  The $N \to \Delta$ weak vertex provides an important contribution to the one
pion production in neutrino-nucleon and neutrino-nucleus scattering for $\pi N$
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
$N \to \Delta$ 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 $\nu_{\mu} p \to \mu^- p\pi^+$ total cross sections within
the two approaches, which include the resonant $\Delta^{++}$ and other
background contributions in a coherent way, are compared to experimental data.
