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(Nc) 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.
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Acknowledgments
The authors would like to thank D. Alvestad and D. Sexty for valuable discussions regarding the basic foundations of complex Langevin and rescalings as part of the kernel freedom. We are further grateful to J.M. Pawlowski, A. Rebhan and F.P.G. Ziegler for very useful discussions and comments, and to A. Ipp for technical input regarding code development. This research was funded by the Austrian Science Fund (FWF) project P 34455-N. Moreover, Paul Hotzy expresses his gratitude to the Doktoratskolleg Particles and Interactions (DK-PI, FWF doctoral program No. W-1252-N27). The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC).
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Boguslavski, K., Hotzy, P. & Müller, D.I. Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel. J. High Energ. Phys. 2023, 11 (2023). https://doi.org/10.1007/JHEP06(2023)011
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DOI: https://doi.org/10.1007/JHEP06(2023)011