Two-photon-exchange effects in the electro-excitation of the $\mathrm{\Delta }$ resonance

We evaluate the two-photon exchange contribution to the $e N \to e \Delta(1232) \to e \pi N$ process at large momentum transfer with the aim of a precision study of the ratios of electric quadrupole (E2) and Coulomb quadrupole (C2) to the magnetic dipole (M1) ${\gamma }^{\ast }N\mathrm{\Delta }$ transitions. We relate the two-photon exchange amplitude to the $N\to \mathrm{\Delta }$ generalized parton distributions and obtain a quantitative estimate of the two-photon effects. The two-photon exchange corrections to the C2/M1 ratio depend strongly on whether this quantity is obtained from an interference cross section or from the Rosenbluth-type cross sections, in similarity with the elastic, $e N \to e N$, process.