Wilson loops with neural networks

TL;DR

The paper tackles the problem of extracting static energies from Wilson loops in lattice QCD, hindered by excited-state contamination and noisy signals. It introduces a gauge-equivariant neural-network framework that replaces the spatial Wilson line with a learnable, gauge-invariant interpolator, trained via a physics-informed loss based on the transfer-matrix formalism and spectral representation. The approach yields competitive ground-state results compared to Coulomb-gauge Wilson lines, improves signals for the static force, and automatically identifies excited-state interpolators (notably the $ ext{$oldPi_u$}$ hybrid) while remaining compatible with the multilevel algorithm. This method provides a robust, gauge-invariant path-integral observable with broad applicability to generalized Wilson loops and Born-Oppenheimer EFT calculations in lattice QCD.

Abstract

Wilson loops are essential objects in QCD and have been pivotal in scale setting and demonstrating confinement. Various generalizations are crucial for computations needed in effective field theories. In lattice gauge theory, Wilson loop calculations face challenges, including excited-state contamination at short times and the signal-to-noise ratio issue at longer times. To address these problems, we develop a new method by using neural networks to parametrize interpolators for the static quark-antiquark pair. We construct gauge-equivariant layers for the network and train it to find the ground state of the system. The trained network itself is then treated as our new observable for the inference. Our results demonstrate a significant improvement in the signal compared to traditional Wilson loops, performing as well as Coulomb-gauge Wilson-line correlators while maintaining gauge invariance. Additionally, we present an example where the optimized ground state is used to measure the static force directly, as well as another example combining this method with the multilevel algorithm. Finally, we extend the formalism to find excited-state interpolators for static quark-antiquark systems. To our knowledge, this work is the first study of neural networks with a physically motivated loss function for Wilson loops.

Figures (17)

• Figure 1: Sketch of the actions of linear (top), bilinear (middle), and convolutional (bottom) layers, showing how gauge equivariance is built in. The black dots represent static quark-antiquark pairs, while the arrowed lines show the path connecting them. The linear layer creates clover-leaf-like insertions; the bilinear layer generates structure along the separation axis; and the convolutional layer creates structure orthogonal to the separation axis.
• Figure 2: Comparing the training history for the ConvBilin training procedure. Blue vertical lines indicate the insertion of a convolutional layer, while the red lines indicate the insertion of a bilinear layer. The $x$-axis represents the Epoch, i.e., the training time, while the $y$-axis displays the value for $L^\mathrm{phys}$ only (not the full loss function). A lower value for $L^\mathrm{phys}$ corresponds to a more optimized result.
• Figure 3: The training histories for each $r$ of a single network. Blue vertical lines indicate the insertion of a convolutional layer, while the red lines indicate the insertion of a bilinear layer. The $x$-axis represents the Epoch, i.e., the training time, while the $y$-axis displays the value for $L^\mathrm{phys}$ only (not the full loss function). A lower value for $L^\mathrm{phys}$ corresponds to a more optimized result.
• Figure 4: The training histories for each $r$ of three different network architecture concepts. We obtain similar peak structures for the "ConvExp" as for the previous "ConvBilin" network. In contrast, the "ConvLimNeighbor" exhibits stability over a wide range in the training history.
• Figure 5: The bare, normalized correlator from the final measurement for $r=4$ and $r=7$ in a logarithmic plot. The data points are connected with lines to guide the eye.