Complex Langevin kernels for real-time SU(N) lattice gauge theory in 3+1 dimensions

Abstract

We investigate complex Langevin (CL) dynamics for real-time SU($N$) gauge theories in 3+1 dimensions and introduce new kernel modifications tailored for lattice gauge theory (LGT). Using the anisotropic kernel as a baseline, we demonstrate that on a rectangular regularized Schwinger-Keldysh time contour, stable simulations are possible for pure SU(2) gauge theory up to a real-time extent of $2β$ at a bare coupling $g=0.5$, where $β$ denotes the inverse temperature. We show that a contour-dependent phase kernel extends this reach to $3β$, while a local, field-dependent kernel based on traced plaquettes achieves $3.5β$. These constructions represent the first application of such kernels to LGTs, especially in 3+1 dimensions. To validate the approach, we generalize the CL correctness criterion based on the drift magnitude to kernelled dynamics, providing a practical tool to assess correct convergence. Our results substantially advance the applicability of CL to real-time simulations in Minkowski space, establishing a framework for controlled studies of non-Abelian gauge theories at unprecedented time scales. A forthcoming paper applies these developments to the computation of unequal-time correlation functions.

Real-time non-Abelian lattice gauge simulations with complex Langevin (CL) dynamics can become unstable or converge incorrectly as the Schwinger–Keldysh contour is extended. This paper investigates whether contour-dependent and field-dependent kernels can enlarge the accessible time window in $3+1$ dimensions while preserving correct expectation values. Two kernel constructions are introduced: a field-independent phase kernel with conjugate phases on the forward and backward contour branches, and a local, gauge-invariant field-dependent kernel affine in traced plaquettes. Both supplement an existing anisotropic kernel and gauge cooling. For pure SU(2) theory at bare coupling $g=0.5$, anisotropy $\xi=32$, and fixed contour tilt $\tan\alpha=0.01$, a rectangular contour supports simulations to $t_{\max}=2\beta$ with the baseline kernel. The phase kernel extends this to $3\beta$ on a $224\times16^3$ lattice; the plaquette-dependent kernel reaches $3.5\beta$ on a $256\times16^3$ lattice, where $\beta$ is the inverse temperature. Validation uses a proposed extension of the exponential drift-tail correctness criterion to kernelled CL, bounded unitarity norms, and agreement of spatial Wilson loops with Euclidean benchmarks. A stochastic Runge–Kutta update accommodates field-dependent diffusion. The demonstrations use restricted parameter scans, one coupling, and a finite contour tilt; neither the zero-tilt limit nor continuum extrapolation is performed. Kernel selection remains empirical, without a general constructive convergence guarantee. Several alternative kernels fail to improve stability. Unequal-time correlation functions and applications to physical real-time observables are deferred to a forthcoming study.