Lepton-pair production in dipion τ lepton decays

We study the ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\nu}}_{\ensuremath{\tau}}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ ($\ensuremath{\ell}=e$, $\ensuremath{\mu}$) decays, which are $O({\ensuremath{\alpha}}^{2})$ suppressed with respect to the dominant di-pion tau decay channel. Both the inner bremsstrahlung and the structure- (and model-) dependent contributions are considered. In the $\ensuremath{\ell}=e$ case, structure-dependent effects are $\mathcal{O}(1%)$ in the decay rate, yielding a clean prediction of its branching ratio, $2.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, measurable with BABAR or Belle(-II) data. For $\ensuremath{\ell}=\ensuremath{\mu}$, both contributions have a similar magnitude, and we get a branching fraction of $(1.6\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$, reachable by the end of Belle-II operation. These decays allow one to study the dynamics of strong interactions with simultaneous weak and electromagnetic probes; their knowledge will contribute to reducing backgrounds in lepton flavor and lepton number violating searches.