hep-latApr 14, 2026

Machine learning for four-dimensional SU(3) lattice gauge theories

Authors: Urs Wenger

Organizations: Institute for Theoretical Physics Albert Einstein Center for Fundamental Physics University of Bern CH-3012 Bern Switzerland

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Abstract

In this review I summarize how machine learning can be used in lattice gauge theory simulations and what ap-proaches are currently available to improve the sampling of gauge field configurations, with a focus on applications in four-dimensional SU(3) gauge theories. These include approaches based on generative machine-learning models such as (stochastic) normalizing flows and diffusion processes, and an approach based on renormalization group (RG) transformations, more specifically the machine learning of RG-improved gauge actions using gauge-equivariant convolutional neural networks. In particular, I present scaling results for a machine-learned fixed-point action in four-dimensional SU(3) gauge theory towards the continuum limit. The results include observables based on the classically perfect gradient-flow scales, which are free of tree-level lattice artefacts to all orders, and quantities related to the static potential and the deconfinement transition.

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Figure 1: Illustration of the continuum limit for asymptotically free lattice field theories: the lattice spacing decreases from right to left, a<a′<a′′a<a^{{}^{\prime}}<a^{{}^{\prime\prime}} , as the coupling decreases, g<g′<g′′g<g^{{}^{\prime}}<g^{{}^{\prime\prime}} , or equivalently β>β′>β′′\beta>\beta^{{}^{\prime}}>\beta^{{}^{\prime\prime}} increases. In the limit β→∞\beta\rightarrow\infty the lattice spacing a→0a\rightarrow 0 vanishes, which for a fixed physical length scale ξ\xi is equivalent to ξ/a→∞\xi/a\rightarrow\infty , i.e., the continuum limit is realized as a second-order (continuous) phase transition.
Figure 2: Illustration of the forward diffusion process where stochastic noise is added and the backward denoising process employed in generative diffusion models. Figure taken from Ref. [ 51 ] .
Figure 3: Illustration of the stochastic non-equilibrium MCMC scheme. Left plot: Configurations obtained on a lattice with open boundary conditions (OBC) using standard MCMC updates are connected via non-equilibrium MCMC updates to configurations on a lattice with periodic boundary conditions (PBC). Right plot: Connecting OBC to PBC on a localized defect using machine-learned stochastic normalizing flows on the links and plaquettes affected by the defect (plot taken from [ 15 ] ).
Figure 4: Illustration of the RGT flow. The plane at β→∞\beta\rightarrow\infty represents the critical surface where all the irrelevant couplings {cα}\{c_{\alpha}\} flow to the fixed point (FP). An action perturbed from the FP in direction of the only relevant coupling g2∼1/βg^{2}\sim 1/\beta flows, under repeated (continuous) RGTs, along the renormalized trajectory (RT) defining quantum perfect actions with no lattice artefacts at finite lattice spacing. The action along the straight line in direction 1/β1/\beta emanating from the FP defines a classically perfect action with no tree-level lattice artefacts O(a2n){\cal O}(a^{2n}) for all nn .
Figure 5: Illustrations of the L-CNN: the convolutional layer (L-Conv) in the left panel parallel transports gauge-covariant objects to a common position (red dot), where they are combined in the bilinear layer (L-Bilin) in the middle panel and eventually mapped to gauge invariant objects with the trace layer (L-Tr) in the right panel. The lower row depicts simple examples of applying the sequence of the layers above.
Figure 6: Illustrations of the architecture search. Shown are the relative action error ( top row ) and derivative error ( bottom row ) of the L-CNN w.r.t. to the exact FP values as a function of the architecture parameters, i.e., the number of layers (left plots), the total number of channels ( middle plots ), and the kernel size ( right plots ).
Figure 8: Left plot: Comparison of various PDFs obtained from AIC-weighted continuum limits of the ratio t0.5/w0.52t_{0.5}/w_{0.5}^{2} using a variety of fit functions and ranges. Right plot: Comparison of continuum predictions for four-dimensional SU(3) gauge theory from MC simulations using either the FP, Wilson or tree-level Symanzik improved lattice action. The β\beta -function results are rescaled by a factor of 50 for visibility.
Figure 9: Left plot: Static quark-antiquark potential obtained on lattices with lattice spacings ranging between 0.08 fm ≤a≤0.3\leq a\leq 0.3 fm. Right plot: Thermodynamic limit of the critical coupling βc\beta_{c} obtained from the position of the Polyakov loop susceptibility at a lattice spacing of a≃0.3a\simeq 0.3 fm corresponding to a temporal lattice extent of Lt=2L_{t}=2 .
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