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 useful tool for interpreting experimental data in dark sector and dark matter searches and represents a contribution to elucidate the possible role of resonances in GeV-scale phenomenology.