Axial meson exchange and the $Z_c(3900)$ and $Z_{cs}(3985)$ resonances as heavy hadron molecules

Early speculations about the existence of heavy hadron molecules were grounded on the idea that light-meson exchanges forces could lead to binding. In analogy to the deuteron, the light-mesons usually considered include the pion, sigma, rho and omega, but not the axial meson $a_1(1260)$. Though it has been argued in the past that the coupling of the axial meson to the nucleons is indeed strong, its mass is considerably heavier than that of the vector mesons and thus its exchange ends up being suppressed. Yet, this is not necessarily the case in heavy hadrons molecules: we find that even though the contribution to binding from the axial meson is modest, it cannot be neglected in the isovector sector where vector meson exchange cancels out. This might provide a natural binding mechanism for molecular candidates such as the $Z_c(3900)$, $Z_c(4020)$ or the more recently observed $Z_{cs}(3985)$. However the $Z_{cs}(3985)$ is more dependent on a mixture of different factors, which (besides axial meson exchange) include $\eta$ exchange and the nature of scalar meson exchange. Together they point towards the existence of two $Z_{cs}(3985)$-like resonances instead of one, while the observations about the role of scalar meson exchange in the $Z_{cs}(3985)$ might be relevant for the $P_{cs}(4459)$. Finally, the combination of axial meson exchange and flavor symmetry breaking effects indicates that the isovector $J^{PC} = 0^{++}$ $D^*\bar{D}^*$ and the strange $J^P = 2^{+}$ $D^*\bar{D}_s^*$ molecules are the most attractive configurations and thus the most likely molecular partners of the $Z_c(3900)$, $Z_c(4020)$ and $Z_{cs}(3985)$.