Perturbative QCD for B_s \to a_1(1260)(b_1(1235))P(V) Decays
  Within the framework of perturbative QCD approach, we study the charmless two-body decays $B_s \to a_1(1260)(b_1(1235))P(V)$ ($P, V$ represent the light pseudo-scalar and vector mesons, respectively.). Using the decays constants and the light-cone distribution amplitudes for these mesons derived from the QCD sum rule method, we find the following results: (a) The decays $\bar B^0_s\to a^{-}_1K^{+}(K^{*+})$ have the contributions from the factorization emission diagrams with a large Wilson coefficient $C_2+C_1/3$ (order of 1), so they have the largest branching ratios and arrive at $10^{-5}$ order. While for the decays $\bar B^0_s\to a^{0}_1 K^{0}(K^{*0})$, the Wilson coefficient is $C_1+C_2/3$ in tree level and color suppressed, so their branching ratios are small and fall in the order of $10^{-7}\sim10^{-8}$. For the decays $\bar B^0_s\to b_1K(K^*)$, all of their branching ratios are of order few times $10^{-6}$. (b) For the pure annihilation type decays $\bar B^0_s\to a_1(b_1)\rho$ except the decays $\bar B^0_s\to a_1\pi$ having large branching ratios of order few times $10^{-6}$, the most other decays have the branching ratios of $10^{-7}$ order. The branching ratios of the decays $\bar B^0_s\to a^0_1(b^0_1)\omega$ are the smallest and fall in the order of $10^{-8}\sim10^{-9}$. (c)The branching ratios and the direct CP-asymmetries of decays $\bar B^0_s\to a^0_1(b_1^0)\eta^{(\prime)}$ are very sensitive to take different Gegenbauer moments for $\eta^{(\prime)}$. (d) Except for the decays $\bar B^0_s\to a^{0}_1 K^{*0}, a^{0}_1\omega, b^{0}_1\omega$, the longitudinal polarization fractions of other $\bar B^0_s\to a_1(b_1)V$ decays are very large and more than 90%. (e) Compared with decays $\bar B^0_s\to a_1(b_1)P$, most of $\bar B^0_s\to a_1(b_1)V$ decays have smaller direct CP asymmetries. 
Zhi-Qing Zhang
hep-ph