Analytic Regularization and Renormalization of Nonperturbation Theories

Abstract

Quantum field theories suffer from infinities. In perturbation theories, these infinities manifest themselves as ultraviolet (UV) divergences in Feynman integrals. In massless theories, there are also Infrared (IR) divergences to contend with. In perturbation expansion, the order-by-order removal of these infinities - the renormalization program¹) - is well understood. The program has been tremendously simplified since the advent of dimensional regularization^{2, 3}), a technique whereby the infinities are analytically isolated as poles in the complex ω-plane, where 2ω is the generalized dimension of Euclidean space-time. For nonperturbatlon theories, a general and viable renormalization procedure has not yet been devised. The problem with which we shall be concerned here is the regularization and renormalization of a nonperturbatlon theory as represented by the nonlinear equations derived from it. We will describe a technique that should allow one to analytically regulate and ultimately renormalize the equation.