Rescattering effects in nucleon-to-meson form factors and application to tau-lepton-induced proton decay

Nucleon decays put extremely stringent bounds on baryon-number-violating interactions. However, in case the corresponding operators involve only $\tau$ leptons, the direct two-body decays, e.g., $p\to\pi^0 \tau^+$, are kinematically not allowed and nucleon decay can only proceed via an off-shell $\tau$, leading to $p\to \pi^0\ell^+\nu_\ell\bar\nu_\tau$. To calculate such processes, the momentum dependence of the form factors for nucleon-to-meson transitions, which describe the hadronization of the underlying process at the quark level, is needed. In this work, we point out new isospin and Fierz relations among such proton-kaon matrix elements and calculate the momentum dependence of the nucleon-to-meson form factors from the universal final-state interactions in terms of pion-nucleon or kaon-nucleon scattering phase shifts. We use these results to derive novel limits on the Wilson coefficients of the baryon-number-violating dimension-$6$ operators involving a $\tau$ lepton, which were previously unconstrained.