Inclusive hadronic decay rate of the $\tau$ lepton from lattice QCD: the $\bar{u}s$ flavour channel and the Cabibbo angle

We present a lattice determination of the inclusive decay rate of the process $\tau \mapsto X_{us}{\nu }_{\tau }$ in which the $\tau$ lepton decays into a generic hadronic state $X_{us}$ with $\bar{u}s$ flavour quantum numbers. Our results have been obtained in $n_f=2+1+1$ iso-symmetric QCD with full non-perturbative accuracy, without any OPE approximation and, except for the presently missing long-distance isospin-breaking corrections, include a solid estimate of all sources of theoretical uncertainties. This has been possible by using the Hansen-Lupo-Tantalo method [1] that we have already successfully applied in [2] to compute the inclusive decay rate of the process $\tau \mapsto X_{ud}{\nu }_{\tau }$ in the $\bar{u}d$ flavour channel. By combining our first-principles theoretical results with the presently-available experimental data we extract the CKM matrix element $\mathrm{\mid }{V}_{us}\mathrm{\mid }$, the Cabibbo angle, with a $0.9$% accuracy, dominated by the experimental error.