Derivatives as an IR Regulator for Massless Fields

TL;DR

This paper develops a method to regulate infrared divergences in massless scalar field theories by resumming derivatives in the effective action, providing a well-defined expansion even at small fields.

Contribution

It introduces a derivative resummation technique that cures IR divergences in massless scalar theories, extending the standard derivative expansion.

Findings

Resummation of derivatives removes IR divergences.

The method yields a well-defined expansion in the scalar field.

Extension to finite temperature is discussed.

Abstract

The free propagator for the scalar $\lambda \phi^4$--theory is calculated exactly up to the second derivative of a background field. Using this propagator I compute the one--loop effective action, which then contains all powers of the field but with at most two derivatives acting on each field. The standard derivative expansion, which only has a finite number of derivatives in each term, breaks down for small fields when the mass is zero, while the expression obtained here has a well--defined expansion in $\phi$. In this way the resummation of derivatives cures the naive IR divergence. The extension to finite temperature is also discussed.