  We perform an analysis of the $b\to c\tau\nu$ data, including $R(D^{(*)})$,
$R(J/\psi)$, $P_\tau(D^{*})$ and $F_L^{D^*}$, within and beyond the Standard
Model (SM). We fit the $B\to D^{(*)}$ hadronic form factors in the HQET
parametrization to the lattice and the light-cone sum rule (LCSR) results,
applying the general strong unitarity bounds corresponding to $J^P=1^-$, $1^+$,
$0^-$ and $0^+$. Using the obtained HQET relations between helicity amplitudes,
we give the strong unitarity bounds on individual helicity amplitudes, which
can be used in the BGL fits. Using the fitted form factors and taking into
account the most recent Belle measurement of $R(D^{(*)})$ we investigate the
model-independent and the leptoquark model explanations of the $b\to c\tau\nu$
anomalies. Specifically, we consider the one-operator, the two-operator new
physics (NP) scenarios and the NP models with a single $R_2$, $S_1$ or $U_1$
leptoquark which is supposed to be able to address the $b\to c\tau\nu$
anomalies, and our results show that the $R_2$ leptoquark model is in tension
with the limit $\mathcal B(B_c\to \tau\nu)<10\%$. Furthermore, we give
predictions for the various observables in the SM and the NP
scenarios/leptoquark models based on the present form factor study and the
analysis of NP.
