Semiclassical approximation in the relativistic potential model of $B$ and $D$ mesons

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 $V_{\mathrm{Coul}}(r)$ and the long-range Lorentz-scalar and Lorentz-vector linear potentials $S_{\text{l.r.}}(r)$ and $V_{\text{l.r.}}(r)$. In the semiclassical 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 $n_r\sim 1$. 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 $\alpha_{\mathrm{s}}$ and the coefficient $\lambda$ of mixing of the long-range scalar and vector potentials $S_{\text{l.r.}}(r)$ and $V_{\text{l.r.}}(r)$.