Relativistic bound states of fermions and antifermions. I

Abstract

We discuss how to study the relativistic bound states of fermions and antifermions; the purpose is to identify these bound states as the observed hadrons. We express the strong Hamiltonian as the sum of $H_{\mathrm{bound}}$ and $H_{\mathrm{decay}}$, where the eigenstates of $H_{\mathrm{bound}}$ are the observed hadrons and $H_{\mathrm{decay}}$ is responsible for their strong decays. We derive $H_{\mathrm{bound}}$ and $H_{\mathrm{decay}}$ by means of a canonical transformation which amounts to eliminating gluons in the underlying theory in which underlying fermions are interacting universally with a neutral vector gluon and also a neutral axial-vector gluon, both of which are assumed to be quite heavy. We find that $H_{\mathrm{bound}}$ consists of an invariant direct four-fermion interaction without any derivative of the fermion field, and $H_{\mathrm{decay}}$ is also of the form of a direct four-fermion interaction with, however, derivatives of the fermion field. We show that $H_{\mathrm{decay}}$ is also effectively invariant and satisfies the Okubo-Zweig-Iizuka rule automatically. We show also how we solve for the two-body and three-body eigenstates of $H_{\mathrm{bound}}$. We show in particular that the eigenstate which consists of $p$, $\overline{n}$, and $\Lambda$ appears very much like the observed $\Sigma$, where $p$, $n$, and $\Lambda$ are the underlying fermions with the quantum numbers of the observed $p$, $n$, and $\Lambda$.