  We study possible new physics (NP) effects on $B_c \to J/\psi \tau\bar\nu$,
which has been recently measured at LHCb as the ratio of $R_{J/\psi} = \mathcal
B(B_c \to J/\psi \tau\bar\nu)/\mathcal B(B_c \to J/\psi \mu\bar\nu)$. Combining
it with the long-standing $R_{D^{(*)}}$ measurements, in which the discrepancy
with the prediction of the standard model is present, we find possible
solutions to the anomaly by several NP types. Then, we see that adding the
$R_{J/\psi}$ measurement does not improve NP fit to data, but the NP scenarios
still give better $\chi^2$ than the SM. We also investigate indirect NP
constraints from the lifetime of $B_c$ and NP predictions on the $\tau$
longitudinal polarization in $\bar B \to D^* \tau\bar\nu$.
