Testing Lepton Flavour Universality in $\mathrm{{\rm Y}}(4S)$ 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 $\mathrm{{\rm Y}}(4S)\to e^{\pm }{\mu }^{\mp }X$, $\Upsilon(4S) \to \mu^\pm \tau_{\rm had}^\mp X$and$\Upsilon(4S) \to e^\pm \tau_{\rm had}^\mp X$\mathrm{.}Weshowthatsuchameasurementcanbedirectlyrelatedtotheratio$R(X)_{\tau\ell} \equiv \Gamma (b\to X \tau \nu ) / \Gamma (b \to X \ell \nu)$($\ell=e$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 $b\to c\mathrm{\ell }\nu$ transitions, complementary to existing exclusive measurements, accessible in the Belle II environment.