Derivatives as an IR Regulator for Massless Fields

Abstract

The free propagator for the scalar $λϕ^4$--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 $ϕ$. In this way the resummation of derivatives cures the naive IR divergence. The extension to finite temperature is also discussed.