1. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 1 Model independent properties of two photon exchange Egle Tomasi-Gustafsson Saclay, France Frascati, January 20, 2006 Collaboration with M.P. Rekalo Presently with G.I. Gakh and E.A. Kuraev

2. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 2 PLAN Introduction –Generalities on form factors –Electric proton FF (space-like) Two-photon exchange –History –Model independent properties –Observables in time-like region –Signatures of two-photon exchange Search for evidence in the data Alternative explanation for the discrepancy of FFs ratio Perspectives

3. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 3 Hadron Electromagnetic Form factors –Characterize the internal structure of a particle (  point-like) –In a P- and T-invariant theory, the EM structure of a particle of spin S is defined by 2S+1 form factors. –Neutron and proton form factors are different. –Elastic form factors contain information on the hadron ground state. –Playground for theory and experiment.

4. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 4 Space-like and time-like regions FFs are analytical functions. In framework of one photon exchange, FFs are functions of the momentum transfer squared of the virtual photon, t=q 2 =-Q 2. Scattering e - + h => e - + he + + e - => h + h _ Annihilation Annihilation _ Form factors are real in the space-like region and complex in the time-like region. t<0t>0

5. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 5 Crossing Symmetry Scattering and annihilation channels: - Described by the same amplitude : - function of two kinematical variables, s and t p 2 → – p 1 k 2 → – k 2 - which scan different kinematical regions

6. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 6 The Rosenbluth separation (1950) Elastic ep cross section (1-γ exchange) point-like particle:  Mott Linearity of the reduced cross section!

7. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 7 The polarization induces a term in the cross section proportional to G E G M Polarized beam and target or polarized beam and recoil proton polarization The polarization method (1967)

8. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 8 THE RESULTS (>2000) Jlab E93-027, E99-007 Spokepersons: Ch. Perdrisat, V. Punjabi, M. Jones, E. Brash M. Jones et ql. Phys. Rev. Lett. 84,1398 (2000) O. Gayou et al. Phys. Rev. Lett. 88:092301 (2002) Linear deviation from dipole  G Ep  G Mp

9. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 9

10. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 10 Electric proton FF Different results with different experimental methods !! New mechanism: two-photon exchange?

11. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 11 Two-Photon exchange 1  -2  interference is of the order of  =e 2 /4  =1/137 (in usual calculations of radiative corrections, one photon is ‘hard’ and one is ‘soft’) In the 70’s it was shown [J. Gunion and L. Stodolsky, V. Franco, F.M. Lev, V.N. Boitsov, L. Kondratyuk and V.B. Kopeliovich, R. Blankenbecker and J. Gunion] that, at large momentum transfer, due to the sharp decrease of the FFs, if the momentum is shared between the two photons, the 2  contribution can become very large  The 2  amplitude is expected to be mostly imaginary. In this case, the 1  -2  interference is more important in time-like region, as the Born amplitude is complex.

12. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 12 Qualitative estimation of 2  exchange From quark counting rules: F d ~ t -5 and F N ~t -2 For t = 4 GeV 2, For d, 3 He, 4 He, 2  effect should appear at ~1 GeV 2, for protons ~ 10 GeV 2 q/2 For ed elastic scattering:

13. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 13 Two-Photon exchange In 1999 M.P. Rekalo, E. T.-G. and D. Prout found a model-independent parametrization of the 2  contribution and applied to ed-elastic scattering data. → Discrepancy between the results from Hall A [L.C. Alexa et al. Phys. Rev. Lett. 82, 1374 (1999)] and Hall C [D. Abbott et al. Phys. Rev. Lett. 82, 1379 (1999)]. M. P. Rekalo, E. T-G and D. Prout, Phys. Rev. C60, 042202 (1999)

14. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 14 1{g1{g 1  -2  interference { 22 11 { M. P. Rekalo, E. T.-G. and D. Prout Phys. Rev. C (1999)

15. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 15 1  -2  interference M. P. Rekalo, E. T-G and D. Prout, Phys. Rev. C60, 042202 (1999) C/A D/A

16. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 16 The 1  -2  interference destroys the linearity of the Rosenbluth plot!

17. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 17 Model independent properties of two photon exchange

18. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 18 Two (complex) EM FFs Functions of one variable (t) → 4 spin ½ fermions → 16 amplitudes in the general case.  P- and T-invariance of EM interaction,  helicity conservation, Model independent considerations for Three (complex) amplitudes Functions of two variables (s,t) Crossing symmetry, C-invariance, T-reversal connect: e ± + N  e ± + N, N+N  e + + e -, and e + + e -  N+N 1  exchange:

19. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 19 The Matrix Element for Assuming P-invariance, and lepton helicity conservation, the matrix element for 1  +2  exchange is: For 1  -exchange: M. L. Goldberger, Y. Nambu and R. Oehme, Ann. Phys 2, 226 (1957) M.P. Rekalo and E. Tomasi-Gustafsson, EPJA 22, 331 (2004) vectoraxial

20. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 20 Generalized Form Factors 2  terms By analogy with Sachs and Fermi-Dirac FFs: Decomposition of the amplitudes: complex functions of 2 variables Both F 1N and F 2N contain 1  +2  !

21. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 21 Unpolarized cross section 2  exchange induces three new terms, of the order of  A. Zichichi, S. M. Berman, N. Cabibbo, R. Gatto, Il Nuovo Cimento XXIV, 170 (1962) B. Bilenkii, C. Giunti, V. Wataghin, Z. Phys. C 59, 475 (1993).

22. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 22 Odd properties of the 2  amplitudes with respect to the transformation: Symmetry relations One can remove or single out the 2  contribution by doing the sum or the difference of the differential cross section at the angles connected by this transformation cos  = - cos  i.e.,    - 

23. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 23 Sum of the differential cross sections: Remove the 2  contribution Total cross section:

24. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 24 Angular asymmetry Single out the 2  contribution with In terms of amplitudes - only interference terms!

25. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 25 2  contribution 2  contribution The sum: is free from 2  contributions!

26. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 26 e ± Model independent considerations for e ± N scattering 2 real functions 3 complex functions 8 real functions determine the e ± e ± 6 complexe amplitudes for e ± N →e ± N M. P. Rekalo and E. T-G Eur. Phys. Jour. A Relations among the functions! Electron and positron scattering e±e± Ne±e± N

27. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 27 e ±In the same kinematical conditions Sum and Difference of e ± N scattering M. P. Rekalo and E. T-G Eur. Phys. Jour. A Electron and positron scattering e±e± Ne±e± N Model independent considerations which hold at O(  2 ) e+e+e+e+ e-e-e-e-

28. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 28 Single spin observables T-odd observable At 90° expected small (vanishes for 1  exchange) : At threshold (vanishes for 1  exchange due to G E =G M ) : TPE contribution: Small, of the order of  Relative role increases when q 2 increases Does not vanish, in the general case, for 1  exchange

29. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 29 Form Factor determination C-odd properties of nucleon polarization with  is the phase difference of the form factors G E and G M

30. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 30 Form Factor ratio R=|GE| / |GM| Form Factor ratio R=|GE| / |GM| The sum: is free from 2  contributions! The Ratio R can be determined by:

31. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 31 Is there any evidence of presence of a 2  contribution in the existing ep data? NON

32. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 32 From the data: deviation from linearity << 1%! Parametrization of 2  -contribution for e+p E. T.-G., G. Gakh Phys. Rev. C (2005)

33. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 33 Possible explanation for the FFs discrepancy: Radiative Corrections

34. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 34 Radiative corrections  Effects of the order of -few percent on polarization observables, -up to 40% on cross section!  Complete calculations in progress Mo and Tsai (1969) Schwinger (1949)

35. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 35 Structure function method Assumes dipole FFs Change the slope ! Q 2 =1 GeV 2 Q 2 =5 GeV 2 Q 2 =3 GeV 2 SF Born Polarization RC Born E.A Kuraev, V.S. Fadin Sov. J. Nucl. Phys. 41, 466 (1985)

36. CEA DSM Dapnia Egle Tomasi-Gustafsson Frascati, 20 Gennaio 2006 36 We have derived model independent formulas for all experimental observables in presence of 2  exchange, as functions of three complex amplitudes for e + + e -  N+N Using symmetry properties one can remove or single out 2  contributions Crossing symmetry, C-invariance, T-reversal connect: e ± + N  e ± + N, N+N  e + + e -, and e + + e -  N+N Conclusions New data welcome in next future! Revise Radiative Corrections! Novosibirsk- VEPP3