Two-photon exchange in elastic electron-nucleon scattering

A detailed study of two-photon exchange in unpolarized and polarized elastic electron-nucleon scattering is presented, taking particular account of nucleon finite size effects. Contributions from nucleon elastic intermediate states are found to have a strong angular dependence, which leads to a partial resolution of the discrepancy between the Rosenbluth and polarization transfer measurements of the proton electric to magnetic form factor ratio, $G_E/G_M$. The two-photon exchange contribution to the longitudinal polarization transfer $P_L$ is small, whereas the contribution to the transverse polarization transfer $P_T$ is enhanced at backward angles by several percent, increasing with $Q^2$. This gives rise to a small, $\lesssim 3\%$ suppression of $G_E/G_M$ obtained from the polarization transfer ratio $P_T/P_L$ at large $Q^2$. We also compare the two-photon exchange effects with data on the ratio of $e^{+}p$ to $e^{-}p$ cross sections, which is predicted to be enhanced at backward angles. Finally, we evaluate the corrections to the form factors of the neutron and estimate the elastic intermediate state contribution to the ${}^3\mathrm{He}$ form factors.

Figure 1

Two-photon exchange box and crossed box diagrams for elastic electron-proton scattering.

Figure 2

Difference between the full two-photon exchange correction to the elastic cross section [using the realistic form factors in Eq. (26)] and the commonly used expression (23) from Ref. 13 for $Q^2=1$–6 GeV${}^2$. The numbers labeling the curves denote the respective $Q^2$ values in ${\mathrm{GeV}}^2$.

Figure 3

Model dependence of the difference between the full two-photon exchange correction and the Mo and Tsai approximation: (a) at $Q^2=1$, 6, and 12 GeV${}^2$, using realistic (solid) [16] and dipole (dashed) form factors; (b) at $Q^2=6$ GeV${}^2$ using the form factor parametrizations from Refs. 16 (solid), [26] (dashed), and [25] with $G_E^p$ constrained by the LT-separated (dot-dashed) and polarization transfer (long-dashed) data.

Figure 4

(Color online) Reduced cross section ${\sigma }_R$ (scaled by the dipole form factor $G_D^2$) versus $\varepsilon$ for several values of $Q^2$: (a) SLAC data [27] at $Q^2=3.25$ (open squares), 4 (filled circles), 5 (open circles), and 6 GeV${}^2$ (filled squares); (b) JLab data [4] at $Q^2=2.64$ (filled squares), 3.2 (open squares), and 4.1 GeV${}^2$ (filled circles). The dotted curves are Born cross sections evaluated using a form factor parametrization [26] with $G_E^p$ fitted to the polarization transfer data [5], whereas the solid curves include 2γ contributions. The curves in the bottom panel have been shifted by ($+1.0\%$, $+2.1\%$, $+3.0\%$) for $Q^2=(2.64,3.2,4.1)$ GeV${}^2$.

Figure 5

(Color online) The ratio of proton form factors ${\mu }_p{G}_E/G_M$ measured using LT separation (open diamonds) [2] and polarization transfer (PT) (open circles) [5]. The LT points corrected for 2γ exchange are shown assuming a linear slope for $\varepsilon =0.2$–0.9 (filled squares) and $\varepsilon =0.5$–0.8 (filled circles) (offset for clarity).

Figure 6

(Color online) Finite 2γ contributions (defined with respect to the Mo-Tsai IR result [13]) to the real parts of the $G_M$ (dashed), $G_E$ (dot-dashed), and $Y_{2\gamma }$ (solid) form factors of the proton at $Q^2=1$, 3, and 6 GeV${}^2$. Note the larger scale in the bottom figure.

Figure 7

Ratio of elastic $e^{+}p$ to $e^{-}p$ cross sections. The data are from SLAC [31, 32], with $Q^2$ ranging from 0.01 to 5 GeV${}^2$. The results of the 2γ exchange calculations are shown by the curves for $Q^2=1$ (dotted), 3 (dashed), and 6 GeV${}^2$ (solid).

Figure 8

Ratio of the finite part [with respect to the IR contribution in Eq. (22)] of the Born+2γ correction relative to the Born term, for (a) longitudinal and (b) transverse recoil proton polarization, at $Q^2=1$ (dotted), 3 (dashed), and 6 GeV${}^2$ (solid). Note the different scales on the vertical axes.

Figure 9

(Color online) Proton electric to magnetic form factor ratio obtained from the polarization transfer measurements [5], with (solid circles) and without (open circles) the 2γ exchange corrections. The corrected values have been offset for clarity. The LT-separated ratio (open diamonds) from Fig. 5 is shown for comparison.

Figure 10

Ratio of the 2γ contribution to the normal polarization, to the unpolarized Born contribution, as a function of $\varepsilon$, for $Q^2=1$ (dotted), 3 (dashed), and 6 GeV${}^2$ (solid).

Figure 11

Normal polarization asymmetry, expressed as a percentage, as a function of the center-of-mass scattering angle, ${\Theta }_{\mathrm{cm}}$, for $Q^2=1$ (dotted), 3 (dashed), and 6 GeV${}^2$ (solid).

Figure 12

(Color online) 2γ contribution to the unpolarized electron-neutron elastic scattering cross section, at $Q^2=1$ (dotted), 3 (dashed), and 6 GeV${}^2$ (solid and dot-dashed). The dot-dashed curve corresponds to the form factor parametrization of Ref. 41, whereas the others are from Ref. 16 (as fitted by the parameters in Table ).

Figure 13

(Color online) Effect of 2γ exchange on the ratio of neutron form factors ${\mu }_n{G}_E^n/G_M^n$ using LT separation. The uncorrected points (open circles) are from the form factor parametrization in Ref. 16, whereas the points corrected for 2γ exchange are obtained from linear fits to ${\delta }^{\mathrm{full}}$ in Fig. 12 for $\varepsilon =0.2-0.9$ (filled squares) and $\varepsilon =0.5-0.8$ (filled circles) (offset for clarity).

Figure 14

(Color online) Effect of 2γ exchange on the ratio of neutron form factors ${\mu }_n{G}_E^n/G_M^n$ using polarization transfer. The uncorrected points (open circles) are from the parametrization in Ref. 16, and the points corrected for 2γ exchange correspond to $\varepsilon =0.3$ (filled squares) and $\varepsilon =0.8$ (filled circles) (offset for clarity).

Figure 15

2γ contribution to the unpolarized electron-${}^3\mathrm{He}$ elastic scattering cross section, with the ${}^3\mathrm{He}$ elastic intermediate state, as a function of $\varepsilon$, for $Q^2=0.05$ (solid), 0.2 (dot-dashed), 0.5 (dashed), and 1 GeV${}^2$ (dotted). A parametrization of the form factor from Ref. 44 is used in all cases, except for the upper solid and dot-dashed curves, for which a dipole shape with mass ${\Lambda }_{{}^3\mathrm{He}}=0.37$ GeV is used.