Estimation of Semileptonic Decays of $B_c$ Meson to S-wave Charmonia with NRQCD

We study the semileptonic differential decay rates of $B_c$ meson to S-wave charmonia, $\eta_c$ and $J/\Psi$, at the next-to-leading order accuracy in the framework of NRQCD. In the heavy quark limit, $m_b \to \infty$, we obtain analytically the asymptotic expression for the ratio of NLO form factor to LO form factor. Numerical results show that the convergence of the ratio is perfect. At the maximum recoil region, we analyze the differential decay rates in detail with various input parameters and polarizations of $J/\psi$, which can now be checked in the LHCb experiment. Phenomenologically, the form factors are extrapolated to the minimal recoil region, and then the $B_c$ to charmonium semileptonic decay rates are estimated.