billion scales: a consequence of
soft Pomeron exchange, and hard collinear photon exchange

39. Beam SSA in the resonance region

40. Normal Beam Asymmetry on Nuclei
Important systematic correction for parity-violation experiments (HAPPEX on 4He, PREX on Pb)
For diffractive 2γ-exchange, scales as ~A/Z~2
Five orders of magnitude enhancement for (HAPPEX) due to excitation of inelastic
intermediate states in 2γ-exchange (AA, Merenkov, to be published)

41. Summary on SSA in Elastic ep-Scattering
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 crucial in reducing model dependence of calculations at small scattering angles, in particular for beam asymmetry
Strong-interaction dynamics for BNSA small-angle ep-scattering above the resonance region is soft diffraction
For the diffractive mechanism An is a) not supressed with beam energy and b) does not grow with Z
If confirmed experimentally → first observation of diffractive component in elastic electron-hadron scattering

42. 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 angular dependence of polarization and/or asymmetries from Born behavior at fixed Q2 (JLab E04-019)
Elastic single-spin asymmetry or induced polarization (JLab E05-015)
2γ additions for parity-violating measurements (HAPPEX, G0)
Through active theoretical support emerged a research program of
Testing precision of the electromagnetic probe
Double-virtual VCS studies with two space-like photons