$|V_{ub}|$ determination and testing of lepton flavour universality in semileptonic $B_c \rightarrow D^{(\ast)}$ decays

In light of prospects for measurements of $B_c \rightarrow D^{(\ast)} l \nu$ decays in the upcoming Upgrade II of the LHC, we show that by using calculated $B_c \rightarrow D^{(\ast)}$ form factors competitive extraction of the $|V_{ub}|$ CKM matrix element from the $B_c \to D \mu \bar{\nu}_{\mu}$ decay might be possible. To minimize experimental and theoretical uncertainties we provide the ratio $|V_{ub}|/|V_{cb}|$ by normalizing the $B_c \rightarrow D^{(\ast)} \mu \bar{\nu}_{\mu}$ to $B_c \to J/\psi \mu \bar{\nu}_{\mu}$ decay. We also briefly examine the suggestion to extract $|V_{ub}|/|V_{cs}|$ from the theoretically interesting ratio of $B_c \rightarrow D^0 e \bar{\nu}_{e}$ and $B_c \rightarrow B_s e \bar{\nu}_{e}$ decay rates in the zero-recoil limit. With the present average value of $|V_{ub}|$, the predicted branching ratios are estimated to be $BR(B_c \to D^0 \mu \bar{\nu}_{\mu}) = (2.4\pm 0.4)\cdot 10^{-5} $ and $BR(B_c \to D^{\ast} \mu \bar{\nu}_{\mu}) = (7\pm3)\cdot 10^{-5} $, and the semileptonic ratios for testing the lepton flavour universality in these $B_c$ decays are $R_c(D^0) =0.64 \pm 0.05$ and $R_c(D^{\ast}) = 0.55 \pm 0.05$. We also provide $q^2$ distributions and various angular observables of $B_c \rightarrow D^{(\ast)} l \nu$ decays.