Relativistic bound states of fermions and antifermions. II

Abstract

We determine the eigenstates of the bound-state Hamiltonian proposed in the preceding article using the covariant Bethe-Salpeter (BS) formalism. The interaction Hamiltonian is the invariant direct four-fermion interaction, and we use the invariant cutoff function. We evaluate the two-body irreducible kernel, considering a very wide class of Feynman diagrams including the ones with the infinite order, which we approximate and express in terms of parameters. This kernel is expected to be adequate for the $S$-wave bound states. We then solve the BS equation for the bound state exactly. We derive in this way a set of eigenvalue equations, which is formally identical to the set obtained in the preceding article. There appears no difficulty in explaining all the observed pseudoscalar and vector mesons by adjusting the parameters in the eigenvalue equations. These mesons include $\pi$, $\rho$, and all the way up to ${\eta }_C$ and $\psi \mathrm{}\left(3100\right)\mathrm{}$. In particular we can achieve this fitting independently of the cutoff parameters. Assuming that $\Upsilon$ is also a two-body bound state of $B\overline{B}$, where $B$ is the underlying bottom fermion, we determine not only the physical mass of $B$ but also the masses of all the expected pseudoscalar and vector bottom mesons. The confirmation of these masses is to be considered as a strong support to this approach.