Revisiting Axial-Vector Meson Mixing

Various phenomenological studies indicate that the mixing angle ${\theta }_{K_1}$ of $K_{1A}$ and $K_{1B}$, the strange partners of the axial-vector mesons $a_1(1260)$ and $b_1(1235)$, respectively, lies in the vicinity of ${35}^{\circ }$ or ${55}^{\circ }$, but whether this angle is larger or smaller than ${45}^{\circ }$ still remains controversial. When the $f_1(1285)$-$f_1(1420)$ mixing angle ${\theta }_{{}^3{P}_1}$ and the $h_1(1170)$-$h_1(1380)$ mixing angle ${\theta }_{{}^1{P}_1}$ are determined from the mass relations, they depend on the masses of $K_{1A}$ and $K_{1B}$, which in turn depend on the mixing angle ${\theta }_{K_1}$. We show that the approximate decoupling of the light $q\bar{q}$ state from the heavier $s\bar{s}$ state, which is empirically valid for vector, tensor and $3^{--}$ mesons, when applied to isoscalar axial-vector mesons, will enable us to discriminate different solutions of ${\theta }_{{}^3{P}_1}$ and ${\theta }_{{}^1{P}_1}$ and pick up ${\theta }_{K_1}\sim {35}^{\circ }$. Indeed, for ${\theta }_{K_1}\sim {55}^{\circ }$, the predicted ${\theta }_{{}^1{P}_1}$ disagrees sharply with the recent lattice calculation and the implied large $s\bar{s}$ content of $h_1(1170)$ and $q\bar q$componentof$h_1(1380)$ cannot explain the observation of their strong decays. We conclude that ${\theta }_{K_1}$ is smaller than ${45}^{\circ }$.