1. Andrei Afanasev, Trento workshop, 31 October 2012 Radiative Corrections for Elastic Scattering of Muons and Electrons on a Proton Andrei Afanasev The George Washington University, Washington, DC, USA

2. Andrei Afanasev, Trento workshop, 31 October 2012 Plan of talk Radiative corrections for elastic charge lepton scattering. Model-independent and model-dependent; soft and hard photons. Refined bremsstrahlung calculations. Role of the masses Two-photon exchange effects. Formulation of a problem. Soft-photon exchange approximation and IR regularization. Beyond soft-photon approximation Summary

3. Andrei Afanasev, Trento workshop, 31 October 2012 Elastic Nucleon Form Factors Based on one-photon exchange approximation Two techniques to measure Latter due to: Akhiezer, Rekalo; Arnold, Carlson, Gross

4. Andrei Afanasev, Trento workshop, 31 October 2012 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) Assumed Q 2 /m e 2 >>1 Multiple soft-photon emission: solved by exponentiation, Yennie-Frautschi-Suura (YFS), 1961

5. Andrei Afanasev, Trento workshop, 31 October 2012 Complete radiative correction in O(α em ) 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 Meister&Yennie; Mo&Tsai 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 Log-enhanced but calculable (a,c,d)

6. Andrei Afanasev, Trento workshop, 31 October 2012 Bremsstrahlung for Relativistic vs Nonrelativistic Lepton Scattering. Accelerated charge always radiates, but the magnitude of the effect depends on kinematics. See Bjorken&Drell (Vol.1, Ch.8):. For large Q 2 >>m e 2 the rad.correction is enhanced by a large logarithm, log(Q2/m e 2 ) ~15 for GeV 2 momentum transfers. For small Q 2 <<m e 2, rad.correction suppressed by Q 2 /m e 2. For intermediate Q 2 ~m e 2, neither enhancement nor suppression, rad correction of the order 2α/π

7. Andrei Afanasev, Trento workshop, 31 October 2012 Basic Approaches to QED Corrections. L.W. Mo, Y.S. Tsai, Rev. Mod. Phys. 41, 205 (1969); Y.S. Tsai, Preprint SLAC-PUB-848 (1971).. Considered both elastic and inelastic inclusive cases. No polarization.. D.Yu. Bardin, N.M. Shumeiko, Nucl. Phys. B127, 242 (1977).. Covariant approach to the IR problem. Later extended to inclusive, semi- exclusive and exclusive reactions with polarization.. E.A. Kuraev, V.S. Fadin, Yad.Fiz. 41, 7333 (1985); E.A. Kuraev, N.P.Merenkov, V.S. Fadin, Yad. Fiz. 47, 1593 (1988).. Developed a method of electron structure functions based on Drell-Yan representation; currently widely used at e + e - colliders. Applied for polarized electron-proton scattering by AA et al, JETP 98, 403 (2004).

8. Andrei Afanasev, Trento workshop, 31 October 2012 Separating soft 2-photon exchange. Tsai; Maximon & Tjon (k→0); similar to Coulomb corrections at low Q 2. 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 for elastic ep. Also done for pion electroproduction in AA, Aleksejevs, Barkanova, arXiv:1207.1767 (there we also used Passarino-Veltman parameterization of loop integrals, inclusion of lepton masses is straightforward) ε δ Soft Q 2 = 6 GeV 2 q 1 →q q 2 →0 Lepton mass is not essential for TPE calculation in ultra-relativistic case; Two-photon effect below 1% for lower energies and Q 2 <0.1GeV 2

9. Andrei Afanasev, Trento workshop, 31 October 2012 Calculations using Generalized Parton Distributions Hard interaction with a quark Model schematics: Hard eq-interaction GPDs describe quark emission/absorption Soft/hard separation Use Grammer-Yennie prescription AA, Brodsky, Carlson, Chen, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004; Phys.Rev.D72:013008,2005 GPD e-e- q

10. Andrei Afanasev, Trento workshop, 31 October 2012 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

11. Andrei Afanasev, Trento workshop, 31 October 2012 Updated Ge/Gm plot AA, Brodsky, Carlson, Chen, Vanderhaeghen, Phys.Rev.Lett.93:122301, 2004; Phys.Rev.D72:013008, 2005 Significant part of the discrepancy is removed by the TPE mechanism Verification coming soon from VEPP, OLYMPUS, JLAB CLAS experiments

12. Andrei Afanasev, Trento workshop, 31 October 2012 Soft TPE for low-energy electron scattering. Soft TPE correction for ep scattering using Mo-Tsai formalism for three fixed electron energies: 115 MeV/c (top), 153 MeV/c (middle) 210 MeV/c (bottom curve); no large logs involved, muon correction is about the same: 0.25% max. Used kinematics for MUSE experiment proposal at PSI

13. Andrei Afanasev, Trento workshop, 31 October 2012 TPE for Proton Charge Radius Extraction. Electron scattering: R ch increase by +0.0015 fm, according to Blunden and Sick, Phys. Rev. C 72, 057601 (2005). “The radius after two- photon correction amounts to 0.897+/-0.018 fm”

14. Andrei Afanasev, Trento workshop, 31 October 2012 Hard Bremsstrahlung. Need to include radiative lepton tensor in a complete form: AA et al, Phys.Rev. D64 (2001) 113009; PLB 514, 269 (2001): terms ~ k emitted photon momentum) usually neglected in rad.correction calculations, but can lead to ~1% effect for Rosenbluth slope at high Q 2

