1. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 1 High order radiative corrections in Electron-Proton scattering Egle Tomasi-Gustafsson Saclay, France JLab, August 5, 2008 In collaboration with Yu. Bystriskiy, V. Bytev and Prof. E.A. Kuraev

2. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 2 Cross section of (quasi)elastic ep-scattering Elastic Inelastic Higher order inelastic double brehmstrahlung, pair production.. Classify radiative corrections:

3. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 3 Unpolarized case

4. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 4 Feynman diagrams, for the scattering amplitude Elastic scattering

5. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 5 Single bremstrahlung

6. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 6 Double Bremstrahlung and Pair Production Lowest order radiative corrections (RC): Mo and Tsai (1969) Infrared divergences : « photon mass », Logarithmic enhancement:

7. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 7 Vacuum Polarization Main contribution: vacuum polarization due to electron positron pair

8. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 8 Vertex function The contribution to F 2 (Q 2 ) is suppressed by compared to F 1 (Q 2 ) The « photon mass »,, is an auxiliary parameter, which does not enter in the final answer for the cross section

9. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 9 Collinear Emission of Photons Contains logarithmic enhancement Suffers from infrared divergences

10. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 10

11. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 11 Scattered electron energy All orders of PT needed  beyond Mo & Tsai approximation! Initial state emission final state emission Quasi-elastic scattering 3% Y0Y0 Not so small! Shift to LOWER Q 2

12. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 12 Short history (I) Schwinger: corrections to cross section for electron scattering in external field  =  0 (1+  ) (1)  Yennie, Frauchi, Suura: cross section of any pure process (without real photon emission) is zero.  Kessler, Ericsson, Baier, Fadin, Khoze, Y. Tsai : quasi real electron method. Emission of hard photon is described in terms of a convolution of a radiative function with Born cross section.   (1) Not adequate

13. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 13 Short history (II) 1977: Altarelli Parisi Gribov Lipatov, Dokshitzer: (DGLAP) Asymptotic freedom, evolution equation, Collins factorization theorem. Drell-Yan picture of hard processed in QED : application of QCD ideas to QED: radiative corrections in form of structure functions and Drell-Yan picture Leading terms: and non leading terms explicitely taken into account in DGLAP evolution equations. In QED known as Lipatov equations (1975).

14. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 14 Structure Function method SF method applied to QED processes: calculation of radiative corrections with precision ~ 0.1%. Takes into account the dynamics of the process Formulated in terms of parton densities (leptons, antileptons, photons) Many applications to different processes E. A. Kuraev and V.S. Fadin, Sov. J. of Nucl. Phys. 41, 466 (1985) Lipatov equations (1975)

15. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 15 Structure Function Method (Applications) –e+e-  hadrons( J/  width) E. A. KURAEV and V.S. FADIN, Sov. J. of Nucl. Phys. 41, 466 (1985) –ep  e’X (elastic and inelastic scattering) E. A. KURAEV ;N.P. MERENKOV and V.S. FADIN, Sov. J. of Nucl. Phys. 47,1009 (1988) –Decay width of mesons (FSI) E. A. KURAEV, JETP Lett.65, 127 (1997) –Radiative corrections for LEP beam (small angle BHABHA scattering) A.B.Arbuzov, E.A.Kuraev et al, Phys. Lett.B 399, 312 (1997) –Compton and double Compton scattering A.N.Ilyichev, E.A. Kuraev, V.Bytev and Y. P. Peresun'ko, J. Exp. Theor. Phys.100 31 (2005) –Structure function method applied to polarized and unpolarized electron-proton scattering: A solution of the GE(p)/GM(p) discrepancy. Y. Bystricky, E.A.Kuraev, E. Tomasi-Gustafsson, Phys. Rev. C75, 015207 (2007). – Radiative corrections to DVCS electron tensor. V.Bytev, E.A.Kuraev, E. Tomasi-Gustafsson, Phys. Rev. C (2008)

16. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 16 The Structure Function Method Distinguish: -leading contributions of higher order -non leading ones E. A. K. and V.S. FADIN, Sov. J. of Nucl. Phys. 41, 466 (1985) The SF method is based on: Renormalization group evolution equation Drell-Yan parton picture of the cross section in QCD Electron SF: probability to ‘find’ electron in the initial electron, with energy fraction x and virtuality up to Q 2

17. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 17 LSF: ‰ precision E. A. K. and V.S. FADIN, Sov. J. of Nucl. Phys. 41, 466 (1985) LLA (Leading Logarithm Approximation) Precision of LLA Even when corrections in first order PT are  ~100%, the accuracy of higher order RC (LSF) is  1% ! Including K-factor

18. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 18 The LSF cross section (for ep ) If the electron is detected in a calorimeter:  the cross section is integrated over the scattered electron energy fraction: The K-factor includes all non leading contributions:

19. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 19 Results Both calculations assume dipole FFs The slope changes (due to different RC) Q 2 =1 GeV 2 Q 2 =5 GeV 2 Q 2 =3 GeV 2 SF Born Polarization RC Born ……… E.T-G, Phys. Part. Nucl. Lett. 4, 281-288 (2007).

20. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 20 Unpolarized Cross section Born +dipole FFs (=unpolarized experiment+Mo&Tsai) SF (with dipole FFs) SF+2  exchange Q 2 =3 GeV 2 Q 2 =5 GeV 2 SF: change the slope! Q 2 =1 GeV 2 2  exchange very small!

21. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 21 Polarization ratio Born SF SF+2  exchange  =60° 2  destroys linearity! 2  exchange very small!  =80°  =20° Yu. Bystricky, E.A.Kuraev, E. T.-G, Phys. Rev. C 75, 015207 (2007)

22. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 22 Radiative Corrections (SF method) Polarization data JLab data SLAC data Yu. Bystricky, E.A.Kuraev, E. Tomasi-Gustafsson, Phys. Rev. C75, 015207 (2007) SF corrected Rosenbluth parameters highly correlated! E.T-G, Phys. Part. Nucl. Lett. 4, 281 (2007)

23. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 23 Interference DVCSBethe-Heitler

24. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 24 Charge Asymmetry RC (LSF) RC(1st order)

25. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 25 HELICITY Asymmetry

26. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 26 Conclusions High precision experiments need highly precise Radiative Corrections Higher order corrections become more and more important at large Q 2 The lepton structure function method can be applied to different electromagnetic processes with permille precision Higher order corrections depend on the relevant kinematical variables

27. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 27 Radiative Corrections (first order) The cross section: The correction ( in powers of Z): Z0: electron emission and vacuum polarization Z1: interference 1  -2  exchange  Z2: target emission L.C. Maximon and J.A Tjon, PRC 62, 054320(2000)

28. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 28 LSF Corrections (High orders included) The cross section: The correction ( Leading Logarithm Approximation): The vacuum polarization: The K-factor:

29. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 29 Radiative Corrections Yu. Bystricky, E.A.Kuraev, E. T-G, Phys. Rev. C75, 015207 (2007) Q 2 =1 GeV 2 L.C. Maximon and J.A Tjon, PRC 62, 054320(2000)

30. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 30 LSF electron (not LLA) MT (Z0) electron MT total LSF LLA MT (Z) two photon LSF protonMT (Z2) proton LSF total Radiative Corrections

31. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 31 LSF total LSF LLA MT (Z0) total MT (Z0) electron MT (Z0) two photon LSF proton MT proton LSF electron (not LLA)

32. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 32

33. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 33 LSF correction (SLAC data) point by point Q 2 =5 GeV 2

34. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 34 The Pauli and Dirac Form Factors Normalization F 1p (0)=1, F 2p (0)= κ p G Ep (0)=1, G Mp (0)=μ p =2.79 The electromagnetic current in terms of the Pauli and Dirac FFs: Related to the Sachs FFs :

35. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 35 Analytical properties of Compton amplitude Elastic form factors and inelastic channels are not independent  SUM RULES

36. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 36 Analytical properties of Compton amplitude Neglect left contribution and close contour on the right side (10% accuracy): Cancellation of strong interaction effects in FFs and inelastic channels! Cancellation proved exactly in QED: the L 2  * - contribution to FFs is cancelled by soft photon emission

37. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 37  resonance (example of inelastic channels) Small contribution ~0.5% Opposite sign with respect to proton intermediate state Cancellation of contributions in elastic and inelastic channels

38. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 38 Interference of 1   2  exchange –Enhancement due to the fast decreasing of form factors (transferred momentum equally shared between the two photons). –Dipole approximation of FFs: Q 0 2 =0.71GeV 2

39. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 39 Interference of 1   2  exchange Explicit calculation for structureless proton –The contribution is small, for unpolarized and polarized ep scattering –Does not contain the enhancement factor L –The relevant contribution to K is ~ 1 E.A.Kuraev, V. Bytev, Yu. Bystricky, E.T-G, Phys. Rev. D74, 013003 (1076)

40. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 40 Two Photon Exchange No exact calculation for ep scattering ( inelastic intermediate states..) but electron-muon scattering constitutes an upper limit!

41. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 41 QED versus QCD Imaginary part of the 2  amplitude electron proton

42. CEA DSM Dapnia Egle TOMASI-GUSTAFSSONAugust 5, 2008 42 QED versus QCD Q 2 =0.05 GeV 2 Q 2 =1.2 GeV 2 Q 2 =2 GeV 2