Hadron Mass Spectrum in a Lattice Gauge Theory

Abstract

We perform the strong coupling expansion in a lattice gauge theory and obtain the hadron mass spectrum. We develop a theory in the Hamiltonian formalism following Kogut and Susskind, but our treatment of quark fields is quite different from theirs. Thus our results largely differ from theirs. In our model and approximation, the pseudoscalar mesons have the same mass as the vectors. The baryon decuplet and the octet are also degenerate. The excited meson states are studied in detail. studied in the path integral formalism. Kogut and Susskind 2> reformulated the theory on a three-dimensional spatial lattice in the Hamiltonian formalism. Then they tried to explain the properties of hadrons, in particular, the mass spectrum based on this model.8>-5> In the lattice gauge theory it is possible to study the strong coupling limit by means of the ordinary perturbation theory. In the present paper we show how the lattice gauge theory describes the bound states of quarks and gives the hadron mass spectrum. We use the Hamiltonian formalism rather than the path integral formalism in order to clarify the physical picture. The hadron mass spectrum is obtained by solving the eigenvalue problem of the lattice Hamiltonian. Our starting Hamiltonian is the same as that of Kogut et al. except for the treatment of the quark fields. There are several methods to introduce the quark fields on a lattice. 8>.5>.e> We use the method of Wilson from the standpoint of simplicity and naturalness. As for flavor symmetry we assume SU (3). The generalization to SU ( 4) etc. does not accompany any difficulties. This article is organized as follows: In § 2 we mention the method to intro­ duce the quark fields and review the lattice gauge theoty in the Hamiltonian formalism. In § 3 we develop the perturbation theory, where the expansion param­ eter is the inverse square of the gauge coupling constant g, and is denoted by x. The masses of the pseudoscalar and the vector mesons are calculated to second order in x, while those of the baryons to third order. As for the excited states, only the meson states are considered. In § 4 our results are briefly summarized. In § 5 we compare our method and results with those of Wilson et al. and those of Kogut et al.