Lepton Flavour Universality in $\tau$ decays

The evidence for Lepton Flavour Universality (LFU) violation in semileptonic $B$-decays has been rising over the past few years. Relying on generic effective field theory (EFT) results, it has been shown that models addressing the $B$-anomalies necessarily lead, at one-loop, to deviations from LFU in $\tau$ decays at the few per-mil level. Once a (renormalizable) UV model is specified, the leading-log EFT result receives finite corrections from the matching at the UV scale. We discuss such corrections in a motivated class of models for the B-anomalies, based on an extended $SU(4)\times SU(3)\times SU(2)\times U(1)$ gauge sector. In this scenario, we obtain precise predictions for the effective $W$-boson and $Z$-boson couplings to leptons in terms of the masses and couplings of the new heavy fields. We confirm a few per-mil deviation from universality, within reach of future high-precision experiments.