Abstract
The mass spectra, mixing angle and decay constants of the $J^P=1^+$ heavy-light mesons are systematically studied within the framework of the Bethe-Salpeter equation (BSE). The full $1^+$ Salpeter wave function is given for the first time. The mixing between the $1^{+-}$ and $1^{++}$ in the $1^+$ heavy-light systems are automatically determined by the dynamics in the equation without any man-made mixing. The results indicate that in a rigorous study there exists the phenomenon of mixing angle inversion or mass inversion within $1^{+}$ heavy-light doublet, which is sensitive to the $s$-quark mass for the charmed mesons and $u$- or $d$-quark masses for the bottomed mesons. This inversion phenomenon can answer the question of why we have confused mixing angles in the literature and partly explain the lower mass of $D_{s1}(2460)$ compared to that of $D_{s1}(2536)$. The decay constants are also presented and can behave as a good quantity to distinguish the $1^+$ doublet in heavy-light mesons. This study indicates that the light-quark mass may play an important role in deciding the mass order, mixing angle, and decay constant relation between the $\ket{j_l=\frac{3}{2}}$ and $\ket{j_l=\frac{1}{2}}$ heavy-light mesons.
Highlights
All the physical mesons have definite JP spin parity or JPC for quarkonia
The results indicate that in a rigorous study there exists the phenomenon of mixing angle inversion or mass inversion within 1þ heavy-light doublet, which is sensitive to the s-quark mass for the charmed mesons and u- or d-quark masses for the bottomed mesons
The obtained mass spectra, decay constants, and mixing angles are presented in Table II, in which we use the symbols θnP and θnH to denote the mixing angles defined in Eqs. (1) and (21), respectively, in order to indicate the different radially excited states
Summary
All the physical mesons have definite JP spin parity or JPC for quarkonia. The unnatural parity 1þ heavy-light mesons were usually studied by two methods: one is the heavy-quark effective theory [11,12], and another makes a man-made mixing between the 1P1 and 3P1 states For the former one, which works in the approximation mQ → þ∞, it does not hold well when the light-quark mass is comparable with the heavy quark, such as in the (cs) and (bc) systems. This work is studied within the framework of the instantaneous Bethe-Salpeter (BS) methods [27,28], which have been widely used and have achieved good performance in the strong decays of heavy mesons [29,30,31], hadronic transition [32,33,34], decay constants calculations, and annihilation rates [35,36,37].
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