Meson exchange effects in elastic $ep$ scattering at loop level and the electromagnetic form factors of the proton

A new form of two-photon exchange (TPE) effect is studied to explain the discrepancy between unpolarized and polarized experimental data in elastic $ep$ scattering. The mechanism is based on a simple idea that apart from the usual TPE effects from box and crossed-box diagrams, the mesons may also be exchanged in elastic $ep$ scattering by two-photon coupling at loop level. The detailed study shows such contributions to reduced unpolarized cross section (${\sigma }_{\mathrm{un}}$) and polarized observables ($P_t,P_l$) at fixed $Q^2$ are only dependent on proton's electromagnetic form factors $G_{E,M}$ and a new unknown universal parameter $g$. After combining this contribution with the usual TPE contributions from box and crossed-box diagrams, the ratio ${\mu }_p{G}_E/G_M$ extracted from the recent precise unpolarized and polarized experimental data can be described consistently.

Figure 1

(a) The Born diagram in elastic $ep$ scattering. (b,c,d) The $s$, $u$, $t$ channels in photoproduction of vector meson, the similar diagrams in VCS are not shown.

Figure 2

TPE contributions in $ep$ scattering. (a) box diagram; (b) crossed-box diagram; (c) meson-exchange diagram by two-photon coupling; (d) effective direct meson-exchange diagram.

Figure 3

Extracted $R$ by the LT and PT methods. $R_{\mathrm{LT},\mathrm{Ex}}^{1\gamma }$ refers to the extracted $R$ by Eq. (1) from the experimental data without any TPE corrections, $R_{\mathrm{LT},\mathrm{Ex}}^{1\gamma +2\gamma (N+\Delta )}$ and $R_{\mathrm{LT},\mathrm{Ex}}^{1\gamma +2\gamma (N+\Delta +M)}$ refer to the extracted $R$ by Eqs. (1) and (6) after applying the usual TPE corrections from Figs. 2 and 2 to the experimental data, respectively. The unpolarized experimental data are taken from [5] and $R_{\mathrm{PT},\mathrm{Ex}}^{1\gamma }$ are taken from [2]. The error bar of experimental data is taken as the weight in the fitting.

Figure 4

Theoretical estimations of TPE corrections to $R_{\mathrm{PT}}$. $\Delta R_{\mathrm{PT},\mathrm{th}}^{N,\Delta ,M,N+\Delta +M}$ refer to the corresponding theoretical estimations of TPE contributions from $N,\Delta$ intermediate states, meson-exchange, and their sum, respectively. The experimental results are taken from [3] and normalized at $\epsilon =0.785$.

Figure 5

Theoretical estimations of TPE corrections to $P_l$. $\Delta P_{l,\mathrm{th}}^{N,\Delta ,M,N+\Delta +M}$ refer to the theoretical estimations of TPE corrections from $N,\Delta$ intermediate states, meson-exchange, and their sum, respectively. The experimental results are taken from [3] and normalized at $\epsilon =0.152$.

Figure 6

The theoretical estimation of ratio $R_{e^{+}/e^{-}}$ at $Q^2=2.64,3.2,4.1{\mathrm{GeV}}^2$ after considering the full TPE corrections from $N,\Delta$ intermediate states and meson-exchange, the experimental data is taken from [18].