Phenomenological study of the $B_{c}$ ${\to}$ $BP$, $BV$ decays with perturbative QCD approach

Inspired by the recent LHCb measurements and forthcoming great potential on $B_{c}$ meson, we study the exclusive $B_{c}$ ${\to}$ $B_{q}P$, $B_{q}V$ decays with the perturbative QCD approach, where $q$ $=$ $u$, $d$, $s$; $P$ and $V$ denote the lightest pseudoscalar and vector $SU(3)$ nonet meson, respectively. By retaining the quark transverse momentum, employing the Sudakov factors, and choosing the typical scale as the maximum virtualities of the internal particles, we calculate the $B_{c}$ ${\to}$ $B$ transition from factors, and our results show that about 90\% contribution to form factors come from the ${\alpha}_{s}/{\pi}$ $<$ $0.3$ region. The contributions of penguin and annihilation to branching ratios are very small due to the serious suppression by the CKM factors. There is some hierarchy relations among the $B_{c}$ ${\to}$ $BP$, $BV$ decays. The branching ratios for $B_{c}$ ${\to}$ $B_{d,s}{\pi}$, $B_{d,s}{\rho}$, $B_{s}K$ are large and could be measured by the running LHCb.