Study on the possible molecular state composed of $D^*_s\bar D_{s1} $ within the Bethe-Salpeter framework

Abstract: Recently a vector charmonium-like state $Y(4626)$ was observed in the portal of $D^+sD{s1}(2536)^-$. It intrigues an active discussion on the structure of the resonance because it has obvious significance for gaining a better understanding on its hadronic structure with suitable inner constituents. It indeed concerns the general theoretical framework about possible structures of exotic states. Since the mass of $Y(4626)$ is slightly above the production threshold of $D^+_s\bar D_{s1}(2536)^-$ whereas below that of $D^*_s\bar D_{s1}(2536)$ with the same quark contents as that of $D^+_s\bar D_{s1}(2536)^-$, it is natural to conjecture $Y(4626)$ to be a molecular state of $D^{*}_s\bar D_{s1}(2536)$, as suggested in literature. Confirming or negating this allegation would shed light on the goal we concern. We calculate the mass spectrum of a system composed of a vector meson and an axial vector i.e. $D^*_s\bar D_{s1}(2536)$ within the framework of the Bethe-Salpeter equations. Our numerical results show that the dimensionless parameter $\lambda$ in the form factor which is phenomenologically introduced to every vertex, is far beyond the reasonable range for inducing an even very small binding energy $\Delta E$. It implies that the $D^*_s\bar D_{s1}(2536)$ system cannot exist in the nature as a hadronic molecule in this model, so that we may not think the resonance $Y(4626)$ to be a bound state of $D^*_s\bar D_{s1}(2536)$, but something else, for example a tetraquark and etc.