Effective field theory approach to lepton number violating $\tau$ decays

We continue our endeavor to investigate lepton number violating (LNV) processes at low energy in the framework of effective field theory (EFT). In this work we study the LNV tau decays $\tau^+\rightarrow \ell^-P_i^{+}P_j^{+}$, where $\ell=e,~\mu$ and $P^+_{i,j}$ are the lowest-lying charged pseudoscalars $\pi^+,~K^+$. We analyze the dominant contributions in a series of EFTs from high to low energy scales, namely, the standard model effective field theory (SMEFT), the low-energy effective field theory (LEFT), and the chiral perturbation theory ($\chi$PT). The decay branching ratios are expressed in terms of the Wilson coefficients of dimension-five and -seven operators in SMEFT and hadronic low energy constants. These Wilson coefficients involve the first and second generations of quarks and all generations of leptons and thus cannot be explored in low energy processes such as nuclear neutrinoless double decay or LNV kaon decays. Unfortunately, the current experimental upper limits on the branching ratios are too weak to set useful constraints on those coefficients. Or, if we assume the new physics scale is larger than 1 TeV, the branching ratios are well below the current experimental bounds. We also estimate hadronic uncertainties incurred in applying $\chi$PT to $\tau$ decays by computing one-loop chiral logarithms and attempt to improve convergence of chiral perturbation by employing dispersion relations in the short-distance part of the decay amplitudes.