Predictions on the second-class current decays ${\tau }^{-}\to {\pi }^{-}{\eta }^{(\prime )}{\nu }_{\tau }$

Abstract

We analyze the second-class current decays ${\tau }^{-}\to {\pi }^{-}{\eta }^{(\prime )}{\nu }_{\tau }$ in the framework of Chiral Perturbation Theory with resonances. Taking into account ${\pi }^0$-$\eta$-${\eta }^{\prime }$ mixing, the ${\pi }^{-}{\eta }^{(\prime )}$ vector form factor is extracted, in a model-independent way, using existing data on the ${\pi }^{-}{\pi }^0$ one. For the participant scalar form factor, we have considered different parameterizations ordered according to their increasing fulfillment of analyticity and unitarity constraints. We start with a Breit-Wigner parameterization dominated by the $a_0(980)$ scalar resonance and after we include its excited state, the $a_0(1450)$. We follow by an elastic dispersion relation representation through the Omnès integral. Then, we illustrate a method to derive a closed-form expression for the ${\pi }^{-}\eta$, ${\pi }^{-}{\eta }^{\prime }$ (and $K^{-}{K}^0$) scalar form factors in a coupled-channels treatment. Finally, predictions for the branching ratios and spectra are discussed emphasizing the error analysis. An interesting result of this study is that both ${\tau }^{-}\to {\pi }^{-}{\eta }^{(\prime )}{\nu }_{\tau }$ decay channels are promising for the soon discovery of second-class currents at Belle-II. We also predict the relevant observables for the partner ${\eta }_{\ell 3}^{(\prime )}$ decays, which are extremely suppressed in the Standard Model.

32 pages, 17 figures. This version contains substantial changes and improvements as compared to the first one

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• New-physics signatures via $CP$ violation in ${\eta }^{(\prime )}\to {\pi }^0{\mu }^{+}{\mu }^{-}$ and ${\eta }^{\prime }\to \eta {\mu }^{+}{\mu }^{-}$ decays
• CP violation in two-meson Tau decays induced by heavy new physics
• ${\tau }^{-}\to {\pi }^{-}\eta {\nu }_{\tau }$ decay induced by QED one-loop effects
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