Testing Lepton Flavour Universality in $Υ(4S)$ 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 $Υ(4S) \to e^\pm μ^\mp X$, $Υ(4S) \to μ^\pm τ_{\rm had}^\mp X$ and $Υ(4S) \to e^\pm τ_{\rm had}^\mp X$. We show that such a measurement can be directly related to the ratio $R(X)_{τ\ell} \equiv Γ(b\to X τν) / Γ(b \to X \ell ν)$ ($\ell=e$ or $μ$), 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 $b\to c\ellν$ transitions, complementary to existing exclusive measurements, accessible in the Belle II environment.