  The $R(D^{(\ast)})$ anomalies observed in $B\to D^{(\ast)}\tau\nu$ decays
have attracted much attention in recent years. In this paper, we study the
$B\to D^{(\ast)}\tau\nu$, $\Lambda_b\to\Lambda_c\tau\nu$, $B_c\to
(J/\psi,\,\eta_c)\tau\nu$, $B\to X_c\tau\nu$, and $B_c\to\tau\nu$ decays, all
being mediated by the same quark-level $b\to c\tau\nu$ transition, in the
Standard Model Effective Field Theory. The most relevant dimension-six
operators for these processes are $Q_{lq}^{(3)}$, $Q_{ledq}$, $Q^{(1)}_{lequ}$,
and $Q^{(3)}_{lequ}$ in the Warsaw basis. Evolution of the corresponding Wilson
coefficients from the new physics scale $\Lambda=1$~TeV down to the
characteristic scale $\mu_b\simeq m_b$ is performed at three-loop in QCD and
one-loop in EW/QED. It is found that, after taking into account the constraint
${\cal B}(B_c\to\tau\nu)\lesssim 10\%$, a single
$\left[C_{lq}^{(3)}\right]_{3323}(\Lambda)$ or
$\left[C^{(3)}_{lequ}\right]_{3332}(\Lambda)$ can still be used to resolve the
$R(D^{(\ast)})$ anomalies at $1\sigma$, while a single
$\left[C^{(1)}_{lequ}\right]_{3332}(\Lambda)$ is already ruled out by the
measured $R(D^{(\ast)})$ at more than $3\sigma$. By minimizing the
$\chi^2(C_i)$ function constructed based on the current data on $R(D)$,
$R(D^\ast)$, $P_\tau(D^\ast)$, $R(J/\psi)$, and $R(X_c)$, we obtain eleven most
trustworthy scenarios, each of which can provide a good explanation of the
$R(D^{(\ast)})$ anomalies at $1\sigma$. To further discriminate these different
scenarios, we predict thirty-one observables associated with the processes
considered under each NP scenario. It is found that most of the scenarios can
be differentiated from each other by using these observables and their
correlations.
