Lattice artifacts proportional to the quark mass in the QCD running coupling

This paper presents a two-loop lattice perturbative analysis of $\mathcal{O}(am)$ discretization artifacts in the QCD running coupling using the background field method with clover-improved Wilson fermions and Symanzik-improved gauge actions, providing general analytical results and numerical coefficients to improve the precision of strong-coupling determinations by removing mass-dependent cutoff effects.

The Gist

Imagine you are trying to measure the strength of a force that holds the smallest building blocks of the universe together (the strong force). Physicists call this the "strong coupling constant," or $\alpha_s$. To do this, they use a giant, invisible digital grid called a Lattice.

Think of this Lattice like a giant chessboard where the universe is played out. Instead of smooth, continuous space, everything happens on the squares of the board.

The Problem: The "Pixelated" Universe

Because the universe is forced onto a grid, it's not perfectly smooth. It's "pixelated." This pixelation creates tiny errors, called artifacts.

Usually, these errors are small. But there's a specific type of error that gets worse the heavier the particles are. Imagine trying to walk across a smooth floor versus walking across a floor covered in thick, heavy rugs. The heavier the rug (the heavier the quark), the more your steps get messed up.

In physics terms, these are $O(am)$ artifacts. They are errors proportional to the size of the grid square ($a$) multiplied by the mass of the particle ($m$). If you have heavy particles (like the "charm" or "bottom" quarks), these errors become huge, ruining your measurement of the strong force.

The Solution: A "Magic Filter"

To fix this, physicists need a special tool called an improvement coefficient (named $b_g$ in the paper). Think of this coefficient as a magic filter or a correction formula.

If you know exactly how the grid messes up your measurements, you can write a formula to subtract that messiness out.

• Old way: Scientists knew the first part of this formula (the "one-loop" calculation). It was like knowing the filter works for light rugs, but they weren't sure if it worked for heavy, thick rugs.
• New way: This paper calculates the second part of the formula (the "two-loop" calculation). This is the heavy-duty version of the filter that accounts for the thick rugs.

How They Did It: The "Background Field" Trick

Calculating this is incredibly hard because it involves summing up billions of tiny interactions. To make it manageable, the authors used a clever trick called the Background Field Method.

Imagine you are trying to study how wind blows through a forest.

1. The Hard Way: You try to simulate every single leaf moving, every branch swaying, and every gust of wind simultaneously. It's chaotic and impossible to calculate.
2. The Background Field Way: You pretend the forest (the background) is a solid, unchanging statue. You only simulate the wind (the quantum fluctuations) moving around the statue. Because the statue doesn't move, the math becomes much simpler, but you still get the exact answer for how the wind behaves.

The authors used this method to isolate the specific errors caused by heavy particles.

The Results: It Depends on the Grid

The paper found that the "magic filter" isn't the same for every type of grid.

• Some grids are simple squares (Wilson action).
• Some are more complex shapes designed to be smoother (Symanzik and Iwasaki actions).

The authors discovered that the correction needed depends heavily on which grid you are using.

• Analogy: Imagine you are correcting a blurry photo. If you took the photo with a cheap camera (simple grid), you need one type of sharpening filter. If you took it with a high-end camera (improved grid), you need a different filter.
• The paper provides the exact numbers for these different filters. They found that for the most advanced grids, the correction needed is actually larger than for the simple grids. This was a surprise! It means that even though advanced grids are "better" in many ways, they have their own unique quirks that must be fixed with this new formula.

Why Does This Matter?

Without this new formula, scientists trying to measure the strong force with heavy particles would be like a chef trying to bake a cake with a broken scale. They might think they have the right amount of sugar, but the heavy particles would throw off the measurement, leading to a "soggy" (inaccurate) result.

By providing this two-loop improvement coefficient, the authors have given the physics community a precise recipe to remove these errors. This allows for:

1. More accurate measurements of the strong force.
2. Better predictions for how heavy particles behave.
3. Higher precision in testing the Standard Model of physics.

In short, they built a better "error-correcting lens" so we can see the universe's fundamental forces more clearly, even when looking at the heaviest particles.

Technical Summary

1. Problem Statement

High-precision determinations of the strong coupling constant ($\alpha_s$) in Lattice QCD are increasingly limited by discretization artifacts, specifically those proportional to the quark mass ($O(am)$).

• Context: In practical lattice simulations, particularly those involving heavy quarks or scale evolution strategies, the dimensionless quantity $am_q$ can become large.
• The Issue: In mass-independent renormalization schemes, linear mass-dependent cutoff effects ($O(am)$) are typically absorbed into a redefined bare coupling $\tilde{g}_0^2$. This redefinition relies on an improvement coefficient, $b_g(g_0^2)$.
• The Gap: Prior to this work, $b_g(g_0^2)$ was only known to one-loop order in perturbation theory. This lack of higher-order precision introduces non-negligible systematic uncertainties in the extraction of $\alpha_s$, especially when using improved fermion actions (clover-improved Wilson fermions) and various gauge actions.

2. Methodology

The authors perform a two-loop lattice perturbative analysis to determine the mass-dependent components of the renormalization factor $Z_g$.

