Off-Mass-Shell Parity-Violating Pion-Nucleon Interaction

In the usual theoretical description of parity violation, off-mass-shell effects of the weak pion-nucleon interaction are neglected. In the Cabibbo theory, the weak parity-violating pion-nucleon vertex is then proportional to ${sin}^2\theta \approx 0.05$ as it arises only from strangeness-changing weak currents. If off-mass-shell effects are taken into account, then a vertex proportional to ${cos}^2\theta$ is possible. In this work, we derive an effective ${cos}^2\theta$ pion-nucleon interaction by means of the renormalizable $\sigma$ model. We obtain an effective dimensionless coupling estimate which is roughly given by $\mathrm{Gg}\overrightarrow{\tau }·\hat{\phi }\frac{\Delta M}{M}$ if the nucleon is timelike and half off its mass shell by $\Delta M$. Thus neutral as well as charged weak pion-nucleon couplings of sizable strengths (∼ ${10}^{-6\mathrm{}}$) are predicted by this model even though no neutral weak currents are invoked.