Beam normal spin asymmetry for the $ep\to e\mathrm{\Delta }(1232)$ process

We calculate the single spin asymmetry for the $ep\to e\mathrm{\Delta }(1232)$ process, for an electron beam polarized normal to the scattering plane. Such single spin asymmetries vanish in the one-photon exchange approximation, and are directly proportional to the absorptive part of a two-photon exchange amplitude. As the intermediate state in such two-photon exchange process is on its mass shell, the asymmetry allows one to access for the first time the on-shell $\mathrm{\Delta }\to \mathrm{\Delta }$ as well as $N^{\ast }\to \mathrm{\Delta }$ electromagnetic transitions. We present the general formalism to describe the $e p \to e \Delta$ beam normal spin asymmetry, and provide a numerical estimate of its value using the nucleon, $\mathrm{\Delta }(1232)$, $S_{11}(1535)$, and $D_{13}(1520)$ intermediate states. We compare our results with the first data from the Qweak@JLab experiment and give predictions for the A4@MAMI experiment.