Physics of the gauged four fermi model in (1 + 1) dimensions

Solvable models have greatly contributed to our understanding of the dynamics of quantum field theories. Two such (1 + 1)–dimensional models solvable in the large N limit which are “classics” in this regard, are the ’t Hooft model [1], gauge theory with fermionic matter, and the Gross–Neveu model [2], a model with a four fermi coupling. (For reviews on the subject, see, for instance, [3].) Both these models are “solvable” from first principles yet they are not integrable models in the usual sense, except for the case of massless Gross–Neveu model [4]. These types of models are hard to come by and we believe that they contribute to the understanding of more realistic theories. Indeed, the ’t Hooft model and the Gross–Neveu model have a wide area of applicability, as evidenced by the contribution of these models in many areas of physics, including but not restricted to particle and nuclear physics, even recently. In this paper, we solve a model in the large–N limit wherein the four fermi couplings and the gauge coupling coexist with arbitrary strengths, thereby enlarging this class of models in an essential way. We extend and generalize the work of Burkardt [5], who derived the meson bound state equation in the gauged four fermi model and analyzed the equation from a somewhat different approach from ours. The contents of this paper are as follows: First, in section 2, we solve the Gross–Neveu model using the Bethe– Salpeter equation, which, to our knowledge, has not been presented previously. In this section, we fix the notation and summarize the physics of the Gross–Neveu model, which will be useful later on. In section 3, we analyze the theory of gauged fermions with four fermi couplings in the large–N limit. The model combines and generalizes the model of ’t Hooft and the model of Gross and Neveu. The model is more general than the Gross–Neveu model even when the gauge coupling is zero. Next we derive analytic equations for the meson states of the theory. Renormalization prescription is given and equations involving only finite physical parameters are derived. In section 4, methods are presented in detail for solving the meson state equations systematically. Using these methods, we derive various results on the physical properties of the model which are analyzed in section 5. We end with a summary and a more general discussion regarding the model in Section 6.