Justification of the complex Langevin method with the gauge cooling procedure

TL;DR

This paper provides a rigorous theoretical justification for the complex Langevin method with gauge cooling, demonstrating its validity for gauge-invariant observables and its potential to improve convergence in complex action simulations.

Contribution

It formulates gauge cooling as an extra term in the Langevin equation and proves it does not affect gauge-invariant observables, validating its use for complex Langevin simulations.

Findings

Gauge cooling can be formulated as an extra term in the Langevin equation.

The extra term does not affect the Fokker-Planck equation for gauge-invariant observables.

Gauge cooling satisfies convergence conditions for the complex Langevin method.

Abstract

Recently there has been remarkable progress in the complex Langevin method, which aims at solving the complex action problem by complexifying the dynamical variables in the original path integral. In particular, a new technique called the gauge cooling was introduced and the full QCD simulation at finite density has been made possible in the high temperature (deconfined) phase or with heavy quarks. Here we provide a rigorous justification of the complex Langevin method including the gauge cooling procedure. We first show that the gauge cooling can be formulated as an extra term in the complex Langevin equation involving a gauge transformation parameter, which is chosen appropriately as a function of the configuration before cooling. The probability distribution of the complexified dynamical variables is modified by this extra term. However, this modification is shown not to affect the Fokker-Planck equation for the corresponding complex weight as far as observables are restricted to gauge invariant ones. Thus we demonstrate explicitly that the gauge cooling can be used as a viable technique to satisfy the convergence conditions for the complex Langevin method. We also discuss the "gauge cooling" in 0-dimensional systems such as vector models or matrix models.