Axial-vector $f_1(1285)-f_1(1420)$ mixing and $B_s \to J/\psi   (f_1(1285), f_1(1420))$ decays
  Inspired by the very recent LHCb measurements of $B_s \to J/\psi f_1(1285)$ and the good agreement between the perturbative QCD predictions and the data for many $B \to J/\psi V$ decays, we here investigate the $B_s \to J/\psi f_1(1285)$ and $B_s \to J/\psi f_1(1420)$ decays for the first time by employing the perturbative QCD (pQCD) approach, in which the $1^3P_1$ states $f_1(1285)$ and $f_1(1420)$ are believed to be the mixture of flavor singlet $f_1$ and octet $f_8$ or of quark-flavor states $f_{1q}$ and $f_{1s}$. We show that the pQCD predictions for the branching ratio of $B_s \to J/\psi f_1(1285)$ agree well with the data within errors for the mixing angle $\theta_{^3P_1} \approx 20^\circ (\phi_{^3P_1} \approx 15^\circ)$ between $f_1 (f_{1q})$ and $f_8 (f_{1s})$ states. Furthermore, the branching ratio of $B_s \to J/\psi f_1(1420)$ and the large transverse polarization fractions in these two considered channels are also predicted and will be tested by the LHC and the forthcoming Super-B factory experiments. Based on the decay rates of $B_s \to J/\psi f_1(1285)$ and $B_s \to J/\psi f_1(1420)$ decay modes predicted in the pQCD approach, the extracted mixing angle between $f_1(1285)$ and $f_1(1420)$ is basically consistent with currently available experimental measurements and lattice QCD analysis within still large theoretical errors. 
Xin Liu and Zhen-Jun Xiao
hep-ph hep-ex