Towards determination of the strong coupling $\alpha_s(m_Z)$ from four-flavor lattice QCD using the continuous $\beta$-function method

This paper reports the current status of a program aiming to determine the strong coupling constant $\alpha_s(m_Z)$ with approximately 0.3% precision using four-flavor lattice QCD simulations in the infinite volume gradient flow scheme, while analyzing cutoff and finite-mass effects and outlining the necessary extrapolations for the final result.

The Gist

The Big Picture: Measuring the "Glue" of the Universe

Imagine the universe is built out of tiny Lego bricks. Some bricks stick together easily, but others are held together by a super-strong, invisible glue. In physics, this glue is called the Strong Force, and the strength of that glue is measured by a number called the Strong Coupling ($\alpha_s$).

This number is crucial. It tells us how particles behave inside protons and neutrons. If we know this number perfectly, we can predict how the universe works with extreme precision. However, right now, our measurement of this "glue strength" is a little fuzzy. The scientists in this paper want to sharpen that measurement from a "rough sketch" to a "high-definition photograph."

The Problem: The "Pixelated" Universe

To measure this glue, the scientists use Lattice QCD. Imagine trying to map a smooth, curved coastline using a grid of square tiles.

• The Grid: The "lattice" is a 3D grid of points where they simulate the particles.
• The Problem: Because the grid is made of squares (pixels), it distorts the smooth curves of reality. This is called a discretization error. It's like trying to measure the circumference of a circle using a ruler made of square blocks; your measurement will be slightly off.

Additionally, the simulation uses "fake" particles that are slightly heavy (massive) to make the math easier to run on computers. But in the real world, these particles are massless. This creates a mass error.

The Solution: The "Continuous Beta-Function" Method

The team is using a clever technique called the Continuous Beta-Function Method. Here is how it works, using an analogy:

The Analogy: The Rolling Ball on a Hill
Imagine the strength of the strong force is like a ball rolling down a hill.

• The Beta-Function is the map that tells you exactly how steep the hill is at every single point.
• If you know the shape of the hill perfectly, you can calculate exactly where the ball started and where it will end up.
• In physics, if you know how the "glue strength" changes as you zoom in or out (change the energy scale), you can calculate its exact value at the scale of the Z-boson (a heavy particle found in nature).

What This Paper Actually Did

This paper is a "status report." The scientists haven't finished the whole puzzle yet, but they have built the most important pieces of the machine. Here are the three main steps they tackled:

1. Smoothing Out the Pixels (Tree-Level Cutoff Effects)

• The Issue: Because their simulation grid is made of squares, the "glue" looks a bit jagged.
• The Fix: They applied a mathematical "filter" (called Tree-Level Normalization).
• The Result: Think of it like applying a "smooth" filter in Photoshop. Before the filter, the image looked blocky and depended on which specific "pixel" you looked at. After the filter, the image became smooth and consistent, regardless of the grid size. This removed a huge source of error.

2. Removing the Weight (Chiral Extrapolation)

• The Issue: To make the computer simulation run fast, they used "heavy" particles. But the real universe uses "light" (massless) particles.
• The Fix: They ran the simulation with three different weights of particles (light, medium, heavy). Then, they drew a line through those points and extended it to where the weight is zero.
• The Result: This is like weighing a suitcase with a heavy rock inside, then weighing it with a lighter rock, and finally calculating what the suitcase weighs with no rock at all. They successfully figured out how to predict the behavior of the massless particles.

3. Checking the Map (The Beta-Function)

• They plotted the "steepness" of the hill (the Beta-Function) across a wide range of energies.
• They checked their map against the "universal laws" of physics (theoretical predictions).
• The Discovery: At very high energies (weak coupling), their map matched the universal laws perfectly. At lower energies (strong coupling), the map started to curve in ways that only their complex simulation could capture. This proves their method is working correctly in the tricky, non-linear zones.

What's Next?

The paper ends with a "To-Do" list for the future:

1. Expand the Room: Currently, they are simulating the particles in a small box. They need to make the box infinitely large to ensure the walls aren't squishing the particles.
2. Refine the Grid: They need to make the grid pixels smaller and smaller until the "blockiness" disappears completely (the continuum limit).
3. The Final Calculation: Once the box is big enough and the pixels are tiny enough, they will integrate their map to get the final, ultra-precise number for the Strong Coupling.

Why Should We Care?

The goal is to get the uncertainty of this "glue strength" down to 0.3%.

• Currently, we are at about 0.6%.
• The scientists want to cut that error in half.

Why? Because the Standard Model (our best theory of the universe) is incredibly sensitive. If we know the "glue" strength better, we can:

• Predict the mass of the Top Quark more accurately.
• Understand how the Higgs Boson is created in colliders.
• Look for cracks in the Standard Model that might reveal New Physics (like Dark Matter).

In short: These scientists are building a super-precise ruler to measure the most fundamental force in the universe, and they have just successfully calibrated the ruler to remove the "pixelation" and "weight" errors. They are getting ready to take the final measurement.

Technical Summary

1. Problem Statement

The strong coupling constant, $\alpha_s(m_Z)$, defined at the $Z$-boson mass scale, is a fundamental parameter of the Standard Model. Its precise value is critical for:

• High-energy phenomenology and precision tests of Quantum Chromodynamics (QCD).
• Determining the top-quark mass and calculating Higgs boson production rates at hadron colliders.
• Reducing uncertainties in theoretical predictions for hadronic $Z$-boson decay widths.

While the 2024 Flavor Lattice Averaging Group (FLAG) report cites a lattice QCD uncertainty of $\sim 0.6\%$, the phenomenology community requires an uncertainty well below $1\%$, with a target precision of $\lesssim 0.2\%$. This project aims to bridge that gap by providing a new, highly precise determination of $\alpha_s(m_Z)$ using a specific non-perturbative lattice approach.

