Scale Setting and Strong Coupling Determination in the Gradient Flow Scheme for 2+1 Flavor Lattice QCD

This paper presents a scale setting and strong coupling determination for 2+1 flavor lattice QCD using HISQ ensembles, reporting precise gradient flow scales ($\sqrt{t_0}$ and $w_0$) and a potential scale ($r_1$) while providing a polynomial ansatz for predicting lattice spacings and estimating $\Lambda_{\overline{\mathrm{MS}}}$.

The Gist

Imagine you are trying to build a perfect model of the universe's smallest building blocks using a giant, digital LEGO set. This is what physicists do in Lattice QCD (Quantum Chromodynamics). They simulate the strong force that holds protons and neutrons together.

But here's the catch: In their digital world, everything is just numbers. There are no "meters" or "seconds." To make their simulation match the real world, they need a ruler. They need to answer the question: "How big is one LEGO brick in real life?"

This paper is about finding the most accurate, reliable ruler possible for this digital universe.

The Problem: Finding the Right Ruler

In the past, physicists used a few different "rulers" to measure their digital bricks.

• The Old Rulers: Some used the distance between two heavy quarks (like a stretched rubber band) or the mass of specific particles. These were okay, but sometimes they were a bit wobbly or hard to calculate precisely.
• The New Ruler (Gradient Flow): A few years ago, a new method called Gradient Flow was introduced. Think of this as a "smoothing filter" for the digital universe. Imagine you have a noisy, static-filled TV screen. Gradient flow is like turning a knob to smooth out the static. As you smooth it out, you reach a point where the image is perfectly clear and stable. The "amount of smoothing" needed to reach that clarity becomes your ruler.

This paper is about calibrating this new, high-tech ruler with extreme precision.

The Experiment: Smoothing the Digital Universe

The authors (a team of scientists from the US, Germany, and China) took a massive set of digital simulations generated by the HotQCD Collaboration. These simulations represent the universe with 2+1 flavors of quarks (up, down, and strange).

They applied the "smoothing filter" (Gradient Flow) to these simulations at different levels of detail (different lattice spacings). They looked for two specific "sweet spots" in the smoothing process:

1. $t_0$ (The "Time" Ruler): A specific point where the smoothing creates a standard amount of energy.
2. $w_0$ (The "Width" Ruler): A slightly different point that measures how fast the smoothing changes.

The Calibration: Checking the Ruler

Just because you have a ruler doesn't mean it's accurate. You have to check it against known objects. The team checked their digital ruler against three real-world "objects":

1. Bottomonium Splittings: Think of these as the "rings" of a heavy atom (like a heavy version of hydrogen). The difference in energy between these rings is a known, precise number in the real world.
2. Decay Constants: These are like measuring how fast a specific type of particle (a Kaon) falls apart.
3. The Phi Meson: A specific particle that acts like a resonance, similar to a tuning fork vibrating at a specific note.

By comparing their digital measurements of these objects to the real-world values, they could finally say: "Okay, our digital ruler says $t_0$ is exactly 0.14428 femtometers (a femtometer is a quadrillionth of a meter)."

The Big Discovery: The "Charm" Difference

One of the most interesting findings in this paper is a comparison with other teams.

• Team A (This Paper): Used simulations with 3 types of quarks (Up, Down, Strange).
• Team B (Other recent papers): Used simulations with 4 types of quarks (Up, Down, Strange, and Charm).

The authors found that the "ruler" size is different depending on whether the "Charm" quark is included in the simulation.

• Analogy: Imagine you are measuring a room. If you measure it with a tape measure that includes a heavy backpack, the room looks slightly different than if you measure it without the backpack.
• The Result: The presence of the heavy "Charm" quark changes the fundamental scale of the universe slightly. This paper confirms that the "ruler" for a 3-quark world is slightly different than the "ruler" for a 4-quark world. This is a crucial detail for future calculations.

The Side Quest: The "Running" Force

The paper also looked at the Strong Force (the glue holding the universe together). In quantum physics, this force changes strength depending on how close you look (like a zoom lens).

• The team calculated how this force "runs" (changes) using their new ruler.
• They found that their digital results matched the theoretical predictions (math formulas) very well, up to a certain point. It's like driving a car and checking if your speedometer matches the GPS; they matched perfectly up to a specific speed.

Why Does This Matter?

This might sound like abstract math, but it's the foundation for understanding the universe.

1. Precision: Future calculations about the early universe (just after the Big Bang) or the inside of neutron stars need a perfect ruler. If the ruler is off by even a tiny bit, the whole model is wrong.
2. Consistency: By proving that the "Charm" quark changes the scale, they help other scientists know which ruler to use for which job.
3. High-Temperature Physics: The HotQCD collaboration uses these results to study what happens when matter gets incredibly hot (like in the early universe). They need this precise scale to know exactly when matter melts into a "soup" of quarks and gluons.

