The Covariant Spectator Theory (CST) is used to calculate the mass spectrum and vertex functions of heavy-light and heavy mesons in Minkowski space. The covariant kernel contains Lorentz scalar, pseudoscalar, and vector contributions. The numerical calculations are performed in momentum space, where special care is taken to treat the strong singularities present in the confining kernel. The observed meson spectrum is very well reproduced after fitting a small number of model parameters. Remarkably, a fit to a few pseudoscalar meson states only, which are insensitive to spin-orbit and tensor forces and do not allow to separate the spin-spin from the central interaction, leads to essentially the same model parameters as a more general fit. This demonstrates that the covariance of the chosen interaction kernel is responsible for the very accurate prediction of the spin-dependent quark-antiquark interactions.

Relativistic equations of Bethe-Salpeter type for hadron structure are most conveniently formulated in momentum space. The presence of confining interactions causes complications because the corresponding kernels are singular. This occurs not only in the relativistic case but also in the nonrelativistic Schrödinger equation where this problem can be studied more easily. For the linear confining interaction the singularity reduces to one of Cauchy principal value form. Although this singularity is integrable, it still makes accurate numerical solutions difficult. We show that this principal value singularity can be eliminated by means of a subtraction method. The resulting equation is much easier to solve and yields accurate and stable solutions. To test the method's numerical efficiency, we performed a three-parameter least-squares fit of a simple linear-plus-Coulomb potential to the bottomonium spectrum.

We use the covariant spectator theory with an effective quark-antiquark interaction, containing Lorentz scalar, pseudoscalar, and vector contributions, to calculate the masses and vertex functions of, simultaneously, heavy and heavy-light mesons. We perform least-square fits of the model parameters, including the quark masses, to the meson spectrum and systematically study the sensitivity of the parameters with respect to different sets of fitted data. We investigate the influence of the vector confining interaction by using a continuous parameter controlling its weight. We find that vector contributions to the confining interaction between 0 % and about 30 % lead to essentially the same agreement with the data. Similarly, the light quark masses are not very tightly constrained. In all cases, the meson mass spectra calculated with our fitted models agree very well with the experimental data. We also calculate the mesons wave functions in a partial wave representation and show how they are related to the meson vertex functions in covariant form.