Three-dimensional relativistic structure for quarkonium states

A three-dimensional relativistic equation is used to calculate the quarkonium spectra. The kernel is the one obtained by projecting the Bethe-Salpeter (BS) kernel with both fermions being on the mass shell. The potential in the momentum representation is assumed to have a Coulomb part with highmomentum cutoff and a confinement part regularized by a small mass parameter. A complete Dirac bilinear covariant set is used to describe the spin structure of the potential. The quarkonium wave function is expanded in terms of the free Dirac spinors which are classified by the irreducible representation of the Lorentz group. Most of the quarkonium states are embedded in the continuum which exists due to the finite barrier of the confinement potential. Several sets of parameters for qualitatively fitting the experimental data are considered. The results show that the axial-vector and tensor bilinear covariants have to be included in the spin structure of the original BS kernel to explain the spectra of the quarkonium system. This property is different from that in the Schrödinger formalism, where the spin-dependent forces originate from relativistic corrections of vector covariants. The mixing of $S$ and $D$ components in the wave function of the $\frac{J}{\psi }$ system is shown to be sensitive to the presence of the axial-vector and tensor components of the potential.