Study of the pure annihilation $B_c \to A_2 A_3$ decays

In this work, we calculate the {\it CP}-averaged branching ratios and the polarization fractions of the charmless hadronic $B_c \to A_2 A_3$ decays within the framework of perturbative QCD(pQCD) approach, where $A$ is either a light $^3P_1$ or $^1P_1$ axial-vector meson. These thirty two decay modes can occur through the annihilation topology only. Based on the perturbative calculations and phenomenological analysis, we find the following results: (a) the branching ratios of the considered thirty two $B_c\to A_2 A_3$ decays are in the range of $10^{-5}$ to $10^{-8}$; (b) $B_c\to a_1 b_1$, $\ov{K}_1^0 K_1^+$ and some other decays have sizable branching ratios and can be measured at the LHC experiments; (c) the branching ratios of $B_c \to A_2(^1P_1) A_3(^1P_1)$ decays are generally much larger than those of $B_c \to A_2(^3P_1) A_3(^3P_1)$ decays with a factor around (10 $\sim$ 100); (d) the branching ratios of $B_c \to \ov{K}_1^0 K_1^+$ decays are sensitive to the value of $\theta_K$, which will be tested by the running LHC and forthcoming SuperB experiments; (e) the large longitudinal polarization contributions govern most considered decays and play the dominant role.