Two-photon exchange corrections to ${\gamma }^{*}N\mathrm{\Delta }$ form factors for $Q^2\le 4{(\mathrm{GeV}/c)}^2$

We evaluate the corrections of the two-photon exchange (TPE) process on the ${\gamma }^{*}N\mathrm{\Delta }$ transition form factors. The contributions of the TPE process to $eN\to e\mathrm{\Delta }\left(1232\right)\to eN\pi$ are calculated in a hadronic model with the inclusion of only the elastic nucleon intermediate states, to estimate its effects on the multipoles $M_{1^{+}}^{(3/2)},E_{1^{+}}^{(3/2)},S_{1^{+}}^{(3/2)}$ at the $\mathrm{\Delta }$ peak. We find that TPE effects on $G_M^{*}$ are very small. $G_E^{*},$ and $G_C^{*}$ are also little affected at small $Q^2$. For $G_E^{*}$, the TPE effects reach about $3–8\%$ near $Q^2\sim 4{\mathrm{GeV}}^2$, depending on the model, MAID or SAID, used to emulate the data. For $G_C^{*}$, the TPE effects decrease rapidly with increasing $\epsilon$ while growing with increasing $Q^2$ to reach $\sim 6–15\%$ with $Q^2\sim 4{\mathrm{GeV}}^2$ at $\epsilon =0.2$. Sizeable TPE corrections to $G_E^{*}$ and $G_C^{*}$ found here point to the need to include TPE effects in the multipole analysis in the region of high $Q^2$ and small $\epsilon$. The TPE corrections to $R_{EM}$ and $R_{SM}$ obtained in our hadronic calculation are compared with those obtained in a partonic calculation for moderate momentum transfer of $2<Q^2<4{\mathrm{GeV}}^2$.

Figure 1

TPE box (a) and contact (b) diagrams for $eN\to e\mathrm{\Delta }\to eN\pi$. The cross-box diagram is not shown.

Figure 2

The TPE corrections to $E_{1^{+}}^{(3/2)}$ and $S_{1^{+}}^{(3/2)}$ vs $\epsilon$ at fixed $Q^2$. The labels MAID and SAID are used to indicate that the results are obtained with using either MAID or SAID to emulate the experimental cross sections, respectively, as explained in the text. The solid and dotted curves (red) refer to the results with MAID, and the dashed and dashed-dotted curves (blue) denote the results with SAID.

Figure 3

The TPE corrections to $E_{1^{+}}^{(3/2)}$ and $S_{1^{+}}^{(3/2)}$ vs $Q^2$ at fixed $\epsilon$. The notation is the same as in Fig. 2.

Figure 4

The TPE corrections to the extracted $R_{EM}$ and $R_{SM}$ vs $Q^2$ at fixed $\epsilon$. The notation for curves is the same as in Fig. 2. The black triangles denote the results of the partonic calculation of [12].