Two-photon exchange contribution to elastic electron-nucleonscattering at large momentum transfer

We estimate the two-photon exchange contribution to elastic electron-proton scattering at large momentum transfer by using a quark-parton representation of virtual Compton scattering. We thus can relate the two-photon exchange amplitude to the generalized parton distributions which also enter in other wide-angle scattering processes. We find that the interference of one- and two-photon exchange contribution is able to substantially resolve the difference between electric form factor measurements from Rosenbluth and polarization transfer experiments. Two-photon exchange has additional consequences which could be experimentally observed, including nonzero polarization effects and a positron-proton/electron-proton scattering asymmetry. The predicted Rosenbluth plot is no longer precisely linear; it acquires a measurable curvature, particularly at large laboratory angle.

Figure 1

Handbag approximation for the elastic lepton-nucleon scattering at large momentum transfers. In the partonic scattering process (indicated by $H$), the lepton scatters from quarks in the nucleon, with momenta $p_q$ and $p_q^{\prime }$. The lower blob represents the GPD’s of the nucleon.

Figure 2

Direct and crossed box diagrams to describe the two-photon exchange contribution to the lepton-quark scattering process, corresponding with the blob denoted by $H$ in Fig. 1.

Figure 3

Rosenbluth plots for elastic $ep$ scattering: ${\sigma }_R$ divided by $({\mu }_p{G}_D{)}^2$, with $G_D=(1+Q^2/0.71{)}^{-2}$. Dotted curves: Born approximation using $G_{Ep}/G_{Mp}$ from polarization data [1, 2]. Solid curves: full calculation using the modified Regge GPD, for the kinematical range $-u>M^2$. Dashed curves: same as solid curves but using the Gaussian GPD. The data are from Ref. 6.

Figure 4

A Rosenbluth plot for elastic $en$ scattering. The curves with the two-photon corrections are plotted only for $-u>M^2$. For $G_M^n$ and $G_E^n$ we used fits from 44, 45, respectively. There is little slope change for the neutron case, for reasons noted in the text.

Figure 5

Nucleon analyzing power, which is equal to normal recoil polarization. The elastic contribution (nucleon intermediate state in the two-photon exchange box diagram) is shown by the dotted curve [27]. The GPD calculation for the inelastic contribution is shown by the dashed curve for the Gaussian GPD, and by the solid curve for the modified Regge GPD. The GPD calculation is cut off in the backward direction at $-u=M^2$. In the forward direction the modified Regge GPD result goes down to $Q^2=2{\mathrm{GeV}}^2$ and the Gaussian GPD result to $Q^2=M^2$.

Figure 6

Recoil proton polarization components $P_s$ and $P_l$ and their ratios relative to the $1\gamma$ exchange results (lower panels) for elastic $ep$ scattering at $Q^2=5{\mathrm{GeV}}^2$. The dotted curves in the upper panels are the Born approximation ($1\gamma$ exchange) results. The solid curves include the $2\gamma$ exchange correction using the GPD calculation, for the kinematical range where both $s,-u>M^2$.

Figure 7

Recoil neutron polarization components $P_s$ and $P_l$ and their ratios relative to the $1\gamma$ exchange results (lower panels) for elastic $ep$ scattering at $Q^2=4.3{\mathrm{GeV}}^2$. See Fig. 6 for notation.

Figure 8

Ratio of $e^{+}/e^{-}$ elastic cross sections on the proton. The GPD calculations for the $2\gamma$ exchange correction are for three fixed $Q^2$ values of 2, 5, and $9{\mathrm{GeV}}^2$, for the kinematical range where $-u$ is above $M^2$. Also shown are all known data, from 48, with $Q^2$ above $1.5{\mathrm{GeV}}^2$ (the missing central value is at 1.111). The numbers near the data give $Q^2$ for that point in ${\mathrm{GeV}}^2$.

Figure 9

Rosenbluth determinations of $G_E/G_M$ including the two-photon corrections. The polarization data is from Jones et al. [1] and Gayou et al. [2], and the Rosenbluth determinations without the two-photon corrections are from 51. The Rosenbluth $G_E/G_M$ are based on data from Andivahis et al. [6]. Some of our points for the Rosenbluth results are slightly offset horizontally for clarity.

Figure 10

Quark mass correction plots for the analyzing power or normal polarization, for fixed electron incoming lab energies of 3 and 6 GeV.

Figure 11

Quark mass correction plot for the reduced cross section. The curves are as labeled. In this plot, all curves have $G_M$ set to 0.99 times the Brash et al. value [43]. The data is from 6.