Neural network interpolators for Wilson loops

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a specific, quiet song playing in a very noisy room. In the world of particle physics, that "song" is the energy of a quark and an antiquark (two fundamental particles) sitting still next to each other. The "noisy room" is the chaotic quantum foam of the universe, and the "listening device" is a mathematical tool called a Wilson loop.

For decades, physicists have struggled to hear this song clearly because, as time goes on, the noise drowns out the signal. This paper introduces a clever new solution: teaching a computer (a neural network) to build a better listening device.

Here is the story of how they did it, broken down into simple concepts:

1. The Problem: The Static Noise

In the old days, to measure the energy between these particles, physicists used a "straight line" of connections (like a taut string) to link them.

The Issue: This straight line is like a generic microphone. It picks up the main song (the ground state), but it also picks up a lot of static and background chatter (excited states).
The Consequence: As you try to listen longer to get a clearer picture, the static gets so loud you can't hear anything. It's like trying to hear a whisper in a hurricane.

2. The Solution: A Shape-Shifting Microphone

Instead of using a fixed, straight string, the author built a Neural Network Interpolator. Think of this not as a fixed string, but as a smart, shape-shifting microphone made of digital clay.

The Clay: The network starts with the straight string and a bunch of tiny loops (plaquettes) that can be attached to it.
The Sculptor: The neural network is the sculptor. It has the power to twist, turn, and rearrange these loops into any shape it wants.
The Rule: There is one strict rule: The shape must respect the "gauge symmetry" of the universe. Imagine this as a rule that says, "No matter how you rotate the room, the microphone must still work the same way." The network is built to obey this rule automatically.

3. The Training: Tuning the Radio

The authors didn't just guess the shape. They let the computer learn it through a process called training.

The Goal: The computer wants to find the shape that makes the "song" (the ground state) as loud as possible and the "static" (excited states) as quiet as possible.
The Scorecard: They used a "loss function," which is like a scorecard. If the computer picks a shape that hears the song clearly, the score goes down (good!). If it hears too much static, the score goes up (bad!).
The Twist: The computer didn't just learn to hear the main song. It was also taught to find other songs (excited states) that are hidden in the noise. It learned to create different "microphone shapes" that are perfectly tuned to different frequencies.

4. The Magic Trick: Orthogonal States

How do you hear two different songs at once without them mixing? You use orthogonal states.

Analogy: Imagine trying to listen to a violin and a cello in the same room. If you use one microphone, they blend together. But if you have two microphones tuned to completely different "directions" (orthogonal), one can hear only the violin, and the other only the cello.
The Result: The neural network automatically learned to build these "directional microphones." It found the shape for the main ground state, and then it found shapes for the excited states (hybrid particles) without the physicists having to tell it exactly what those shapes should look like.

5. The Outcome: A Clearer Symphony

When they tested this new method:

The Ground State: They got a crystal-clear picture of the static quark-antiquark potential (the main song).
The Excited States: They successfully isolated the "hybrid" states (particles where the glue between the quarks is vibrating). These are usually very hard to see, but the neural network found them automatically.
Efficiency: By using a technique called "multilevel algorithms," they could calculate these results much faster and with less noise, like using noise-canceling headphones on a jet engine.

Summary

In the past, physicists had to hand-craft the shapes of their measuring tools, often guessing wrong. This paper shows that we can teach a computer to design its own measuring tools.

The neural network acts like a master chef who, instead of following a fixed recipe, tastes the soup and automatically adjusts the spices until the flavor is perfect. In this case, the "flavor" is the energy of the particles, and the "spices" are the shapes of the loops. The result is a much clearer, more accurate understanding of how the fundamental forces of nature hold the universe together.

1. Problem Statement

In lattice Quantum Chromodynamics (QCD), extracting the static quark-antiquark potential from Wilson loops is hindered by a poor signal-to-noise ratio at large Euclidean times.

Ground State Extraction: Standard methods use "smearing" (e.g., APE smearing) or Coulomb gauge fixing to improve the overlap of the trial state with the ground state. However, smearing relies on hand-crafted shapes, and gauge fixing can introduce ambiguities (Gribov copies).
Excited State Extraction: Identifying excited states (such as hybrid states containing valence gluons) typically requires constructing trial states with specific, complex geometric shapes to generate specific quantum numbers. This is a manual and non-trivial process.
Goal: The paper aims to automate the construction of optimal interpolators for both ground and excited states using a neural network (NN) that learns the optimal spatial connection within the Wilson loop without manual bias regarding quantum numbers.

2. Methodology

A. Neural Network Architecture

The authors replace the straight spatial Wilson line in the Wilson loop definition with a neural network parametrization, $\tilde{S}(x, x+r, t)$ .

