A dispersive analysis of the pion vector form factor and $\tau^{-}\to K^{-}K_{S}\nu_{\tau}$ decay.

We explore the potential of a combined analysis of the decays $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ and $\tau^{-}\to K^{-}K_{S}\nu_{\tau}$ in the determination of the $\rho(1450)$ and $\rho(1700)$ resonance properties in the frame of resonance Chiral Theory supplemented by dispersion relations. On the one hand, we take advantage of the very precise data on the modulus squared of the pion vector form factor $|F_{V}^{\pi}|^{2}$ obtained by Belle to carry out a very dedicated analysis of the region where these resonances come up into play. Our study provides an improved treatment of the systematic theoretical errors and, as a most important result, we conclude that they dominate over the fit uncertainties in the determination of the $\rho(1450)$ and $\rho(1700)$ pole parameters and tend to be larger than in other determinations quoted in the literature where these errors were ignored or underestimated. The results of our analysis are summarized in numerical tables for the form factor modulus and phase, including both statistical and systematic errors, that can be found as ancillary material of this paper. As a byproduct, we also determine the low-energy observables of the pion vector form factor and the $\rho$-pole position. On the other hand, we benefit from the recent experimental data for the transition $\tau^{-}\to K^{-}K_{S}\nu_{\tau}$ released by BaBar to perform a first analysis of its decay spectrum and discuss the role of these resonances in this decay. We point out that higher-quality data on the $K^{-}K_{S}$ decay channel will allow compete with the $|F_{V}^{\pi}|^{2}$ ones and improve the determination of the $\rho(1450)$ and $\rho(1700)$ resonance parameters as a result of a combined analysis. We hope our study to be of interest for present and future experimental analysis of these decays.