Lepton-pair production in dipion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>?</mml:mi></mml:math> lepton decays

We study the $\tau^- \to \nu_{\tau} \pi^{-}\pi^{0}\ell^{+}\ell^{-}$ ($\ell=e,\,\mu$) decays, which are $O(\alpha^2)$-suppressed with respect to dominant di-pion tau decay channel. Both inner-bremsstrahlung and structure- (and model-)dependent contributions considered. In $\ell=e$ case, structure-dependent effects $\mathcal{O}(1\%)$ in rate, yielding a clean prediction of its branching ratio, $2.3\times10^{-5}$, measurable BaBar or Belle(-II) data. For $\ell=\mu$, both have similar magnitude we get fraction $(1.6\pm0.3)\times10^{-7}$, reachable by end Belle-II operation. These decays allow dynamics strong interactions simultaneous weak electromagnetic probes; their knowledge will contribute reducing backgrounds lepton flavor/number violating searches.