Quasiclassical theory of the Dirac equation with a scalar-vector interaction and its applications in the physics of heavy-light 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-Coul(r) = -xi/r and the long-range Lorentz-scalar and Lorentz-vector linear potentials S-l.r.(r) = (1 - lambda)(sigma r + V-0) and V-l.r.(r) = lambda(sigma r + V-0) where 0 <= 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) similar to 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(s) and the coefficient lambda of mixing of the long-range scalar and vector potentials S-l.r.(r) and V-l.r.(r). The quasiclassical formulas for asymptotic coefficients of wave functions at zero and infinity are obtained.