$\Upsilon$ and $\psi$ leptonic decays as probes of solutions to the $R_D^{(*)}$ puzzle

Experimental measurements of the ratios $R(D^{(*)})\equiv\frac{\Gamma(B\to D^{(*)}\tau\nu)}{\Gamma(B\to D^{(*)}\ell\nu)}$ ($\ell=e,\mu$) show a $3.9\sigma$ deviation from the Standard Model prediction. In the absence of light right-handed neutrinos, a new physics contribution to $b\to c\tau\nu$ decays necessarily modifies also $b\bar b\to\tau^+\tau^-$ and/or $c\bar c\to\tau^+\tau^-$ transitions. These contributions lead to violation of lepton flavor universality in, respectively, $\Upsilon$ and $\psi$ leptonic decays. We analyze the constraints resulting from measurements of the leptonic vector-meson decays on solutions to the $R(D^{(*)})$ puzzle. Available data from BaBar and Belle can already disfavor some of the new physics explanations of this anomaly. Further discrimination can be made by measuring $\Upsilon(1S,2S,3S)\to\tau\tau$ in the upcoming Belle II experiment.