Path integration and perturbation theory with complex Euclidean actions

The Euclidean path integral quite often involves an action that is not completely real, i.e. a complex action. This occurs when the Minkowski action contains $t$-odd $CP$-violating terms. This usually consists of topological terms, such as the Chern-Simons term in odd dimensions, the Wess-Zumino term, the $\theta$ term or Chern character in 4-dimensional gauge theories, or other topological densities. Analytic continuation to Euclidean time yields an imaginary term in the Euclidean action. It also occurs when the action contains fermions, the fermion path integral being in general a sum over positive and negative real numbers. Negative numbers correspond to the exponential of $i\pi$ and hence indicate the presence of an imaginary term in the 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, this procedure is not followed. Rather, only the real part of the Euclidean action is considered, and its critical points are used to define the perturbation theory, a procedure that can lead to incorrect results. In this article we present a simple example to illustrate this point. The example consists of $N$ scalar fields in $0+1$ dimensions interacting with a $U(1)$ gauge field in the presence of a Chern-Simons term. In this example 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 that the standard method does not give a correct perturbative expansion. The implications of our work include perturbation theory in the standard model which is $CP$-violating, but also for calculations in the presence of topological terms which have given rise to radical changes in the spectrum of the theory.