Scrutinizing New Physics in Semi-leptonic $B_{c}\rightarrow J/\psi \tau\nu$ Decay

We perform a global analysis of the $b\to c\tau\nu$ data using the recent lattice results on the $B_c\to J/\psi$ vector and axial-vector form factors. To explore the effects from the tensor operator of new physics beyond the standard model, we determine the tensor form factors by using the non-relativistic QCD (NRQCD) relations between tensor and (axial-)vector form factors. Based on the lattice+NRQCD form factors, we fit the Wilson coefficients and the new physics couplings in $R_2$, $S_1$ and $U_1$ leptoquark models by including the recently measured $R(\Lambda_c)$ and imposing the relaxed constraint $\mathcal B(B_c\to J/\psi)<30\%$ in light of the recent studies on LEP1 data and $B_c$ lifetime. We give predictions for the experimental observables including $R(J/\psi)$, $P_\tau(J/\psi)$, $F_L(J/\psi)$ and $\mathcal A_{FB}(J/\psi)$ as well as their $q^2$ distribution in new physics scenarios/models. Our results suggest that the longitudinal $\tau$ polarization fraction $P_\tau(J/\psi)$ and the forward-backward asymmetry $\mathcal A_{FB}(J/\psi)$ are useful for testing the $R_2$ leptoquark model.