Infra-Red Asymptotic Dynamics of Gauge Invariant Charged Fields: QED versus QCD

TL;DR

This paper investigates the infrared behavior of gauge-invariant charged fields in QED and QCD, demonstrating finite two-point functions in QED and exploring divergence factorization in QCD.

Contribution

It constructs an infrared-finite electron field in QED up to fourth order and analyzes the failure of similar construction in QCD, proposing a factorization approach for divergences.

Findings

QED electron two-point function is infrared finite with on-shell normalization.

In QCD, infrared divergences do not cancel at fourth order but follow a factorization pattern.

Speculation on the implications of divergence factorization for QCD's asymptotic dynamics.

Abstract

The freedom one has in constructing locally gauge invariant charged fields in gauge theories is analyzed in full detail and exploited to construct, in QED, an electron field whose two-point function W(p), up to the fourth order in the coupling constant, is normalized with on-shell normalization conditions and is, nonetheless, infra-red finite; as a consequence the radiative corrections vanish on the mass shell $p^2=\mu^2$ and the free field singularity is dominant, although, in contrast to quantum field theories with mass gap, the eigenvalue $\mu^2$ of the mass operator is not isolated. The same construction, carried out for the quark in QCD, is not sufficient for cancellation of infra-red divergences to take place in the fourth order. The latter divergences, however, satisfy a simple factorization equation. We speculate on the scenario that could be drawn about infra-red asymptotic dynamics of QCD, should this factorization equation be true in any order of perturbation theory.