  Understanding single pion production reactions on free nucleons is the first
step towards a correct description of these processes in nuclei, which are
important for signal and background contributions in current and near future
accelerator neutrino oscillation experiments. In this work, we reanalyze our
previous studies of neutrino-induced one-pion production on nucleons for
outgoing $\pi N$ invariant masses below 1.4 GeV. Our motivation is to get a
better description of the $\nu_\mu n\to\mu^- n\pi^+$ cross section, for which
current theoretical models give values significantly below data. This channel
is very sensitive to the crossed $\Delta (1232)$ contribution and thus to spin
1/2 components in the Rarita-Schwinger $\Delta$ propagator. We show how these
spin 1/2 components are nonpropagating and give rise to contact interactions.
In this context, we point out that the discrepancy with experiment might be
corrected by the addition of appropriate extra contact terms and argue that
this procedure will provide a natural solution to the $\nu_\mu n\to\mu^-
n\pi^+$ puzzle. To keep our model simple, in this work we propose to change the
strength of the spin 1/2 components in the $\Delta$ propagator and use the
$\nu_\mu n\to\mu^- n\pi^+$ data to constraint its value. With this
modification, we now find a good reproduction of the $\nu_\mu n\to\mu^- n\pi^+$
cross section without affecting the good results previously obtained for the
other channels. We also explore how this change in the $\Delta$ propagator
affects our predictions for pion photoproduction and find also a better
agreement with experiment than with the previous model.
