First determination of $f_{+}(0)|V_{us}|$ from a combined analysis of $\tau \to K\pi {\nu }_{\tau }$ decay and $\pi K$ scattering with constraints from $K_{\ell 3}$ decays

Abstract

We perform a combined analysis of $\tau \to K\pi {\nu }_{\tau }$ decay and $\pi K$ scattering with constraints from $K_{\ell 3}$ data using a $N/D$ approach that fulfills requirements from unitarity and analyticity. We obtain a good fit of the $I=1/2$ $\pi K$ amplitude in the $P$ wave using the LASS data above the elastic region while in this region data are generated via Monte Carlo using the FOCUS results based on $D_{\ell 4}$ decay. The spectrum and branching ratio of $\tau \to K\pi {\nu }_{\tau }$ constrained by $K_{\ell 3}$ decays are also well reproduced leading to $f_{+}(0)|V_{us}|=0.2163\pm 0.0014$. Furthermore, we obtain the slope of the vector form factor ${\lambda }_{+}=(25.56\pm 0.40)\times 10^{-3}$ while the value of the scalar form factor at the Callan-Treiman point is $\ln C=0.2062\pm 0.0089$. Given the experimental precision our results are compatible with the Standard model.

Comments: 30 pages, 7 figures, Updated numerics and discussion slightly extended. Published version