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

TL;DR

This paper analyzes second-class current decays of tau leptons into pion and eta mesons using Chiral Perturbation Theory with resonances, providing predictions for branching ratios and spectra to aid future experimental searches.

Contribution

It introduces a model-independent extraction of the vector form factor and explores various parameterizations of the scalar form factor, including dispersion relations and coupled-channels methods.

Findings

Predicted branching ratios suggest these decays are detectable at Belle-II.

Provided detailed spectra and form factor parameterizations for these decays.

Highlighted the potential for discovering second-class currents in upcoming experiments.

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 Omn\`{e}s 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 $\eta^{(\prime)}_{\ell 3}$ decays, which are extremely suppressed in the Standard Model.