meson and the light - heavy quarkonium spectrum

We compute the c ¯ b spectrum from a first principle Salpeter equation obtained in a preceding paper. For comparison we report also the heavy-light quarko-nium spectrum and the hyperfine separations previously presented only in a graphical form. Notice that all results are parameter free. The spectrum and properties of the c ¯ b systems have been calculated various times in the past in the framework of the heavy quarkonium theory [1]. However the recent experimental observation of the B + c meson [2] has arisen new theoretical interest on the problem [3–5]. The mentioned spectrum has been considered again either from the potential and the lattice simulation point of view. A particular interesting quantity should be the hyperfine splitting that as for the c¯ c case seams to be sensible to relativistic and subleading corrections in α s. For the above reasons it seems to us worthwhile to present in this paper a calculation of the c ¯ b spectrum based on an effective mass operator with full relativistic kinematics which we have obtained in previous works and applied with a certain success to a fit of the entire quarkonium spectrum, heavy-heavy, light-light and light-heavy cases with the exception however of the c ¯ b case [6,7]. For comparison and completeness we report also numerical results for the light-heavy spectrum which we have given previously only in a graphical form. The mass operator was obtained by a three dimensional reduction of the q ¯ q Bethe-Salpeter equation introduced in [8]. It has the quadratic form M 2 = M 2 0 + U, with a kinetic part M 0 = w 1 + w 2 = m 2 1 + k 2 + m 2 2 + k 2 and a " potential " that in terms of the istantaneous approximation of the B-S kernel is given by k|U|k ′ = 1 (2π) 3