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

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.

28 pages + references and appendices, 7 figures, 1 table; v2: revised, data and plot files attached, published version

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• Simulating the Berezinskii-Kosterlitz-Thouless Transition with Complex Langevin
• Correctness criteria for complex Langevin
• Combining complex Langevin dynamics with score-based and energy-based diffusion models