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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 thatfulfills requirements from unitarity and analyticity. We obtain a good fit ofthe $I=1/2$ $\pi K$ amplitude in the $P$ wave using the LASS data above theelastic region while in this region data are generated via Monte Carlo usingthe FOCUS results based on $D_{\ell 4}$ decay. The spectrum and branching ratioof $\tau\to K\pi \nu_\tau$ constrained by $K_{\ell3}$ decays are also wellreproduced leading to $f_+(0) |V_{us}|= 0.2163 \pm 0.0014 $. Furthermore, weobtain the slope of the vector form factor $\lambda_+=(25.56 \pm 0.40) \times10^{-3} $ while the value of the scalar form factor at the Callan-Treiman pointis $\ln C=0.2062 \pm 0.0089$. Given the experimental precision our results arecompatible with the Standard model.

the journal of high energy physics/the journal of high energy physics

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