Gradient corrections to the quantum effective action

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.