Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel

TL;DR

This paper introduces a novel anisotropic kernel and a symmetric discretization approach for the complex Langevin method, enabling stable real-time gauge theory simulations that surpass previous limitations, with potential implications for QCD phenomenology.

Contribution

The authors develop a new anisotropic kernel and a symmetric discretization scheme for complex Langevin equations, allowing stable simulations of non-Abelian gauge theories on complex time contours.

Findings

Achieved stable complex Langevin simulations of SU(2) Yang-Mills in 3+1D.

Validated results using multiple observables.

Enabled simulations on extended real-time contours exceeding inverse temperature.

Abstract

The complex Langevin (CL) method is a promising approach to overcome the sign problem that occurs in real-time formulations of quantum field theories. Using the Schwinger-Keldysh formalism, we study SU($N_c$) gauge theories with CL. We observe that current stabilization techniques are insufficient to obtain correct results. Therefore, we revise the discretization of the CL equations on complex time contours, find a time reflection symmetric formulation and introduce a novel anisotropic kernel that enables CL simulations on discretized complex time paths. Applying it to SU(2) Yang-Mills theory in 3+1 dimensions, we obtain unprecedentedly stable results that we validate using additional observables and that can be systematically improved. For the first time, we are able to simulate non-Abelian gauge theory on time contours whose real-time extent exceeds its inverse temperature. Thus, our approach may pave the way towards an ab-initio real-time framework of QCD in and out of equilibrium with a potentially large impact on the phenomenology of heavy-ion collisions.