Renormalization of Massless Feynman Amplitudes in Configuration Space

TL;DR

This paper develops a systematic method for renormalizing massless Feynman amplitudes in both Euclidean and Minkowski spaces using distribution extension techniques, providing criteria for convergence that include divergence cancellations.

Contribution

It introduces a covariant extension-based renormalization approach for massless QFT amplitudes, refining convergence criteria beyond traditional power counting.

Findings

Defines a renormalization invariant residue for subdivergence-free amplitudes.

Establishes a necessary and sufficient condition for amplitude convergence.

Extends the convergence notion to non-primitively divergent amplitudes.

Abstract

A systematic study of recursive renormalization of Feynman amplitudes is carried out both in Euclidean and in Minkowski configuration space. For a massless quantum field theory (QFT) we use the technique of extending associate homogeneous distributions to complete the renormalization recursion. A homogeneous (Poincare covariant) amplitude is said to be convergent if it admits a (unique covariant) extension as a homogeneous distribution. For any amplitude without subdivergences - i.e. for a Feynman distribution that is homogeneous off the full (small) diagonal - we define a renormalization invariant residue. Its vanishing is a necessary and sufficient condition for the convergence of such an amplitude. It extends to arbitrary - not necessarily primitively divergent - Feynman amplitudes. This notion of convergence is finer than the usual power counting criterion and includes cancellation of divergences.