Machine-learned RG-improved gauge actions and classically perfect gradient flows

TL;DR

The paper addresses lattice artifacts in nonperturbative gauge theories by using a machine-learned fixed-point (FP) lattice action for SU($3$) gauge theory. A gauge-equivariant CNN parameterizes the FP action, enabling a classically perfect gradient flow with no tree-level lattice artifacts to all orders in the lattice spacing $a$. Gradient-flow observables show discretization effects below 1% up to $a \approx 0.14$ fm, enabling continuum physics to be extracted from relatively coarse lattices. Comparisons with Wilson and Symanzik actions demonstrate consistent continuum limits and illustrate that ML-based FP actions can significantly reduce lattice artifacts, suggesting a path toward quantum perfect actions.

Abstract

Extracting continuum properties of quantum field theories from discretized spacetime is challenging due to lattice artifacts. Renormalization-group (RG)-improved lattice actions can preserve continuum properties, but are in general difficult to parameterize. Machine learning (ML) with gauge-equivariant convolutional neural networks provides a way to efficiently describe such actions. We test a machine-learned RG-improved lattice gauge action, the classically perfect fixed-point (FP) action, for four-dimensional SU(3) gauge theory through Monte Carlo simulations. We establish that the gradient flow of the FP action is free of tree-level discretization effects to all orders in the lattice spacing, making it classically perfect. This allows us to test the quality of improvement of the FP action, without introducing additional artifacts. We find that discretization effects in gradient-flow observables are highly suppressed and less than 1% up to lattice spacings of 0.14 fm, allowing continuum physics to be extracted from coarse lattices. The quality of improvement achieved motivates the use of the FP action in future gauge theory studies. The advantages of ML-based parameterizations also highlight the possibility of realizing quantum perfect actions in lattice gauge theory.

Figures (6)

• Figure 1: Continuum-limit extrapolations for the ratios $t_{0.3}/w_{0.3}^2$ and $t_{0.5}/t_{0.3}$. Results from Wilson and Symanzik MC simulations are shown using plaquette and clover discretizations of the action density.
• Figure 2: The $\beta$-function of the GF renormalized coupling at $g_{\rm GF}^2 = 15.79$ with highly suppressed lattice artifacts in the results for the FP compared to the Wilson and Symanzik gauge actions.
• Figure 3: The 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 shows very good consistency. The $\beta$-function results are rescaled by a factor of 50 for visibility.
• Figure 4: Continuum-limit extrapolations of FP, Wilson and Symanzik action measurements, for the ratios $t_{0.4}/t_{0.3}$ ( top), $t_{0.4}/w_{0.4}^2$ ( middle), and $t_{0.5}/w_{0.5}^2$ ( bottom). Lattice artifacts are small with the FP action and gradient flow (below 1% at $a \simeq 0.14$ fm). A variety of polynomial fits with different ranges and highest power $(a^2/t_0)^n$ are shown, shaded according to their respective AIC weight.
• Figure 5: Comparison of various PDFs for the quantity $\beta(g^2=15.79$ ( top), $t_{0.5}/t_{0.3}$ ( middle), $t_{0.4}/t_{0.3}$ ( bottom). Below the plot, the median, 16$^{\rm th}$ and 84$^{\rm th}$ percentiles are marked.