Toward precise $\xi$ gauge fixing for the lattice QCD

This paper proposes and validates an empirical precision extrapolation method to achieve high-precision $\xi$-gauge fixing in lattice QCD, successfully reproducing $\xi$-dependent renormalization constants for local quark bilinear operators up to $\xi \sim 1$ with 0.3% accuracy.

The Gist

Imagine you are trying to take a perfectly clear photograph of a tiny, invisible particle inside a proton. In the world of physics, this particle is a "quark." To get a clear picture, you need to set up your camera (the mathematical framework) in a very specific way.

In the paper you provided, the $\chi$QCD Collaboration is trying to solve a problem with how they "focus" their camera. Here is the story of what they did, explained simply.

The Problem: The "Fuzzy" Camera Lens

In physics, there are different ways to set up the rules for how particles behave, called "gauges." Think of these like different camera filters.

• The Landau Gauge ($\xi = 0$): This is the "standard filter" everyone uses. It's very easy to focus, and the picture comes out sharp.
• The $\xi$-Gauge: This is a different filter that physicists want to use to see things from a new angle. However, trying to focus this specific filter is incredibly difficult. As you try to make the picture sharper, the camera starts to shake violently. No matter how much you try, you can't get a perfectly clear image; the picture always stays a bit blurry.

For years, scientists were stuck using only the "standard filter" (Landau gauge) because the "new filter" ($\xi$-gauge) was too hard to use precisely. They wanted to use the new filter to understand how particles behave when they aren't perfectly stable (off-shell), but the blurriness made the data unreliable.

The Discovery: A Universal "Blur" Rule

The team noticed something interesting while looking at their blurry photos. They found that the amount of "blur" (mathematical error) didn't happen randomly. Instead, it followed a predictable pattern, like a specific type of fog that thickens in a known way as you zoom in.

They realized: "If we know exactly how the blur behaves at low quality, we can mathematically predict what the picture would look like if it were perfectly sharp."

They call this "Precision Extrapolation." It's like looking at a low-resolution photo, measuring exactly how the pixels are distorted, and then using a computer algorithm to reconstruct the high-resolution image that would have existed if the camera had been perfect.

The Experiment: Testing the Fix

To prove their idea worked, they did two things:

1. The Practice Run (Landau Gauge): First, they tested their "blur correction" method on the easy-to-focus Landau gauge. They intentionally took photos with a very blurry lens (low precision) and used their math to guess what the sharp photo would look like.

   • Result: When they compared their "guessed" sharp photo to an actual photo taken with a super-sharp lens, they matched almost perfectly (within 0.3%). This proved their math was sound.

2. The Real Challenge ($\xi$-Gauge): Next, they applied this same "blur correction" method to the difficult $\xi$-gauge. They took their blurry, hard-to-focus photos and used the formula to extrapolate the "perfect" result.

   • Result: The corrected results matched the theoretical predictions from advanced physics calculations (perturbation theory) with high accuracy.

The Analogy: The Noisy Radio

Think of the $\xi$-gauge like a radio station that is very far away and full of static (noise).

• Normally, you can't hear the music clearly because the static is too loud.
• The authors realized that the static isn't random; it follows a specific rhythm.
• They developed a "noise-canceling" formula. Instead of trying to build a better radio tower (which is hard and expensive), they just listened to the static, analyzed its pattern, and mathematically subtracted it to reveal the clear music underneath.

The Conclusion

The paper claims that they have successfully created a method to get high-precision results from a gauge (camera setting) that was previously too difficult to use.

• What they achieved: They can now study quarks using the $\xi$-gauge with an accuracy of about 0.3%, which is good enough to trust the results.
• The Limit: Their "noise-canceling" trick works well for certain settings, but if the "static" gets too loud (very large $\xi$ values), the method breaks down because there isn't enough clear signal left to analyze.
• The Takeaway: They didn't build a better camera; they built a better way to fix the photos taken with the old, shaky camera. This allows physicists to explore new angles of particle physics that were previously blocked by technical difficulties.

Technical Summary

Technical Summary: Toward Precise $\xi$ Gauge Fixing for the Lattice QCD

Problem Statement
Lattice Quantum Chromodynamics (QCD) provides a first-principles framework for solving the theory, yet its application to off-shell partons (quarks and gluons) has been largely restricted to the Landau gauge ($\xi=0$). While general $\xi$-gauge fixing was proposed years ago, its practical application to realistic configurations is hindered by severe convergence issues, particularly at large $\xi$ and/or strong gauge coupling $g$. Achieving high-precision gauge fixing in these regimes is numerically demanding, often failing to converge even after extensive iterations. This limitation prevents a systematic understanding of how parton properties depend on the gauge parameter $\xi$ and hinders benchmark comparisons between lattice QCD and functional approaches (such as Dyson-Swinger Equations and functional renormalization group methods) that often utilize non-zero $\xi$.

