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We study the semileptonic differential decay rates of $B_c$ meson to S-wavecharmonia, $\eta_c$ and $J/\Psi$, at the next-to-leading order accuracy in theframework of NRQCD. In the heavy quark limit, $m_b \to \infty$, we obtainanalytically the asymptotic expression for the ratio of NLO form factor to LOform factor. Numerical results show that the convergence of the ratio isperfect. At the maximum recoil region, we analyze the differential decay ratesin detail with various input parameters and polarizations of $J/\psi$, whichcan now be checked in the LHCb experiment. Phenomenologically, the form factorsare extrapolated to the minimal recoil region, and then the $B_c$ to charmoniumsemileptonic decay rates are estimated.

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