Testing Lepton Flavour Universality in $\Upsilon (4S)$ Decays

We propose a novel method to probe the persistent hints of Lepton Flavour Universality violation observed in semileptonic $B$ decays. Relying on the specific properties of the Belle II experiment, it consists in comparing the inclusive rates of $\Upsilon(4S) \to e^\pm \mu^\mp X$, $\Upsilon(4S) \to \mu^\pm \tau_{\rm had}^\mp X$ and $\Upsilon(4S) \to e^\pm \tau_{\rm had}^\mp X$. We show that such a measurement can be directly related to the ratio $R(X)_{\tau\ell} \equiv \Gamma (b\to X \tau \nu ) / \Gamma (b \to X \ell \nu)$ ($\ell=e$ or $\mu$), once appropriate experimental cuts are applied to suppress the effects of neutral $B$ mixing and leptons emitted through charm or tau decays. Such a measurement would thus constitute an additional and potentially competitive probe of Lepton Flavour Universality in $b\to c\ell\nu$ transitions, complementary to existing exclusive measurements, accessible in the Belle II environment.