  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.
