LFU violations in leptonic $\tau$ decays and B-physics anomalies

We present a complete analysis of Lepton Flavor Universality (LFU) violations in leptonic $\tau$ decays in motivated models addressing the $B$-physics anomalies, based on the $SU(4)\times SU(3)\times SU(2)\times U(1)$ gauge group. We show that the inclusion of vector-like fermions, required by $B$-physics data, leads to sizable modifications of the leading-log results derived within an Effective Field Theory approach. In the motivated parameter-space region relevant to the $B$-physics anomalies, the models predict a few per-mil decrease of the effective $W$-boson coupling to $\tau$, within the reach of future experiments.