Gauge Equivariance: To preserve the physical properties of the Wilson loop, the NN is constructed using gauge-equivariant layers. This ensures that under a gauge transformation $G(x)$ , the NN output transforms exactly like the original Wilson line: $\tilde{S} \to G(x)\tilde{S}G^\dagger(x+r)$ .
Input Elements: The network starts with the straight Wilson line and all possible spatial plaquette insertions along the line.
Layer Types:
- Linear Layers: Weighted sums of inputs, with a bias term multiplied by the straight Wilson line to maintain equivariance.
- Bilinear Layers: Combine elements bilinearly (e.g., $\phi \phi^\dagger \phi$ ) to introduce non-linearity while preserving gauge properties.
- Convolutional Layers: Convolve elements from neighboring lattice sites using gauge links, effectively mixing spatial information.
Training Strategy: The network is trained using the AdamW optimizer. A "layer-wise" training approach is employed: the network is trained for several hundred epochs, then a new layer is inserted (initialized as an identity map) before the final layer, and training resumes. This prevents optimization instability when increasing network depth.

B. Loss Function and Optimization

The training objective is to maximize the overlap with specific energy eigenstates.

Single State: For a single state, the physical loss function $L_{phys}$ minimizes the weighted sum of Wilson loop expectation values over Euclidean time $t$ . The weights $W_t$ are tuned to focus on the regime where excited-state contamination is high (small $t$ ) but noise is manageable.
Multi-State (Excited States): To find excited states, the method generalizes to a Correlation Matrix $C_{ij}(t)$ $C_{ij} (t)$ constructed from $N$ $N$ output channels of the NN.
- Generalized Eigenvalue Problem (GEVP): The eigenvalues $\lambda_n(t, t_0)$ of the correlation matrix are used to extract energies $E_n$ .
- Loss Function: The loss is modified to sum over multiple eigenvalues $\lambda_n$ , weighted by factors $\propto e^{-0.6n}$ to prioritize lower states.
- Orthogonality: An explicit orthogonality term $L_{ortho}$ is added to the loss function to enforce that the learned interpolators are orthogonal to one another.

C. Simulation Setup

Theory: Quenched QCD (no dynamical fermions).
Lattice: $20^3 \times 40$ lattice sites with Wilson action ( $\beta = 6.281$ ).
Data: 6000 pre-sampled configurations.
Signal Improvement: The trained NN is frozen and used as a new interpolator. The multilevel algorithm is applied to the temporal Wilson lines to drastically reduce noise, allowing for high-precision extraction of excited states.

3. Key Contributions

Automated Interpolator Construction: The paper demonstrates that a gauge-equivariant neural network can automatically learn optimal interpolators for both the ground state and excited states without manual tuning of geometric shapes or quantum number constraints.
Gauge-Equivariant Layers: The specific construction of linear, bilinear, and convolutional layers that strictly obey the gauge transformation rules of static quark-antiquark pairs.
Stable Training Protocol: The introduction of a controlled layer-insertion strategy (identity initialization) and gradient clipping allows for the training of deep networks without the optimization process collapsing due to the large number of parameters.
Excited State Extraction: Successfully extracting hybrid states (excited states with valence gluons) by optimizing a multi-channel correlation matrix and solving the GEVP.

4. Results

Training Stability: Networks with $N_{hidden}=18$ showed the best performance. Larger networks ( $N_{hidden}=26$ ) suffered from memory issues and non-thermalized weight fluctuations, while smaller networks converged less effectively.
Effective Masses:
- Ground State ( $n=0$ ): The effective mass curve is nearly flat, recovering the well-known static quark-antiquark potential.
- Excited States ( $n=1, 2, 3$ ): The first excited states ( $n=1, 2$ ) appear as a degenerate doublet, consistent with the $\Pi_u$ hybrid state. The $n=3$ state resembles the next hybrid state.
- Small $r$ Behavior: As the separation $r \to 0$ , the first two excited states degenerate, converging to the same gluelump state, as theoretically expected.
Energy Differences: The energy difference $\Delta E_n(r) = E_n(r) - E_0(r)$ is a physical quantity that remains finite in the continuum limit, showing clear plateaus for the hybrid states.

5. Significance and Outlook

Paradigm Shift: This work moves away from "hand-crafted" smearing shapes toward data-driven, optimized interpolators. It proves that neural networks can discover complex quantum states (hybrids) that are difficult to isolate manually.
Scalability: The method is compatible with the multilevel algorithm, making it computationally efficient for high-precision calculations.
Future Directions: The authors propose extending this method to:
- Off-axis separations (non-axial Wilson loops).
- Full QCD (including dynamical fermions).
- Direct application to gluelumps and other exotic hadronic systems.

In conclusion, the paper establishes a robust framework for using neural networks to solve the spectral problem in lattice gauge theories, offering a powerful tool for exploring the spectrum of QCD beyond the ground state.