Renormalization of Tamm-Dancoff integral equations.

During the last few years, interest has arisen in using light-front Tamm-Dancoff field theory to describe relativistic bound states for theories such as QCD. Unfortunately, difficult renormalization problems stand in the way. We introduce a general, non-perturbative approach to renormalization that is well suited for the ultraviolet and, presumably, the infrared divergences found in these systems. We reexpress the renormalization problem in terms of a set of coupled inhomogeneous integral equations, the “counterterm equation.” The solution of this equation provides a kernel for the Tamm-Dancoff integral equations which generates states that are independent of any cutoffs. We also introduce a Rayleigh-Ritz approach to numerical solution of the counterterm equation. Using our approach to renormalization, we examine several ultraviolet divergent models. Finally, we use the Rayleigh-Ritz approach to find the counterterms in terms of allowed operators of a theory. Introduction First attempted in the 1950’s by Tamm and Dancoff [1, 2], the idea of describing relativistic bound states in terms of a small number of particles (Fock space truncation) ran into considerable difficulties. There were severe divergences associated with connected Green’s functions as well as divergences associated with particle creation and annihilation from the vacuum and the approach was soon abandoned. In 1966, Weinberg noticed that, in the infinite momentum frame, creation and annihilation of particles from the vacuum is forbidden and the divergences associated with the vacuum itself are removed [3]. This is similarly true for light front quantized field theories. During the last few years, interest has arisen in combining these two ideas, Tamm-Dancoff Fock space truncation and light front quantization, in an attempt to describe relativistic bound states [4, 5]. Although divergences associated with the vacuum are removed, very difficult renormalization problems remain. In perturbation theory, divergences fall into two categories: ultraviolet (UV) divergences and infrared (IR) divergences. One generally associates UV divergences with particles having large energy. In the context of light-front physics, large energies arise from particles having large transverse or small longitudinal momenta, and thus UV divergences and some IR divergences have a common origin. New non-perturbative divergences arise because the Tamm-Dancoff truncation does not include all diagrams for any given order in perturbation theory. To date, these divergences in light-front Tamm-Dancoff calculations have only been renormalized within the context of perturbation theory [6]. What we propose here is a general, nonperturbative approach to renormalization that naturally handles the UV and, we believe, some of the IR divergences found in light-front Tamm-Dancoff field theory. In this approach, it is shown that the renormalization problem can be rewritten as a set of coupled integral equations, much in the same way that the eigenvalue problem is written as a set of coupled integral equations. One can then solve this set of coupled integral equations numerically using the same techniques that are used in solving the eigenvalue problem itself. Although the context of light-front field theory is the motivation for this work, our approach will not depend on any of its special properties. Presumably, our approach to renormalization could be used in other, unrelated contexts. In section 1, we will introduce the renormalization problem using a simple one 2 dimensional UV divergent model that has been discussed by other authors. We will introduce the so-called “high-low analysis” and show how it can be used to renormalize the model. In section 2 we will use the high-low analysis to examine the general case and derive the “counterterm equation,” a set of coupled integral equations which relate the bare and renormalized Hamiltonians. In section 3, we will show how renormalization group ideas are expressed in the context of our renormalization procedure. Next, in section 4, we will discuss a variational, or Rayleigh-Ritz, approach to solving the counterterm equation. In sections 5 and 6, we will apply our renormalization scheme to some simple examples. Finally, in section 7, we use the Rayleigh-Ritz approach to find the counterterm in terms of the allowed operators of a theory. 1. Model A Let us start by looking at model A, a simple toy model that has been studied by a number of authors [6, 7, 8, 9]. Consider the homogeneous integral equation