The vertex corrections due to the triangle singularity mechanism in the light axial vector meson couplings to $K^{\ast }\bar{K}+c\mathrm{.}c\mathrm{.}$

The light axial vector mesons can couple to $K^{\ast }\bar{K}+c\mathrm{.}c\mathrm{.}$ in an $S$ wave at the tree level. Due to the near-threshold $S$-wave interactions the couplings can be affected by the final state interactions. It is of peculiar interest that the pion exchange between $K^{\ast }\bar{K}+c\mathrm{.}c\mathrm{.}$ can go through a triangle diagram which is within the kinematics of the triangle singularity (TS). This mechanism will introduce energy dependence and a $D$-wave amplitude to the vertex couplings. Hence, it should be necessary to investigate the role played by the $K^{\ast }\bar{K}+c\mathrm{.}c\mathrm{.}$ threshold in order to have a better understanding of the light axial vector spectrum.