Improved description of di-lepton production in ${\tau }^{-}\to {\nu }_{\tau }{P}^{-}$ decays

Abstract

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 $W{P}^{-}{\gamma }^{\ast }$ 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 $|V_{ud}/V_{us}{|}^2=0.05$.

Comments: 28 pages, 14 figures, 3 tables