A technique for calculating the γ-matrix structures of the diagrams of a total four-fermion interaction with infinite number of vertices ind=2+ε dimensional regularization

Abstract

It is known [1] that ind=2+ε dimensional regularization any four-fermion interaction generates an infinite number of counterterms of the form

, where

$$\gamma _{\alpha _1 ...\alpha _n }^{(n)} \equiv As[\gamma \alpha _1 ...\gamma \alpha _n ]$$

is an antisymmetrized product of γ matrices. Therefore, a multiplicatively renormalizable complete model must include all such vertices, and the calcultion of the γ-matrix factors of its diagrams is a rather complicated problem. An effective technique for such calculations is proposed here. Its main elements are the realization of the γ matrices by the operators of a fermionic free field, transition to generating functions and functionals, the use of various functional forms of Wick's theorem, and reduction of the generald-dimensional problem to the cased=1. The general method is illustrated by specific calculations of the γ factors of one- and two-loop diagrams with arbitrary set of vertices γ⁽ⁿ⁾⊗γ⁽ⁿ⁾.