Semileptonic $B_c$ meson decays to S-wave charmonium states

We study the semileptonic decays of $B_c$ meson to S-wave charmonium states in the framework of relativistic independent quark model based on an average flavor-independent confining potential $U(r)$ in the scalar-vector harmonic form $U(r)=frac{1}{2}(1+gamma^0)(ar^2+V_0)$, where ($a$, $V_0$) are the potential parameters.The form factors for $B_c^+to eta_c /psi e^+ u_e$ transitions are studied in the physical kinematic range. Our predicted branching ratios (BR) for transitions to ground state charmonia are found comparatively large $sim $ $10^{-2}$, compared to those for transitions to radially excited 2S and 3S states. Like all other mpdel predictions, our predicted BR are obtained in the hierarchy: BR($B_c^+to eta_c /psi (3S)$) $<$ BR($B_c^+to eta_c/ psi (2S)$) $<$ BR($B_c^+to eta_c /psi (1S)$). The longitudinal ($Gamma_L$) and transverse polarization ($Gamma_T$) for $B_c to psi(ns)$ decay modes are predicted in the small and large $q^2$ - region as well as in the whole physical region. The ratios for such transitions are obtained $frac {Gamma_L}{Gamma_T} < 1$ throughout the kinematic range which means the $B_c^+$ meson transitions to vector meson charmonium states take place predominantly in transverse polarization mode. The theoretical predictions on these transitions could be tested in the on-going and forthcoming experiments at LHCb.