Speaker
Description
The inclusive hadronic decays of the $\tau$ lepton provide an alternative avenue for determining the CKM matrix elements $V_{ud}$ and $V_{us}$. In this talk, I will present recent results from the ETM Collaboration for the inclusive hadronic $\tau$ decay rate, obtained in $N_{f}=2+1+1$ QCD using the novel HLT method. This approach circumvents the well-known inverse Laplace transform problem that hinders this calculation, allowing to obtain first-principles results without relying on the operator-product expansion (OPE) or perturbative QCD. Except for isospin-breaking (IB) effects, all systematic uncertainties are under control. In the $\bar{u}s$ channel we obtain $|V_{us}|_{\tau-{\rm latt}-{\rm incl}}= 0.2189(7)_{\rm th}(18)_{\rm exp}$, which reveals a $3\sigma$ tension with purely hadronic determinations of $|V_{us}|$. Since this tension can no longer be attributed to the OPE approximation, it prompts a closer examination of experimental uncertainties and highlights the importance of determining IB corrections from first principles. I will briefly present an update on the ongoing status of our calculation of these corrections.
