Is $f_1(1420)$ the partner of $f_1(1285)$ in the $^3P_1$ $q\bar{q}$   nonet?
  Based on a $2\times 2$ mass matrix, the mixing angle of the axial vector states $f_1(1420)$ and $f_1(1285)$ is determined to be $51.5^{\circ}$, and the theoretical results about the decay and production of the two states are presented. The theoretical results are in good agreement with the present experimental results, which suggests that $f_1(1420)$ can be assigned as the partner of $f_1(1285)$ in the $^3P_1$ $q\bar{q}$ nonet. We also suggest that the existence of $f_1(1510)$ needs further experimental confirmation. 
De-Min Li, Hong Yu and Qi-Xing Shen