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Nucleon decays put extremely stringent bounds on baryon-number-violatinginteractions. However, in case the corresponding operators involve only $\tau$leptons, the direct two-body decays, e.g., $p\to\pi^0 \tau^+$, arekinematically 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 suchprocesses, the momentum dependence of the form factors for nucleon-to-mesontransitions, which describe the hadronization of the underlying process at thequark level, is needed. In this work, we point out new isospin and Fierzrelations among such proton-kaon matrix elements and calculate the momentumdependence of the nucleon-to-meson form factors from the universal final-stateinteractions in terms of pion-nucleon or kaon-nucleon scattering phase shifts.We use these results to derive novel limits on the Wilson coefficients of thebaryon-number-violating dimension-$6$ operators involving a $\tau$ lepton,which were previously unconstrained.

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