2. Methodology

The authors employ the continuous $\beta$-function method within the infinite volume gradient flow scheme. The core strategy involves the following steps:

• Renormalization Group (RG) $\beta$-function: The method relies on integrating the RG $\beta$-function, $\beta(g^2) = \mu^2 \frac{dg^2}{d\mu^2}$, to determine the QCD $\Lambda$-parameter. This parameter is then converted to the $\overline{\text{MS}}$ scheme and evolved to the $Z$-boson mass scale using 5-loop perturbation theory.
• Gradient Flow: The coupling is defined non-perturbatively using gradient flow, a continuous smearing transformation evolving fields along a fictitious flow time $\tau$. The renormalized coupling $g^2_{GF}$ is derived from the expectation value of the flowed Yang-Mills energy density $\langle E(\tau) \rangle$.
• Lattice Action: The study uses degenerate four-flavor lattice QCD simulations employing the Highly Improved Staggered Quark (HISQ) action for fermions and the tree-level Symanzik-improved (Lüscher–Weisz) gauge action.
• Ensembles: Simulations cover a wide range of bare gauge couplings ($7.0 \leq \beta_b \leq 20.0$).
  • Weak coupling ($\beta_b \geq 8.0$): Simulated directly in the massless limit ($am_f = 0$).
  • Strong coupling ($7.0 \leq \beta_b \leq 7.5$): Simulated with three non-zero fermion masses ($am_f = 0.001, 0.0025, 0.005$) to perform a chiral extrapolation to the massless limit.
• Extrapolation Strategy: The analysis pipeline requires:
  1. Chiral Extrapolation: Removing finite-mass effects ($am_f \to 0$).
  2. Infinite-Volume Extrapolation: Removing finite-size effects ($L/a \to \infty$).
  3. Continuum Extrapolation: Removing discretization errors ($a^2/\tau \to 0$).

3. Key Contributions and Technical Analysis

A. Tree-Level Cutoff Effects (Section 3)

A significant portion of the work focuses on mitigating discretization artifacts.

• Problem: At non-zero lattice spacing, the lattice evaluation of the gradient-flow energy density deviates from the continuum expression at tree level.
• Solution: The authors implement Tree-Level Normalization (TLN). They incorporate a correction factor $C(a^2/\tau, a/L)$ into the normalization of the gradient-flow coupling.
• Result: Comparisons of the running coupling $g^2_{GF}$ using three different discretizations of the energy density (Wilson, Clover, and Symanzik) show that without TLN, the results diverge significantly. With TLN applied, the results from all three discretizations converge, demonstrating a substantial reduction in cutoff effects. This is shown for both weak coupling ($\beta_b = 20.0$) and strong coupling ($\beta_b = 7.0$).

B. Chiral Extrapolation (Section 4)

• Context: In the strong-coupling regime, simulations must be performed at non-zero quark masses and extrapolated to the massless limit.
• Method: The authors assume a linear dependence of the coupling on the fermion mass ($g^2_{GF} \propto am_f$) for small masses.
• Result: Preliminary results for $\beta_b = 7.00, 7.25, 7.50$ show a mild dependence on fermion mass. However, at the strongest coupling ($\beta_b = 7.00$), the largest mass ($am_f = 0.005$) introduces a significant slope, necessitating careful extrapolation. This behavior was not observed in smaller volume datasets ($L/a=24$), highlighting volume dependence.

C. $\beta$-Function Behavior (Section 2)

• The authors present the gradient-flow $\beta$-function over a broad range of couplings ($g^2_{GF} \in (0.75, 18)$).
• Strong Coupling: The data deviates from the perturbative one-loop prediction in the non-perturbative strong-coupling regime, as expected.
• Weak Coupling: As the coupling approaches the UV fixed point ($g^2_{GF} \to 0$), the non-perturbative $\beta$-function is expected to converge to perturbative predictions. Preliminary data at weak coupling ($\beta_b \geq 8.0$) still shows deviations from perturbative curves, attributed to the fact that infinite-volume and continuum limits have not yet been taken.

4. Current Results and Status

• Data Scope: The current proceedings present results primarily from ensembles with spatial volume $L/a = 32$ (temporal extent $2L/a = 64$).
• Preliminary Nature: The results are preliminary. The infinite-volume and continuum extrapolations, which are crucial for the final precision, are deferred to future studies.
• Current Limitations:
  • The $\beta$-function data at weak coupling does not yet align perfectly with perturbative expectations due to finite-volume and discretization effects.
  • To ensure the continuum $\beta$-function covers the reference scale $\tau_0$ (defined by $\tau_0^2 \langle E(\tau_0) \rangle = 0.3$), the authors plan to add an additional bare coupling in the $7.00 \leq \beta_b \leq 7.25$ region.
• Future Work:
  • Generating data on larger volumes ($L/a$ up to 64) to perform infinite-volume extrapolations.
  • Performing continuum extrapolations to quantify remaining discretization effects.
  • Executing a blinded analysis to minimize bias before extracting the final $\alpha_s(m_Z)$.

5. Significance

This work represents a critical step toward a $\sim 0.3\%$ determination of $\alpha_s(m_Z)$, aiming to improve upon the current FLAG 2024 average by a factor of two. By rigorously addressing tree-level cutoff effects and establishing a robust pipeline for chiral, infinite-volume, and continuum extrapolations in a four-flavor HISQ system, the Fermilab Lattice and MILC Collaborations are laying the groundwork for a precision test of the Standard Model that meets the stringent requirements of modern collider phenomenology. The successful application of the continuous $\beta$-function method in the gradient flow scheme with four dynamical flavors offers a promising alternative to traditional step-scaling methods.