Investigating $B_c$ semileptonic decays

I present analyses of semileptonic $B_c$ decays. I first discuss the decays induced by $c \to \{s, d \}$ transitions, in particular $B_c \to B_a \, \bar{\ell} \, \nu_\ell$ and $B_c \to B_a^*(\to B_a \, \gamma) \, \bar{\ell} \, \nu_\ell$ decays, with $a = \{s, d \}$ and $\ell = \{ e, \mu \}$, in the Standard Model and in the extension based on the low-energy Hamiltonian comprising the full set of $D = 6$ semileptonic $c \to s, d$ operators with left-handed neutrinos.\\ Moreover, I consider $b \to c$ modes, with particular focus on the determination of the form factors parametrizing the $B_c \to J/\psi,\eta_c$ matrix elements of the operators in a generalized low-energy $b \to c$ semileptonic Hamiltonian. In this case I consider an expansion in nonrelativistic \qcd\ together with an expansion in the inverse heavy-quark mass which allows the form factors to be expressed in terms of universal functions in a selected kinematical range.\\ The Heavy Quark Spin Symmetry is used for both analyses. In the first case, it allows the relevant hadronic matrix elements to be related and the lattice \qcd\ results on $B_c$ form factors to be exploited. Optimized observables are selected, and correlations among them is studied to identify the effects of the various operators in the extended low-energy Hamiltonian. In the second one, using as an input the lattice \qcd\ results for the $B_c \to J/\psi$ matrix element of the Standard Model operator, it is possible to obtain information on other form factors. The extrapolation to the full kinematical range is also presented.