1. Two-photon physics in elastic electron-nucleon scattering
Marc Vanderhaeghen
College of William & Mary / JLab
JLab, May 12th 2004

2. Outline
Introduction : Rosenbluth vs polarization measurements of GE and GM of nucleon
puzzle : different results extracted for GE/GM
Elastic electron-nucleon scattering beyond the one-photon exchange approximation
Partonic calculation of two-photon exchange contribution
generalized parton distributions of nucleon
Results for cross section and polarization transfer
SSA in elastic electron-nucleon scattering
P.A.M. Guichon, M.Vdh PRL 91, (2003)
Y.C. Chen, A.Afanasev, S. Brodsky, C. Carlson, M.Vdh hep-ph/
M. Gorchtein, P.A.M. Guichon, M. Vdh hep-ph/
B. Pasquini, M. Vdh in progress

3. Introduction Rosenbluth separation method
One-photon exchange elastic electron-nucleon cross section
Method : at fixed Q2, vary angle q (or equivalently e)
and plot reduced cross section
versus e

4. One-photon theorist’s view

5. Polarization transfer method
Method : measure ratio of sideways ( ) to
longitudinal ( ) recoil polarization of proton
(absolute normalization drops out !)
in one-photon exchange approximation :

6. Rosenbluth vs polarization transfer measurements of GE/GM of proton
SLAC
Rosenbluth data
Jlab/Hall A Polarization data
Jones et al. (2000)
Gayou et al. (2002)
Two methods, two different results !

7. Speculation : missing radiative corrections
Speculation : there are radiative corrections to Rosenbluth experiments that are important and are not included
missing correction : linear in e, not strongly Q2 dependent
Q2 = 6 GeV2
GE term is proportionally smaller at large Q2
effect more visible at large Q2
if both FF scale in same way

8. Radiative correction diagrams
bremsstrahlung
vertex corrections
2 photon exchange box diagrams

9. Comments on radiative corrections
Radiative corrections at electron side,
well understood and taken care of
Soft bremsstrahlung
involves long-wavelength photons
compositeness of nucleon only enters through
on-shell form factors
Box diagrams involve photons of all wavelengths
long wavelength (soft photon) part is included in radiative correction (IR divergence is cancelled with electron proton bremsstrahlung interference)
short wavelength contributions :
not done in “old” days

10. Status of radiative corrections
Tsai (1961), Mo & Tsai (1968)
box diagram calculated using only nucleon intermediate state and using q1 ¼ 0 or q2 ¼ 0 in both numerator and denominator (calculate 3-point function) -> gives correct IR divergent terms
Maximon & Tjon (2000)
same as above, but make the above approximation only in numerator (calculate 4-point function)
+ use on-shell nucleon form factors in loop integral
Blunden, Melnitchouk, Tjon (2003)
further improvement by keeping the full numerator

11. Elastic eN scattering beyond one-photon exchange approximation
Kinematical invariants :
(me = 0)
equivalently, introduce

12. Observables including two-photon exchange
Real parts of two-photon amplitudes

13. Phenomenological analysis
2-photon exchange is a candidate to explain the discrepancy between both experimental methods
Guichon, Vdh (2003)

14. Partonic calculation of two-photon exchange contribution
“handbag”
“cat’s ears”
main contribution at large Q2 :
handbag diagrams (one active quark)
to reproduce the IR divergent contribution at nucleon correctly (i.e. to satisfy the Low Energy Theorem)
need cat’s ears diagrams (two active quarks)

15. Calculation of hard scattering amplitude
electron helicity
quark helicity
Calculation for em -> em can be found in literature
(e.g. van Nieuwenhuizen (1971) ), which we verified explicitly
IR divergences of boxes must disappear or cancel in the end,
regularize through photon mass l

16. Separation soft-hard parts in electron-quark box
Follow the decomposition of Grammer and Yennie (1973) :
soft part calculated as 3-point function
reproduces Low Energy Theorem
kinematics partonic subprocess :

