$b\to c\tau\nu$ Transitions in the Standard Model Effective Field Theory

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.