15. Andrei Afanasev, Trento workshop, 31 October 2012 Electron Structure Functions. For polarized ep->e’X scattering, AA et al, JETP 98, 403 (2004); elastic ep: AA et al. PRD 64, 113009 (2001).. Resummation technique for collinear photons (=peaking approx.). Difference <0.5% from previous calculation including hard brem. Bystritskiy, Kuraev, Tomasi-Gustafson (2007) claimed this approach resolves Rosenbluth vs polarization discrepancy… but used incorrect energy cutoff ΔE/E of 3% (instead of e.g. 1.5%) =>miscalculated rad.correction by ~5% (absolute)

16. Andrei Afanasev, Trento workshop, 31 October 2012 Rosenbluth vs Super-Rosenbluth. Two experimental settings realized for elastic ep-scattering:. p(e,e’)p: detection of scattered electron, or “Rosenbluth”. p(e,p’)e: detection of recoil proton, or “super-Rosenbluth”. For GeV energies and momentum transfers bremsstrahlung corrections are different, but of the same order ~10-20%.. Caution for “super-Rosenbluth”: hard collinear bremsstrahlung from the scattered electron is allowed, may lead to numerical instabilities in rad.correction calculations (corresponding large logarithms from soft and hard collinear photon emission cancel each other) See AA, Akushevich, Ilychev, PLB 514, 269 (2001).. Third type experiment: JLAB Hall B Positron exp (talk by Brian Raue). Detecting final e’ and p’, beam energy reconstructed

17. Andrei Afanasev, Trento workshop, 31 October 2012 Lepton Mass Effects. Standard approximations keep the lepton mass in the logarithms but neglect it in power terms. May be justified in the ultrarelativistic case and Q 2 >>(lepton mass) 2. Most of analysis codes use exact mass dependence for hard brem, but use above approximations for the “soft” part of brem correction. Revised approach is required that will NOT result in new theoretical uncertainties. New rad.correction codes no longer use peaking approximation (justified for relatively small lepton masses). Formalism and Monte-Carlo generators can be adapted for this analysis (ELRADGEN; MASCARAD, etc; more on www.jlab.org/RC)

18. Andrei Afanasev, Trento workshop, 31 October 2012 Coulomb and Two-Photon Corrections. Coulomb correction calculations are well justified at lower energies and Q2 (see also John Arrington’s slides from this workshop). Hard two-photon exchange (TPE) contributions cannot be calculated with the same level of precision as the other contributions.. Two-photon exchange is independent on the lepton mass in an ultra- relativistic case.. Issue: For energies ~ mass TPE amplitude is described by 6 independent generalized form factors; but experimental data on TPE are for ultrarelativistic electrons, hence independent info on 3 other form factors will be missing.. Theoretical models show the trend that TPE has a smaller effect at lower Q 2. The reason is that “hard” TPE amplitudes do not have a 1/Q 2 Coulomb singularity, as opposed to the Born amplitude.

19. Andrei Afanasev, Trento workshop, 31 October 2012 Beam Single-Spin Asymmetry Parity-conserving. Observed spin-momentum correlation of the type: where k 1,2 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. Beam asymmetry (~10 -5 ) measured at Mainz and JLAB in parallel with parity-violating asymmetries (G0, HAPPEX, PREX, Qweak). See most recent data from PREX&HAPPEX on nuclei, arXiv:1208.6164 (to appear in PRL)

20. Andrei Afanasev, Trento workshop, 31 October 2012 Special Feature of the Transverse Beam Asymmetry. Can we use beam asymmetry to constrain TPE in muon scattering cross section?. Proportional to lepton mass (lepton helicity violation is required for nonzero asymmetry). But the asymmetry is enhanced due to hard collinear photon exchange (at least one of the photons in two-photon exchange mechanism), this feature is unique for transverse polarization; corresponding mechanism does not appear in the cross section as a consequence of electromagnetic current conservation (AA, Merenkov, Phys.Lett. B599 (2004) 48; hep-ph/0407167v2; Phys.Rev. D70 (2004) 073002). Asymmetry is intedepndent of beam energy at fixed Q2 (AA, Merenkov’04; Gorchtein’04). Cannot use the asymmetry to constrain directly lepton helicity- violating TPE contribution to cross section

21. Andrei Afanasev, Trento workshop, 31 October 2012 Helicity-Flip in TPE; estimate of inelastic contribution. Scalar t-channel exchange contributes to TPE if you give up ultra- relativistic limit (no longer setting m lepton to zero!). For Q 2 =10 -2 GeV 2 is about 10 -3, lepton helicity-flip dominates, scales as Can be studied directly in the ratio of μ+ and μ- cross sections

22. Andrei Afanasev, Trento workshop, 31 October 2012 Conclusions. Radiative corrections are standard and well-established in electron scattering, care must be taken in this experiment that the radiative correction calculations are correctly implemented without invalid approximations.. Parts of the radiative corrections are expected to be suppressed for muons due to the larger muon mass.. Two-photon exchange corrections are generally expected to be small, helicity-flip mechanism is enhanced for the muons. Can be studied directly in the ratio of μ+ and μ- cross sections. We evaluate systematic uncertainty to cross sections from radiative corrections at 0.1% for muon at Q 2 ~0.01 GeV 2, due to uncertainty in two-photon exchange effects