17. Calculation of soft part at nucleon level
LET : sum of soft contributions from the partonic calculation has to match the soft contributions at nucleonic level
To satisfy the LET, one has to include the
soft-photon contributions from the cats’ ears diagrams
Pictorially :
soft
soft
soft
soft

18. Calculation of bremsstrahlung
IR finite
soft part of electron-nucleon box
bremsstrahlung contribution : Maximon, Tjon (2000)
Experimentalists do corrections according to Mo-Tsai
relative to Mo-Tsai, the above formula gives
a correction factor (1 + p a) + terms of size 0.001

19. Convolution with GPDs result for handbag amplitude (large Q2 )
work in frame q+ = 0, nm is a Sudakov vector ( n2 = 0, n . P = 1 )
handbag amplitude depends on GPD(x, x = 0, Q2),
which also appear in other wide angle scattering processes (e.g. WACS)

20. Hard part to invariant amplitudes for elastic eN scattering
GPD integrals
“magnetic” GPD
“electric” GPD
“axial” GPD

21. Observables including two-photon exchange
in terms of A, B, C (real parts)

22. Model for GPDs at large Q2
use gaussian-valence model : Radyushkin (1998), Diehl et al. (1999)
s = 0.8 GeV2
Forward parton distributions at m2 = 1 GeV2
MRST2002 NNLO
Leader, Sidorov, Stamenov (2002)

23. Test of GPDs models : form factors
gaussian valence model
non-linear Regge model

24. Form factors used as input in calculation
magnetic proton form factor
Brash et al. (2002)
electric proton form factor :
GE / GM of proton fixed from polarization data Gayou et al. (2002)

25. Cross section :
with
1g + 2g (hard+soft)
1 g
1g + 2g (hard)

26. 1g + 2g (non-linear Regge model )
Cross section :
with
1 g
1g + 2g (non-linear Regge model )
1g + 2g (gaussian valence model )

27. e+ / e- Ratio Direct test of real part of 2g amplitude
data figure from Arrington (2003)

28. Polarization transfer observables
1g + 2g 1g
2 g correction
on is small
2 g correction on
can be tested at small e !

29. SSA in elastic eN scattering
spin of beam OR target NORMAL to scattering plane
on-shell intermediate state (MX = W)
lepton
hadron

30. Integrand : beam normal spin asymmetry
Ee = GeV
MAID

31. Beam normal spin asymmetry
(elastic)
MAMI data (prelim.)

32. Target normal spin asymmetry
general formula, of order e2
involves the imaginary part of two-photon exchange amplitudes

33. Target normal spin asymmetry : partonic calculation
two-photon electron-quark amplitude
“magnetic” GPD
“electric” GPD

34. Target normal spin asymmetry : PROTON results%
GM term
GPD prediction
elastic
GE term
JLab proposals : could reach 0.1% precision

35. Target normal spin asymmetry : NEUTRON results
GE term %
elastic
GPD prediction
GM term
sizeable asymmetry on neutron :
no cancellation effect as on proton

36. Elastic electron-nucleon amplitudes with electron helicity flip
In Born approximation :

37. Elastic electron-quark amplitudes with electron helicity flip
lepton mass
new amplitude

38. Beam normal spin asymmetry : partonic calculation
“magnetic” GPD
“electric” GPD
“magnetic” GPD
“electric” GPD

39. Beam normal spin asymmetry : results
Results of GPD calculation
Note : elastic contribution to Bn is negligibly small
Present PV experimental set-ups (0.1 ppm precision) : opportunity to measure this asymmetry

40. Conclusions
Developed the formalism to describe elastic electron-nucleon scattering beyond the one-photon exchange approximation
Performed a partonic calculation of two-photon exchange contribution in terms of generalized parton distributions of nucleon (handbag calculation)
Able to resolve existing discrepancy between Rosenbluth and polarization transfer observables quantitatively
SSA in elastic electron-nucleon scattering :
promising observables to access doubly (spacelike) virtual Compton scattering