Speaker
Description
The inclusive hadronic decays of the $\tau$ lepton offer an alternative method for extracting the CKM matrix elements $V_{\mathrm{ud}}$ and $V_{\mathrm{us}}$. In this talk, I will discuss recent results from the ETM Collaboration on the inclusive hadronic decay rate of the $\tau$, 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 us to obtain first-principles results without relying on the operator-product expansion or perturbative QCD. Apart from isospin-breaking (IB) effects, all systematic uncertainties are under control. In the $\bar{u}s$ channel we obtain $|V_{us}|_{\tau\mathrm{-latt-incl}} = 0.2189(7)_\mathrm{th}(18)_\mathrm{exp}$, 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.
