Effective Majorana mass matrix from tau and pseudoscalar meson lepton number violating decays

An observation of any lepton number violating process will undoubtedly point towards the existence of new physics and indirectly to the clear Majorana nature of the exchanged fermion. In this work, we explore the potential of a minimal extension of the Standard Model via heavy sterile fermions with masses in the $[ 0.1 - 10]$ GeV range concerning an extensive array of "neutrinoless" meson and tau decay processes. We assume that the Majorana neutrinos are produced on-shell, and focus on three-body decays. We conduct an update on the bounds on the active-sterile mixing elements, $|U_{\ell_\alpha 4} U_{\ell_\beta 4}|$, taking into account the most recent experimental bounds (and constraints) and new theoretical inputs, as well as the effects of a finite detector, imposing that the heavy neutrino decay within the detector. This allows to establish up-to-date comprehensive constraints on the sterile fermion parameter space. Our results suggest that the branching fractions of several decays are close to current sensitivities (likely within reach of future facilities), some being already in conflict with current data (as is the case of $K^+ \to \ell_\alpha^+ \ell_\beta^+ \pi^-$, and $\tau^- \to \mu^+ \pi^- \pi^-$). We use these processes to extract constraints on all entries of an enlarged definition of a $3\times 3$ "effective" Majorana neutrino mass matrix $m_{\nu}^{\alpha \beta}$.