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 $K_{\ell3}$ decays

We perform a combined analysis of $\tau\to K\pi \nu_\tau$ decay and $\pi K$ scattering with constraints from $K_{\ell3}$ 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 $K_{\ell3}$ 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.