Analytic renormalization via continuous space dimension

Abstract

In recent ycars analytic techniques have been extensively used in the renormalization of finite-order perturbation theory (1). This letter provides yet another analytic approach, which will be shown immediately to be equivalent to the wcll-knoval additive renormalization by counter terms (2). To place this work in perspective, wc shall s tar t by briefly reviewing the current situation and the motivations for further study. Roughly speaking, renormalization may be described as follows. A generic Feynman integrM is formMly defined by a (possibly diw'~rgent) integral, which is then modified into a well-defined, regularized integral by introducing additional parameters. A specific prescription is then given to remove the regularizing parameters and obtain a finite result, called renormalized integral. Because of their role, the reglflarizh~g parameters may be called generalized cut-offs, even in the schemes, such as analytic renormalization, where they may take complex values. Within any scheme, it must be shown that the difference between the original formal Feynman integral and the renormMized one, can be traced to the introduction of Lagrangian counter terms. This is necessary for the physical interpretat ion of the manipulat ions and also, perhaps more important , to guarantee, at least at a formal level, that the resulting renorrealized pertnrbat ive series originates a uni tary S-matrix. Any almlytic renormalization is an obvious candidate to renormalize the so-called nonrenormalizable theories, where the additive renormalization fails for the occurrence of infinite counter terms. Very recently the equivalence between analytic renormalization and additive renormalization has been proved; each recipe may be transformed into the other one, modu/us a finite renormaliz~ation (a).