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The charmless $B_{c}$ $\to$ $PP$, $PV$~(where $P$ and $V$ denote the lightpseudoscalar and vector mesons, respectively) decays can occur only via theweak annihilation diagrams within the Standard Model and provide, therefore, anideal place to probe the strength of annihilation contribution in hadronic$B_{u,d,s}$ decays. In this paper, we study these kinds of decays in theframework of QCD factorization, by adopting two different schemes: scheme I issimilar to the method usually adopted in the QCD factorization approach, whilescheme II is based on the infrared behavior of gluon propagator and runningcoupling. For comparison, in our calculation, we adopt three kinds of wavefunctions for $B_{c}$ meson. It is found that: (a) The predicted branchingratios in scheme I are, however, quite small and are almost impossible to bemeasured at the LHCb experiment. (b) In scheme II, by assigning a dynamicalgluon mass to the gluon propagator, we can avoid enhancements of thecontribution from soft endpoint region. The strength of annihilationcontributions predicted in scheme II is enhanced compared to that obtained inscheme I. However, the predicted branching ratios are still smaller than thecorresponding ones obtained in the perturbative QCD approach. The largediscrepancies among these theoretical predictions indicate that more detailedstudies of these decays are urgently needed and will be tested by the futuremeasurements performed at the LHCb experiment.

Charmless $B_{c}$ $\to$ $PP$, $PV$ decays in the QCD factorization approach