Dimensional regularization and infra-red divergence

Abstract

The problem of dimensional rcgularization wherein one considers the lowest-order divergent graphs occurring in quantum field theory as functions of the dimensiona~ parameter v was in t roduced by BELLINI and GIA1KBIAGI (1,2), 'T HOOFT and VELT~AN (3), ~,nd by ASHMORE and others (4). These techniques were found to be extremely useful in giving a well-defined procedure of renormalization which also gives a manifestly gauge-invariant t r e a tmen t of vector and tensor fields. This method was later used to deal with chiral symmetries and abnormal amplitudes (5) and also to compensate the Adler anomaly (6) without in t roducing ext ra fields. Following BELLINI and GIA~BIAGI (~) (hereafter referred to as BG), we have re-exaznined the problem of infra-red divergence in v-dimensional quantum electrodynamics. BG havc pointed out tha t the infra-red divergence in this formalism appears as a simple pole of F(2--~/2) at v = 4 in the renormalized electron self-energy X1(p, ~) in the lowest order. Now the ultra-violet divergence, in this approach, is also manifested as a simple pole at v = 4. Thus, the question of how to separate these two singularities at the level of 4-dimensional QED arises. A clue to this is provided by BG who conjecture tha t the infra-red logarithmic singularity will indeed appear when the photon is given a small mass/~. In this note we point out the importance of taking first a finite mass/~, for the photon for this problem, and then passing to the limit /t->0 for the correct disentangling of the two types of singularities.