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One pion production in neutrino-nucleon scattering and the different parametrizations of the weak $N\rightarrowΔ$ vertex

The $N \to Δ$ weak vertex provides an important contribution to the one pion production in neutrino-nucleon and neutrino-nucleus scattering for $πN$ invariant masses below 1.4 GeV. Beyond its interest as a tool in neutrino detection and their background analyses, one pion production in neutrino-nucleon scattering is useful to test predictions based on the quark model and other internal symmetries of strong interactions. Here we try to establish a connection between two commonly used parametrizations of the weak $N \to Δ$ vertex and form factors (FF) and we study their effects on the determination of the axial coupling $C_5^A(0)$, the common normalization of the axial FF, which is predicted to hold 1.2 by using the PCAC hypothesis. Predictions for the $ν_μ p \to μ^- pπ^+$ total cross sections within the two approaches, which include the resonant $Δ^{++}$ and other background contributions in a coherent way, are compared to experimental data.