The argument for justification of the complex Langevin method and the condition for correct convergence

TL;DR

This paper critically examines the justification of the complex Langevin method, identifying a key condition related to the drift term's probability distribution that determines its reliability in complex-action problems.

Contribution

It clarifies the necessary and sufficient condition for the method's correctness and introduces a new gauge cooling approach to improve its applicability.

Findings

The probability of the drift term must decay exponentially or faster for the method to be justified.

The condition effectively distinguishes trustworthy results from unreliable ones.

A new gauge cooling technique can directly reduce the drift term's magnitude.

Abstract

The complex Langevin method is a promising approach to the complex-action problem based on a fictitious time evolution of complexified dynamical variables under the influence of a Gaussian noise. Although it is known to have a restricted range of applicability, the use of gauge cooling made it applicable to various interesting cases including finite density QCD in certain parameter regions. In this paper, we revisit the argument for justification of the method. In particular, we point out a subtlety in the use of time-evolved observables, which play a crucial role in the previous argument. This requires that the probability of the drift term should fall off exponentially or faster at large magnitude. We argue that this is actually a necessary and sufficient condition for the method to be justified. Using two simple examples, we show that our condition tells us clearly whether the results obtained by the method are trustable or not. We also discuss a new possibility for the gauge cooling, which can reduce the magnitude of the drift term directly.