Direct lattice calculation of inclusive hadronic decay rates of the $\tau$ lepton

The inclusive hadronic decay-rates of the $\tau$ lepton are particularly interesting from the phenomenological point of view since they give access to the CKM matrix elements $V_{ud}$ and $V_{us}$. In this talk, we discuss how a recent method for the extraction of smeared spectral densities from Euclidean lattice correlators can be used to obtain a direct lattice determination of inclusive hadronic $\tau$ decay rates. We also present preliminary numerical results obtained by applying this method to correlators measured on two gauge ensembles produced by the ETMC with $N_f=2+1+1$ dynamical flavours at physical pion masses, lattice spacing $a\simeq 0.08$ fm and volumes $L\simeq 5.1$ fm and $L\simeq 7.6$ fm.