Path integration and perturbation theory with complex Euclidean actions

TL;DR

This paper investigates perturbation theory in complex Euclidean actions, demonstrating that standard methods often fail and proposing a contour deformation approach with an explicit 0+1D example involving scalar fields and a Chern-Simons term.

Contribution

It introduces a method to correctly perform perturbation theory with complex actions by deforming the integration contour, illustrated through an explicit example where standard methods are shown to be incorrect.

Findings

Standard perturbation methods fail for complex actions.

Contour deformation to complex critical points is necessary.

Explicit example confirms the failure of traditional approaches.

Abstract

The Euclidean path integral quite often involves an action that is not completely real {\it i.e.} a complex action. This occurs when the Minkowski action contains $t$-odd CP-violating terms. Analytic continuation to Euclidean time yields an imaginary term in the Euclidean action. In the presence of imaginary terms in the Euclidean action, the usual method of perturbative quantization can fail. Here the action is expanded about its critical points, the quadratic part serving to define the Gaussian free theory and the higher order terms defining the perturbative interactions. For a complex action, the critical points are generically obtained at complex field configurations. Hence the contour of path integration does not pass through the critical points and the perturbative paradigm cannot be directly implemented. The contour of path integration has to be deformed to pass through the complex critical point using a generalized method of steepest descent, in order to do so. Typically, what is done is that only the real part of the Euclidean action is considered, and its critical points are used to define the perturbation theory. In this article we present a simple 0+1-dimensional example, of $N$ scalar fields interacting with a U(1) gauge field, in the presence of a Chern-Simons term, where alternatively, the path integral can be done exactly, the procedure of deformation of the contour of path integration can be done explicitly and the standard method of only taking into account the real part of the action can be followed. We show explicitly that the standard method does not give a correct perturbative expansion.