Improved description of di-lepton production in $\tau^-\to\nu_\tau P^-$ decays

Recently, the Belle collaboration reported the first measurements of the $\tau^- \to\nu_{\tau}\pi^-e^+e^-$ branching fraction and the spectrum of the pion-dielectron system. In an analysis previous to Belle's results, we evaluated this branching fraction which turned out to be compatible with that reported by Belle, although with a large uncertainty. This is the motivation to seek for improvements on our previous evaluation of $\tau^- \to \nu_{\tau}\pi^-\ell^+\ell^-$ decays ($\ell=e,\,\mu$). In this paper we improve our calculation of the $WP^-\gamma^*$ vertex by including flavour symmetry breaking effects in the framework of the Resonance Chiral Theory. We impose QCD short-distance behaviour to constrain most parameters and data on the $\pi^-e^+e^-$ spectrum reported by Belle to fix the remaining free ones. As a result, improved predictions for the branching ratios and hadronic/leptonic spectra are reported, in good agreement with observations. Analogous calculations for the strangeness-changing $\tau^- \to \nu_{\tau}K^-\ell^+\ell^-$ transitions are reported for the first time. Albeit one expects the $m_{\pi\mu^+\mu^-}$ spectrum to be measured in Belle-II and the observables with $\ell=e$ can be improved, it is rather unlikely that the $K$ channels can be measured due to the suppression factor $|V_{ud}/V_{us}|^2=0.05$.