Semileptonic $B_c$ meson decays to S-wave charmonia and $X(3872)$ within the covariant light-front approach

In this work, we investigate the semileptonic decays of $B_c$ meson to ${\eta }_c(1S,2S,3S)$, $\psi (1S,2S,3S)$ and $X(3872)$ within the framework of covariant light-front quark model (CLFQM). We combine the helicity amplitudes via the corresponding form factors to obtain the branching ratios of the semileptonic decays $B_c\to {\eta }_c(1S,2S,3S)\mathrm{\ell }{\nu }_{\mathrm{\ell }}$, $B_c\to \psi (1S,2S,3S))\mathrm{\ell }{\nu }_{\mathrm{\ell }}$ and $B_c\to X(3872)\mathrm{\ell }{\nu }_{\mathrm{\ell }}$ with $\mathrm{\ell }=e,\mu ,\tau$. In view of the $R_{J\mathrm{/}\mathrm{\Psi }}$ anomaly released by the LHCb collaboration, it is necessary to calculate the ratios $R_X$ with $X=\psi (1S,2S,3S),{\eta }_c(1S,2S,3S),X(3872)$ systematically, which are helpful to check the lepton flavor universality (LFU). Furthermore, we also take into account another two physical observables, one is the longitudinal polarization fraction $f_L$ and the other is the forward-backward asymmetry $A_{FB}$, which can provide new clues to understand the $R_{J\mathrm{/}\mathrm{\Psi }}$ anomaly. Such theoretical predictions are necessary and interesting, which can be tested in the future LHCb experiments.