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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Updated Ge/Gm plot AA, Brodsky, Carlson, Chen, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004; hep-ph/0502013

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!

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

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 04-116, Spokepersons W. Brooks et al.

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

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

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.B595:250-259,2004; 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)

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) = 1.2673 ± 0.0035 (from b decay)

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) 015501

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

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

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/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

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

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

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/0208260 Transverse beam SSA, units are parts per million Figures from AA et al, hep-ph/0208260 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/0208260

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

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

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/0505022 Confirm quasi-real photon exchange enhancement

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

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:

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

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

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

Beam SSA in the resonance region

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)

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

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