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Two-photon physics in elastic electron-nucleon scattering. Marc Vanderhaeghen College of William & Mary / JLab. JLab, May 12 th 2004. Outline. Introduction : Rosenbluth vs polarization measurements of G E and G M of nucleon puzzle : different results extracted for G E /G M

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two photon physics in elastic electron nucleon scattering
Two-photon physics in elastic electron-nucleonscattering


College of William & Mary / JLab

JLab, May 12th 2004

  • 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 PRL91, 142303 (2003)

Y.C. Chen, A.Afanasev, S. Brodsky, C. Carlson, M.Vdh hep-ph/0403058

M. Gorchtein, P.A.M. Guichon, M. Vdh hep-ph/0404206

B. Pasquini, M. Vdh in progress



One-photonexchange elastic electron-nucleon crosssection

Method : at fixed Q2, vary angle q (or equivalently e)

and plot reduced cross section

versus e


Polarization transfer method

Method : measure ratio of sideways ( ) to

longitudinal ( ) recoil polarization of proton

(absolute normalization drops out !)

in one-photon exchange approximation :

rosenbluth vs polarization transfer measurements of g e g m of proton
Rosenbluth vs polarization transfer measurements of GE/GM of proton


Rosenbluth data

Jlab/Hall A Polarization data

Jonesetal. (2000)

Gayouetal. (2002)

Twomethods, twodifferentresults !


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


Radiative correction diagrams


vertex corrections

2 photon exchange box diagrams


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


Status of radiative corrections


  • Tsai (1961), Mo & Tsai (1968)

box diagram calculated usingonly nucleon intermediate stateandusing q1¼ 0 or q2¼ 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

elastic en scattering beyond one photon exchange approximation
Elastic eN scattering beyond one-photon exchange approximation

Kinematical invariants :

(me = 0)

equivalently, introduce


Observables including two-photon exchange

Real parts of two-photon amplitudes


Phenomenological analysis

2-photon exchange is a candidate to explain the discrepancy between both experimental methods

Guichon, Vdh (2003)

partonic calculation of two photon exchange contribution
Partonic calculation of two-photon exchange contribution


“cat’s ears”

  • maincontributionatlargeQ2:

handbagdiagrams (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)


Calculation of hard scattering amplitude

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


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 :


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 :






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


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)


Hard part to invariant amplitudes

for elastic eN scattering

GPD integrals

“magnetic” GPD

“electric” GPD

“axial” GPD


Observables including two-photon exchange

in terms of A, B, C (real parts)


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


Leader, Sidorov, Stamenov (2002)


Test of GPDs models : form factors

gaussian valence model

non-linear Regge model


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 dataGayou et al. (2002)


Crosssection :


1g + 2g (hard+soft)

1 g

1g + 2g (hard)


Crosssection :


1 g

1g + 2g (non-linear Regge model )

1g + 2g(gaussian valence model )

e e ratio
e+ / e- Ratio

Direct test of real part of 2g amplitude

data figure from Arrington (2003)


Polarization transfer observables

1g + 2g


2 g correction

on is small

2 g correction on

can be tested at small e !

ssa in elastic en scattering
SSA in elastic eN scattering

spin of beam OR target NORMAL to scattering plane

on-shell intermediate state (MX = W)




Beam normal spin asymmetry


MAMI data (prelim.)


Target normal spin asymmetry

general formula, of order e2

involves the imaginary part of two-photon exchange amplitudes


Target normal spin asymmetry : partonic calculation

two-photon electron-quark amplitude

“magnetic” GPD

“electric” GPD


Target normal spin asymmetry : PROTON results


GM term

GPD prediction


GE term

JLab proposals : could reach 0.1% precision


Target normal spin asymmetry : NEUTRON results

GE term



GPD prediction

GM term

sizeable asymmetry on neutron :

no cancellation effect as on proton


Elastic electron-nucleon amplitudes

with electron helicity flip

In Born approximation :


Elastic electron-quark amplitudes

with electron helicity flip

lepton mass

new amplitude


Beam normal spin asymmetry : partonic calculation

“magnetic” GPD

“electric” GPD

“magnetic” GPD

“electric” GPD


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

  • 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 ofgeneralized 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