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

We analyze the second-class current decays ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\eta}}^{(\ensuremath{'})}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ in the framework of chiral perturbation theory with resonances. Taking into account ${\ensuremath{\pi}}^{0}\text{\ensuremath{-}}\ensuremath{\eta}\text{\ensuremath{-}}{\ensuremath{\eta}}^{\ensuremath{'}}$ mixing, the ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\eta}}^{(\ensuremath{'})}$ vector form factor is extracted, in a model-independent way, using existing data on the ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ one. For the participant scalar form factor, we have considered different parametrizations ordered according to their increasing fulfillment of analyticity and unitarity constraints. We start with a Breit-Wigner parametrization 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\`es integral. Then, we illustrate a method to derive a closed-form expression for the ${\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\eta}$, ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\eta}}^{\ensuremath{'}}$ (and ${K}^{\ensuremath{-}}{K}^{0}$) 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 ${\ensuremath{\tau}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\eta}}^{(\ensuremath{'})}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ decay channels are promising for the soon discovery of second-class currents at Belle-II. We also predict the relevant observables for the partner ${\ensuremath{\eta}}_{\ensuremath{\ell}3}^{(\ensuremath{'})}$ decays, which are extremely suppressed in the Standard Model.