Relativistic description of quark-antiquark bound states. Spin-independent treatment

We present the results of a detailed study of light- and heavy-quark-antiquark bound states in the context of the reduced Bethe-Salpeter equation with static vector and scalar interactions. In the present paper, we consider the spin-averaged spectra. Spin effects are considered in a separate paper. We find that this approach, although apparently successful for the heavy-quark $b\overline{b}$ and $c\overline{c}$ states, fails for the $s\overline{s}$, $l\overline{l}$, and light-heavy states. The reasons for the failure are intrinsic to the method, as we discuss. Difficulties are already evident for the $c\overline{c}$ states.