Semileptonic decays of $B_c$ mesons into charmonium states in a relativistic quark model

We use the framework of a relativistic constituent quark model to study the semileptonic transitions of the $B_c$ meson into $(\overline{c}c)$ charmonium states where $(\overline{c}c)={\eta }_c({}^{1}S_0)$, $J/\psi ({}^{3}S_1)$, ${\chi }_{c0}({}^{3}P_0)$, ${\chi }_{c1}({}^{3}P_1)$, $h_c({}^{1}P_1)$, ${\chi }_{c2}({}^{3}P_2)$, $\psi ({}^{3}D_2)$. We compute the $q^2$ dependence of all relevant form factors and give predictions for their semileptonic $B_c$ decay modes including also their $\tau$ modes. We derive a formula for the polar angle distribution of the charged lepton in the $(l{\nu }_l)$-c.m. frame and compute the partial helicity rates that multiply the angular factors in the decay distribution. For the discovery channel $B_c\to J/\psi (\to {\mu }^{+}{\mu }^{-})l\nu$ we compute the transverse/longitudinal composition of the $J/\psi$ which can be determined by an angular analysis of the decay $J/\psi \to {\mu }^{+}{\mu }^{-}$. We compare our results with the results of other calculations.