Resummation of mass terms in perturbative massless quantum field theory

TL;DR

This paper examines the mathematical subtleties of resumming mass terms in massless quantum field theory, revealing insights into perturbation theory convergence and implications for gauge symmetry breaking.

Contribution

It provides a detailed mathematical analysis of mass resummation in massless QFT, highlighting subtle issues and potential generalizations in distribution theory.

Findings

Resummation shifts particle mass from zero to m.

Subtleties arise in rigorous resummation procedures.

Insights into gauge symmetry breaking mechanisms.

Abstract

The neutral massless scalar quantum field $\Phi$ in four-dimensional space-time is considered, which is subject to a simple bilinear self-interaction. Is is well-known from renormalization theory that adding a term of the form $-\frac{m^2}{2} \Phi^2$ to the Lagrangean has the formal effect of shifting the particle mass from the original zero value to m after resummation of all two-leg insertions in the Feynman graphs appearing in the perturbative expansion of the S-matrix. However, this resummation is accompanied by some subtleties if done in a proper mathematical manner. Although the model seems to be almost trivial, is shows many interesting features which are useful for the understanding of the convergence behavior of perturbation theory in general. Some important facts in connection with the basic principles of quantum field theory and distribution theory are highlighted, and a remark is made on possible generalizations of the distribution spaces used in local quantum field theory. A short discussion how one can view the spontaneous breakdown of gauge symmetry in massive gauge theories within a massless framework is presented.