Some of semileptonic and nonleptonic decays of $B_c$ meson in a Bethe-Salpeter relativistic quark model

The semileptonic decays $B_c^{+}\to P(V)+{\mathrm{\ell }}^{+}+{\bar{\nu }}_{\mathrm{\ell }}$ and the nonleptonic decays $B_c^{+}\to P(V)+L$, where $P(V)$ denotes a pseudoscalar (vector) charmonium or ($\bar{b}s$)-meson, and $L$ denotes a light meson, are studied in the framework of improved instantaneous Bethe-Salpeter (BS) equation and the Mandelstam formula. The numerical results (width and branching ratio of the decays) are presented in tables, and in order to compare conveniently, those obtained by other approaches are also put in the relevant tables. Based on the fact that the ratio $\frac{$\mathcal{B}\mathcal{R}$(B_c^+\rightarrow\psi(2S)\pi^+)}{$\mathcal{B}\mathcal{R}$(B_c^+\rightarrow J/\psi \pi^+)}=0.24^{+0.023}_{-0.040}$ estimated here is in good agreement with the observation by the LHCb $\frac{$\mathcal{B}\mathcal{R}$(B_c^+\rightarrow \psi(2S)\pi^+)}{$\mathcal{B}\mathcal{R}$(B_c^+\rightarrow J/\psi \pi^+)}=0.250\pm0.068($\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}$)\pm0.014($\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}$)\pm0.006($\mathcal{B}$)$, one may conclude that with respect to the decays the present framework works quite well.