Light-front quark model analysis of the exclusive rare $B_c\to D_{(s)}({\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-},{\nu }_{\mathrm{\ell }}{\bar{\nu }}_{\mathrm{\ell }})$ decays

We investigate the exclusive rare $B_c\to D_{(s)}{\nu }_{\mathrm{\ell }}\overline{{\nu }_{\mathrm{\ell }}}$ and $B\to D_{(s)}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ ($\mathrm{\ell }=e,\mu ,\tau$) decays within the standard model and the light-front quark model constrained by the variational principle for the QCD motivated effective Hamiltonian. The form factors $f_{\pm }(q^2)$ and $f_T(q^2)$ are obtained from the analytic continuation method in the $q^{+}=0$ frame. While the form factors $f_{+}(q^2)$ and $f_T(q^2)$ are free from the zero-mode, the form factor $f_{-}(q^2)$ is not free from the zero-mode in the $q^{+}=0$ frame. We discuss the covariance(i.e. frame-independence) of our model calculation and quantify the zero-mode contributions to $f_{-}(q^2)$ for $B_c\to D_{(s)}$ decays. The branching ratios and the longitudinal lepton polarization asymmetries are calculated with and without the long-distance contributions. Our numerical results for the non-resonant branching ratios for $B_c\to D(D_s)\sum\nu_{\ell}\bar{\nu_{\ell}}$and$B_c\to D(D_s)\ell^+\ell^-$ are in the order of ${10}^{-8}({10}^{-7})$ and ${10}^{-9}({10}^{-8})$, respectively. The averaged values of the lepton polarization asymmetries obtained from the linear (harmonic oscillator) potential parameters are found to be $-0.99(-0.99)$ for $B_c\to D{\mu }^{+}{\mu }^{-}$ and $-0.16(-0.15)$ for $B_c\to D{\tau }^{+}{\tau }^{-}$, and $-0.98 (-0.98)$for$B_c\to D_s\mu^+\mu^-$and$-0.14 (-0.12)$for$B_c\to D_s\tau^+\tau^-$, respectively.