Lepton-pair production in di-pion $\tau$ lepton decays

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