Inclusive hadronic decay rate of the tau lepton from lattice QCD: the us flavor channel and the Cabibbo angle

We present a lattice determination of the inclusive decay rate of the process 𝜏↦𝑋𝑢⁢𝑠⁢𝜈𝜏 in which the 𝜏 lepton decays into a generic hadronic state 𝑋𝑢⁢𝑠 with 𝑢⁢𝑠 flavor quantum numbers. Our results have been obtained in 𝑛𝑓=2+1+1 isosymmetric QCD with full nonperturbative accuracy, without any operator product expansion 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 [M. Hansen et al., Phys. Rev. D 99, 094508 (2019)] that we have already successfully applied [A. Evangelista et al., Phys. Rev. D 108, 074513 (2023)] to compute the inclusive decay rate of the process 𝜏 ↦𝑋𝑢⁢𝑑⁢𝜈𝜏 in the 𝑢⁢𝑑 flavor channel. By combining our first-principles theoretical results with the presently available experimental data, we extract the Cabibbo-Kobayashi-Maskawa matrix element |𝑉𝑢⁢𝑠|, the Cabibbo angle, with a 0.9% accuracy, dominated by the experimental error.