Combined analysis of the decays $\tau^-\to K_S\pi^-\nu_\tau$ and $\tau^-\to K^-\eta\nu_\tau$

In a combined study of the decay spectra of $\tau^-\to K_S\pi^-\nu_\tau$ and $\tau^-\to K^-\eta\nu_\tau$ decays within a dispersive representation of the required form factors, we illustrate how the $K^*(1410)$ resonance parameters, defined through the pole position in the complex plane, can be extracted with improved precision as compared to previous studies. While we obtain a substantial improvement in the mass, the uncertainty in the width is only slightly reduced, with the findings $M_{K^{*\prime}}=1304 \pm 17\,$MeV and $\Gamma_{K^{*\prime}} = 171 \pm 62\,$MeV. Further constraints on the width could result from updated analyses of the $K\pi$ and/or $K\eta$ spectra using the full Belle-I data sample. Prospects for Belle-II are also discussed. As the $K^-\pi^0$ vector form factor enters the description of the decay $\tau^-\to K^-\eta\nu_\tau$, we are in a position to investigate isospin violations in its parameters like the form factor slopes. In this respect also making available the spectrum of the transition $\tau^-\to K^-\pi^0\nu_\tau$ would be extremely useful, as it would allow to study those isospin violations with much higher precision.