Predictions on the second-class current decays $τ^{-}\toπ^{-}η^{(\prime)}ν_τ$

We analyze the second-class current decays $τ^{-}\toπ^{-}η^{(\prime)}ν_τ$ in the framework of Chiral Perturbation Theory with resonances. Taking into account $π^{0}$-$η$-$η^{\prime}$ mixing, the $π^{-}η^{(\prime)}$ vector form factor is extracted, in a model-independent way, using existing data on the $π^{-}π^{0}$ 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 $π^{-}η$, $π^{-}η^{\prime}$ (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 $τ^{-}\toπ^{-}η^{(\prime)}ν_τ$ decay channels are promising for the soon discovery of second-class currents at Belle-II. We also predict the relevant observables for the partner $η^{(\prime)}_{\ell 3}$ decays, which are extremely suppressed in the Standard Model.

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