Global Determination of $|V_{us}|$

In light of ongoing issues with the first-row unitarity test of the CKM matrix, we showcase a global fit to measurements of $K\to \ell \nu$, $K\to \pi\ell \nu$, $\tau \to K\nu$, and $\tau\to K\pi \nu$ decays for the first time. Fitting the semileptonic and 3-body $\tau$ decay data simultaneously becomes computationally feasible because we employ a simple form factor parametrisation for the $K\pi$ form factor that manifestly connects the semileptonic and pair-production regions. We find good agreement with the data and use our analyses to infer $|V_{us}|$ and the parameters for the $K\pi$ form factors. Our result for $|V_{us}|$ sits close to the results extracted from exclusive $K_{\ell 2}$ and somewhat above the $K_{\ell 3}$ decay and inclusive $\tau$ decay result. Moreover, our results for the $K\to \pi$ vector form factor at zero momentum transfer are compatible with lattice QCD determinations, despite not using lattice QCD inputs for this quantity in our fit of the hadronic matrix elements. We are further able to determine some of the pole parameters of the individual scalar and vector $K\pi$ resonances with masses below the $\tau$ mass. We caution that our results account only for short-distance electromagnetic corrections but not long-distance contributions; this is due to the lack of a consistent description of long-distance corrections to the $K\pi$ matrix elements both above and below threshold for an arbitrary model of the form factors.