Revisiting $B_{c}^-\to J/\psi (\eta_c) L^-$ decays within the SM and beyond in QCD factorization

Motivated by the deviations observed between the data and the SM predictions of $\mathcal{B}(\bar{B}_s^0\to D_s^+ \pi^-)$ and $\mathcal{B}(\bar{B}_d^0\to D^+ K^-)$, we revisit the $B_{c}^{-}\to J/\psi(\eta_{c}) L^{-}$ decays, with $L=\pi, K^{(*)}, \rho$, both within the SM and beyond. Since these processes are also mediated by $b\to c \bar{u} d(s)$ transitions and hence dominated by the colour-allowed tree topology, the QCD factorization (QCDF) is expected to hold in the heavy-quark limit. Firstly, we update the SM predictions of these decays by including the nonfactorizable vertex corrections up to the NNLO in $\alpha_s$. It is found that, relative to the LO results, the branching ratios of these decays up to the NLO and NNLO corrections are always enhanced, with a relative amount given by $\delta_{\text{NLO}} = (\mathcal{B}^\text{NLO}-\mathcal{B}^\text{LO})/\mathcal{B}^\text{LO} \approx +6\%$ and $\delta_{\text{NNLO}} = (\mathcal{B}^\text{NNLO}-\mathcal{B}^\text{LO})/\mathcal{B}^\text{LO} \approx +9\%$, respectively. To minimize the uncertainties brought by $V_{cb}$ and the transition form factors, we construct the ratios $R_{J/\psi(\eta_{c}) L}$, $R_{(s)L}^{(\ast)}$, and $R_{\pi/\mu\nu_{\mu}}$, which are then used to constrain the model-independent new physics (NP) Wilson coefficients. After considering the latest Belle data and the updated $B_{(s)}\to D_{(s)}^{(*)}$ form factors, we find that the deviations can still be explained by the NP four-quark operators with $(1+\gamma_{5}) \otimes (1-\gamma_{5})$ and $(1+\gamma_{5}) \otimes (1+\gamma_{5})$ structures, while the solution with $\gamma^\mu (1+\gamma_{5}) \otimes \gamma_\mu (1-\gamma_{5})$ structure does not work anymore, under the combined constraints from $R_{(s)L}^{(\ast)}$ at the $2\sigma$ level. Furthermore, the ratio $R_{\pi/\mu\nu_{\mu}}$, once measured precisely, could provide complementary constraint.