Methodology
The authors propose an empirical precision extrapolation method to approximate high-precision $\xi$-gauge fixing. The approach relies on the observation that off-shell parton matrix elements exhibit a universal power-law dependence on gauge-fixing precision ($\theta$), a phenomenon previously identified for non-local operators in the Landau gauge.

1. Gauge Fixing Implementation: The study employs the "over-relaxation" method to enforce the $\xi$-gauge condition $\Delta(x) = \Lambda(x)$ on the lattice. The authors investigate three definitions of the bare gauge coupling $g_0$ (naive, full tadpole-improved, and $u_0$-approximation) to define the effective gauge parameter $\xi$. They find that the $u_0$-approximation definition ($g_0^{(c)}$) provides the most consistent effective $\xi$ when compared to perturbative expectations.
2. Precision Extrapolation: The core methodology involves computing observables at various achievable gauge-fixing precisions $\theta$ (where $\theta$ is the residual of the gauge-fixing condition) and extrapolating to the exact limit ($\theta \to 0$) using the empirical ansatz:
   $$X(\theta) = X_{fit} e^{-c(X) \theta^{n(X)}}$$
   Here, $X_{fit}$ represents the extrapolated exact result, while $c$ and $n$ are fit parameters.
3. Validation Strategy:
   • Landau Gauge ($\xi=0$): The method is first validated in the Landau gauge, where high-precision results ($\theta \sim 10^{-14}$) are achievable. The authors verify that the power-law ansatz holds for local quark bilinear operators (scalar $S$ and tensor $T_{\mu\nu}$) across different momenta and lattice spacings.
   • $\xi$-Gauge ($\xi > 0$): The method is then applied to $\xi$-gauges up to $\xi \sim 1$. Since high-precision gauge fixing is unattainable here (residuals plateau at $\theta \sim 10^{-4.5}$), the extrapolation is used to recover the high-precision limit.
   • Consistency Check: The non-perturbative $\xi$-dependence of the RI/MOM renormalization constants ($Z_{S,T}^{RI}$) is compared against known 3-loop perturbative results. The ratio of lattice results to perturbative predictions should be unity (up to discretization errors) if the gauge fixing and extrapolation are successful.

Key Results

• Universal Power Law: The study confirms that the deviation of renormalization constants from their exact values scales as $\sim \theta^{0.5}$ (i.e., $n \approx 0.5$) for local operators in the Landau gauge, consistent with findings for non-local operators. This scaling holds across different fermion discretizations (overlap and clover) and momentum scales.
• Extrapolation Accuracy in Landau Gauge: Using relatively coarse gauge-fixing data ($\theta \ge 10^{-5}$), the extrapolation recovers high-precision results ($\theta \sim 10^{-14}$) with deviations of less than 0.2–0.3%. This demonstrates the efficacy of the method in the Landau gauge.
• $\xi$-Gauge Validation: For $\xi \le 1$, the precision-extrapolated RI/MOM renormalization constants for scalar and tensor operators reproduce the $\xi$-dependent perturbative results at the 0.3% level.
• Discretization Errors: The authors identify that discretization errors in the gauge-fixing condition introduce an $O(a^2\mu^2)$ dependence in the ratio of lattice to perturbative results. By defining an effective gauge parameter $\tilde{\xi} = \xi a^2\mu^2 / (4\sin^2(a\mu/2))$, they suppress this dependence to the 1% level or less for $a^2\mu^2 \le 10$.
• Convergence Limits: The study notes that for $\xi \gtrsim 1.2$, the minimal achievable residual $\theta_{min}$ increases exponentially ($\theta_{min} \approx 10^{2.9\xi - 7.3}$), drastically reducing the number of viable data points and rendering the extrapolation unreliable.

Significance and Claims
The paper claims to provide a practical solution to the long-standing numerical challenge of $\xi$-gauge fixing on the lattice. By establishing an empirical power-law dependence of physical quantities on gauge-fixing precision, the authors introduce a method to approximate high-precision $\xi$-gauge results using data from lower-precision, numerically feasible simulations.

The authors emphasize that while the $\xi$-gauge fixing problem is mathematically ill-posed (unlike the Landau gauge), the precision extrapolation serves as a robust practical tool rather than a rigorously proven limit. The successful reproduction of perturbative $\xi$-dependence at the 0.3% level validates the procedure for $\xi \le 1$.

The work is significant because it:

1. Enables the first systematic lattice study of $\xi$-dependent off-shell parton properties beyond the Landau gauge.
2. Provides a foundation for benchmarking lattice QCD against functional methods (DSE/fRG) in non-Landau gauges.
3. Offers a pathway to investigate the sensitivity of parton features to gauge choices, potentially improving phenomenological models of non-perturbative QCD.
4. Identifies that improved gauge-fixing conditions (beyond the standard discretization) are necessary to further suppress discretization errors in future studies.

The authors conclude that this method paves the way for systematic investigations of $\xi$-dependent quark and gluon propagators and their non-perturbative infrared interactions under controlled gauge-fixing uncertainties, though they remain modest about the applicability of the method for $\xi \gtrsim 1.2$ without more sophisticated algorithms.