Gradient corrections to the quantum effective action

TL;DR

This paper derives the quantum effective action up to second order in gradients and two-loop order for scalar fields, providing insights for cosmological phenomena like bubble nucleation and addressing renormalization issues in resummed perturbation theory.

Contribution

It introduces a gradient expansion of the effective action up to second order, including two-loop calculations and the application of the 2PI formalism for renormalization in scalar field theories.

Findings

Second-order gradient corrections are significant in multi-field scenarios.

Two-loop 1PI effective action is nonrenormalisable without 2PI counterterms.

Gradient expansion is consistent with equations of motion for propagators.

Abstract

We derive the quantum effective action up to second order in gradients and up to two-loop order for an interacting scalar field theory. This expansion of the effective action is useful to study problems in cosmological settings where spatial or time gradients are important, such as bubble nucleation in first-order phase transitions. Assuming spacetime dependent background fields, we work in Wigner space and perform a midpoint gradient expansion, which is consistent with the equations of motion satisfied by the propagator. In particular, we consider the fact that the propagator is non-trivially constrained by an additional equation of motion, obtained from symmetry requirements. At one-loop order, we show the calculations for the case of a single scalar field and then generalise the result to the multi-field case. While we find a vanishing result in the single field case, the one-loop second-order gradient corrections can be significant when considering multiple fields. As an example, we apply our result to a simple toy model of two scalar fields with canonical kinetic terms and mass mixing at tree-level. Finally, we calculate the two-loop one-particle irreducible (1PI) effective action in the single scalar field case, and obtain a nonrenormalisable result. The theory is rendered renormalisable by adding two-particle irreducible (2PI) counterterms, making the 2PI formalism the right framework for renormalization when resummed 1PI two-point functions are used in perturbation theory.