• Theoretical Framework:
  • Background Field Method (BFM): The calculation utilizes the BFM, where the gauge link $U_\mu$ is decomposed into a quantum field $Q_\mu$ and a background field $B_\mu$. This method preserves background gauge invariance, allowing the coupling renormalization to be derived solely from the two-point function of the background field ($Z_B$), avoiding the complexity of three-point functions. The relation is given by $Z_g = Z_B^{-1/2}$.
  • Lattice Actions:
    • Fermions: Clover-improved Wilson action with a free clover coefficient $c_{sw}$ and Wilson parameter $r=1$.
    • Gauge: Symanzik-improved gauge actions, including 1$\times$1 plaquettes and 1$\times$2 rectangles. The study covers three specific cases: Wilson plaquette, Tree-Level Symanzik (TLS), and Iwasaki actions.
• Computational Strategy:
  • Diagrammatics: The calculation involves evaluating one-particle-irreducible (1PI) two-point functions of the background field at non-zero fermion mass. This includes one-loop fermion loops and two-loop diagrams involving internal gluon exchanges and fermion mass counterterm insertions.
  • Handling Singularities: A major technical challenge is the infrared behavior of the integrands, which contain pole structures from massless gluon propagators and mass expansions. The authors demonstrate that while individual diagrams may exhibit severe divergences (e.g., higher-order poles), gauge symmetry ensures cancellations when diagrams are combined, softening singularities to integrable double poles.
  • Numerical Integration: Loop integrals are converted into finite sums over a discretized Brillouin zone (hypercubic lattice). The workflow involves:
    1. Pre-evaluating trigonometric factors.
    2. Exploiting symmetries (reflections, reordering) to reduce the summation domain.
    3. Hierarchical factorization of algebraic expressions to optimize inner loops.
    4. Infinite Volume Extrapolation: Results from finite lattice sizes are extrapolated to the infinite-volume limit using multiple functional Ansätze to control systematic errors.

3. Key Contributions

• First Two-Loop Determination: This work provides the first two-loop calculation of the improvement coefficient $b_g(g_0^2)$ for the QCD running coupling in the presence of massive quarks.
• Generalization: The results are derived for general numbers of colors ($N_c$) and flavors ($N_f$), and as a function of the clover coefficient $c_{sw}$.
• Action Dependence: The study explicitly quantifies how the improvement coefficient depends on the choice of the gluonic discretization (Wilson, TLS, Iwasaki), a factor previously unexplored at two loops.
• Consistency Checks: The authors verify that all external momentum dependence (logarithmic terms) cancels out when the perturbative expansion of $c_{sw}$ is substituted, ensuring the improvement coefficient is purely local and consistent with $O(a)$ improvement frameworks.

4. Key Results

The improvement coefficient is expanded as $b_g = b_g^{(1)} g_0^2 + b_g^{(2)} g_0^4 + O(g_0^6)$.

• One-Loop Coefficient ($b_g^{(1)}$):
  • Depends only on $N_f$ and $c_{sw}$, not on the gauge action or $N_c$.
  • For the tree-level clover value ($c_{sw}=1$), it simplifies to the known result: $b_g^{(1)} = 0.012000 N_f$.
• Two-Loop Coefficient ($b_g^{(2)}$):
  • Exhibits a complex dependence on $N_c$, $N_f$, $c_{sw}$, and the specific gauge action.
  • The authors provide explicit numerical formulas for Wilson, TLS, and Iwasaki actions.
• Numerical Values for $N_c=3$:
  When substituting known one-loop values for $c_{sw}$, the total coefficient $b_g$ for different actions is:
  • Wilson: $N_f (0.012000 g_0^2 - 0.020067 g_0^4)$
  • TLS: $N_f (0.012000 g_0^2 - 0.025347 g_0^4)$
  • Iwasaki: $N_f (0.012000 g_0^2 - 0.04201 g_0^4)$
• Observations:
  • The two-loop corrections ($g_0^4$ term) are substantial, often larger in magnitude than the one-loop term.
  • The magnitude of the two-loop correction increases significantly when moving from the unimproved Wilson action to improved actions (TLS and Iwasaki), indicating that cutoff effects are highly sensitive to the gauge action choice.

5. Significance

• Reduced Systematic Uncertainty: By providing the two-loop term, this work allows for a more accurate definition of the improved bare coupling $\tilde{g}_0^2$. This is crucial for reducing systematic errors in high-precision $\alpha_s$ determinations, particularly in simulations with heavy quarks where $am_q$ is large.
• Benchmark for Non-Perturbative Studies: The results serve as a rigorous perturbative benchmark for future non-perturbative determinations of $b_g$.
• Action Sensitivity: The findings highlight that the choice of gauge action significantly impacts mass-dependent artifacts. Researchers must account for these specific two-loop corrections when comparing results across different lattice formulations.
• Future Outlook: While perturbative improvement alone may not capture all effects at current lattice spacings, combining these two-loop results with future non-perturbative inputs is expected to significantly enhance the control over mass-dependent cutoff effects in Lattice QCD.