Single pion resonant production in BSM scenarios: cross sections and amplitudes

We present a comprehensive theoretical framework describing single pion resonant production through inelastic dark fermion-nucleon interactions mediated by resonances in the GeV-scale regime. Building upon the Rein-Sehgal approach, we derive differential cross sections for processes in which an incoming dark fermion scatters off a nucleon, exciting a resonance that subsequently decays into a nucleon and a pion. Our formulation accommodates various mediator types-namely, dark photons with vector and axial couplings, as well as scalar and pseudoscalar mediators-thereby extending the conventional approach that Rein, Sehgal and Berger performed for neutrino interactions. Transition amplitudes for the nucleon-to-resonance conversion are computed using the relativistic harmonic-oscillator quark model from Feynman, Kislinger and Ravndal, while a Breit-Wigner prescription is employed to incorporate finite resonance widths. This framework offers a robust tool for interpreting experimental data in dark sector and matter searches and represents a contribution to elucidate the role of resonances in GeV-scale phenomenology.