Towards a more complete description of nucleon distortion in lepton-induced single-pion production at low-$Q^2$

Theoretical predictions for lepton-induced single-pion production (SPP) on ${}_{12}C$ are revisited in order to assess the effect of different treatments of the current operator. On one hand we have the asymptotic approximation, which consists in replacing the particle four-vectors that enter in the operator by their asymptotic values, i.e., their values out of the nucleus. On the other hand we have the full calculation, which is a more accurate approach to the problem. We also compare with results in which the final nucleon is described by a relativistic plane wave, to rate the effect of the nucleon distortion. The study is performed for several lepton kinematics, reproducing the SPP contribution to the inclusive and semi-inclusive cross sections belonging to the low-$Q^2$ region (between 0.05 and 1 ${\mathrm{GeV}}^2$), which is of special interest in charged-current (CC) neutrino-nucleus $1\pi$ production. The results of the SPP contribution to the inclusive electron cross section are compared with experimental data. We find nontrivial corrections comparable in size with the effect of the nucleon distortion, namely, corrections up to 6%, either increasing or diminishing the asymptotic prediction, and a shift of the distributions towards higher energy transfer. For the SPP contribution to the semi-inclusive cross section, we observe the correction to be prominent mainly at low values of the outgoing nucleon kinetic energy. Finally, for CC neutrino-induced $1{\pi }^{+}$ production, we find a reduction at low $Q^2$ with respect to both the plane-wave approach and the asymptotic case.