Quasiclassical theory of the Dirac equation with a scalar-vector interaction and its applications in the physics of heavy-light mesons

Abstract

We construct a relativistic potential quark model of $D$, ${D}_{s}$, $B$, and ${B}_{s}$ mesons in which the light quark motion is described by the Dirac equation with a scalar-vector interaction and the heavy quark is considered a local source of the gluon field. The effective interquark interaction is described by a combination of the perturbative one-gluon exchange potential ${V}_{\mathrm{Coul}}(r)=\ensuremath{-}\ensuremath{\xi}/r$ and the long-range Lorentz-scalar and Lorentz-vector linear potentials ${S}_{\mathrm{l}.\mathrm{r}.}(r)=(1\ensuremath{-}\ensuremath{\lambda})(\ensuremath{\sigma}r+{V}_{0})$ and ${V}_{\mathrm{l}.\mathrm{r}.}(r)=\ensuremath{\lambda}(\ensuremath{\sigma}r+{V}_{0})$, where $0\ensuremath{\le}\ensuremath{\lambda}<1/2$. Within the quasiclassical approximation, we obtain simple asymptotic formulas for the energy and mass spectra and for the mean radii of $D$, ${D}_{s}$, $B$, and ${B}_{s}$ mesons, which ensure a high accuracy of calculations even for states with the radial quantum number ${n}_{r}\ensuremath{\sim}1$. We show that the fine structure of $P$-wave states in heavy-light mesons is primarily sensitive to the choice of two parameters: the strong-coupling constant ${\ensuremath{\alpha}}_{s}$ and the coefficient $\ensuremath{\lambda}$ of mixing of the long-range scalar and vector potentials ${S}_{\mathrm{l}.\mathrm{r}.}(r)$ and ${V}_{\mathrm{l}.\mathrm{r}.}(r)$. The quasiclassical formulas for asymptotic coefficients of wave functions at zero and infinity are obtained.