  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.
