# How does Gauge Cooling Stabilize Complex Langevin?

## TL;DR

This paper investigates how gauge cooling stabilizes the complex Langevin method by analyzing exact solutions and stochastic equations, demonstrating its effectiveness in localizing distributions in an $SU(2)$ model.

## Contribution

It provides a detailed analysis of the gauge cooling mechanism, deriving stochastic equations and showing how it prevents excursions in the complex Langevin process.

## Key findings

- Gauge cooling minimizes Frobenius norm of link variables.
- It helps form localized distributions in the Polyakov loop model.
- Gauge cooling prevents excursions away from the real axis.

## Abstract

We study the mechanism of the gauge cooling technique to stabilize the complex Langevin method in the one-dimensional periodic setting. In this case, we find the exact solutions for the gauge transform which minimizes the Frobenius norm of link variables. Thereby, we derive the underlying stochastic differential equations by continuing the numerical method with gauge cooling, and thus provide a number of insights on the effects of gauge cooling. A specific case study is carried out for the Polyakov loop model in $SU(2)$ theory, in which we show that the gauge cooling may help form a localized distribution to guarantee there is no excursion too far away from the real axis.