Dynamical model of weak pion production reactions

The dynamical model of pion electroproduction developed by Sato and Lee [Phys. Rev. C $63,$ 055201 (2001)] has been extended to investigate the weak pion production reactions. With the conserved vector current hypothesis, the weak vector currents are constructed from electromagnetic currents by isospin rotations. Guided by the effective chiral Lagrangian method and using the unitary transformation method developed previously, the weak axial vector currents for $\pi$ production are constructed with no adjustable parameters. In particular, the $N-\Delta$ transitions at $Q^2\to 0$ are calculated from the constituent quark model and their $Q^2$ dependence is assumed to be identical to that determined in the study of pion electroproduction. The main feature of our approach is to renormalize these bare $N-\Delta$ form factors with the dynamical pion cloud effects originating from the nonresonant $\pi$ production mechanisms. The predicted cross sections of neutrino-induced pion production reactions, $N({\nu }_{\mu },{\mu }^{-}\pi )N,$ are in good agreement with the existing data. We show that the renormalized (dressed) axial $N-\Delta$ form factor contains large dynamical pion cloud effects and these renormalization effects are crucial in getting agreement with the data. We conclude that the $N-\Delta$ transitions predicted by the constituent quark model are consistent with the existing neutrino-induced pion production data in the $\Delta$ region, contrary to the previous observations. This is consistent with our previous findings in the study of pion electroproduction reactions. However, more extensive and precise data of neutrino-induced pion production reactions are needed to further test our model and to pin down the $Q^2$ dependence of the axial vector $N-\Delta$ transition form factor.