Pion Production in Neutrino-Nucleon Reactions

We construct a model for the weak pion production off the nucleon, which in addition to the weak excitation of the $\mathrm{\Delta }(1232)$ resonance and its subsequent decay into $N\pi$, it includes also some background terms required by chiral symmetry. We re-fit the $C_5^A(q^2)$ form factor to the flux averaged ${\nu }_{\mu }p\to {\mu }^{-}p{\pi }^{+}$ ANL $q^2-$differential cross section data, finding a substantially smaller contribution of the $\mathrm{\Delta }$ pole mechanism than traditionally assumed in the literature. We also show that the interference between the Delta pole and the background terms produces parity-violating contributions to the pion angular differential cross section.