In a Nutshell

This paper is the ultimate calibration manual for the digital universe. The authors took a new, high-tech method for measuring space (Gradient Flow), tested it against the most reliable real-world objects they could find, and confirmed that it works beautifully. They also discovered a subtle but important difference in how the universe scales when you include the "Charm" quark, ensuring that future physicists have the most accurate map possible for exploring the deepest secrets of matter.

Technical Summary

Here is a detailed technical summary of the paper "Scale Setting and Strong Coupling Determination in the Gradient Flow Scheme for 2+1 Flavor Lattice QCD."

1. Problem Statement

Lattice Quantum Chromodynamics (QCD) calculations require "scale setting" to convert dimensionless lattice quantities into physical units (e.g., femtometers). While various scales exist (e.g., potential scales $r_0, r_1$), the gradient flow scales ($t_0, w_0$) have emerged as highly precise and stable observables due to their renormalization properties and lack of need for large-distance correlation function fits.

However, several critical issues remain:

• Flavor Dependence: There is ongoing tension regarding whether gradient flow scales differ between $N_f=2+1$ (up, down, strange) and $N_f=2+1+1$ (including dynamical charm) QCD. Existing determinations often use the same gauge configurations or suffer from limited independent cross-checks.
• Consistency: Many studies report only $t_0$ or $w_0$, lacking a comprehensive determination of their ratios and their relation to traditional potential scales ($r_1$).
• Strong Coupling: The running of the strong coupling constant in the gradient flow scheme needs precise non-perturbative determination in the continuum limit to test perturbative QCD predictions and extract the QCD $\Lambda$ parameter.
• High-Temperature Applications: The HotQCD Collaboration's high-temperature calculations rely heavily on the $r_1$ scale. An independent, precise determination of $t_0$ and $w_0$ is needed to provide alternative scale-setting methods for these studies.

2. Methodology

The authors performed a comprehensive lattice QCD study using the Highly Improved Staggered Quark (HISQ) action for $N_f=2+1$ flavors.

• Ensembles: They utilized 17 different bare gauge couplings ($\beta = 10/g_0^2$) ranging from 6.423 to 8.400, generated by the HotQCD Collaboration. This covers a wide range of lattice spacings ($a \approx 0.008$ fm to $0.067$ fm).
• Quark Masses: Simulations were performed with two light quark mass ratios: $m_s/m_l = 20$ (near physical pion mass $\sim 160$ MeV) and $m_s/m_l = 5$ (heavier pion mass $\sim 320$ MeV).
• Gradient Flow Implementation:
  • Used Zeuthen flow with a Symanzik improved Lüscher–Weisz gauge action to eliminate tree-level $O(a^2)$ discretization errors.
  • Calculated the action density $E(\tau_F)$ using two discretization schemes for the field strength tensor: clover-type and $a^2$-improved (mixture of square and rectangle plaquettes).
  • Defined scales $t_i$ and $w_i$ via conditions on $\tau_F^2 \langle E \rangle$ and its derivative at flow time $\tau_F$.
• Scale Determination Strategy:
  • Physical Inputs: The dimensionless products of flow scales with physical observables were computed and extrapolated to the continuum limit. Inputs included:
    • Bottomonium mass splittings ($\Upsilon$ and $\eta_b$ states).
    • Pseudo-scalar meson decay constants ($f_K$ and $f_{\eta_s}$).
    • $\phi$-meson mass ($m_\phi$).
  • Continuum Extrapolation: Performed using Ansätze linear in $a^2$ (for clover) and $\alpha_s a^2$ or $a^4$ (for improved discretization). Systematic uncertainties were estimated by varying the range of lattice spacings included in the fits.
• Strong Coupling: The gradient flow coupling $g^2_{\text{flow}}$ was calculated at small flow times ($\sqrt{8\tau_F} < 0.2$ fm) and extrapolated to the continuum. The results were compared with perturbative expansions to extract $\Lambda_{\overline{MS}}$.

3. Key Contributions

1. Precise Determination of $t_0$ and $w_0$ in $N_f=2+1$: Provided high-precision values for gradient flow scales using multiple independent physical inputs (bottomonium, decay constants, $\phi$ mass) and two discretization schemes.
2. Resolution of Flavor Dependence: Confirmed that gradient flow scales in $N_f=2+1$ QCD are distinct from those in $N_f=2+1+1$ QCD, validating the impact of the dynamical charm quark on scale setting.
3. Re-evaluation of $r_1$: Re-determined the potential scale $r_1$ using the same ensembles, providing a consistent cross-check with the gradient flow scales.
4. Perturbative Matching: Demonstrated that the gradient flow coupling running is compatible with perturbative QCD up to flow radii of $\sqrt{8\tau_F} \approx 0.15$ fm, provided higher-order corrections are included.
5. Systematic Checks: Rigorously addressed systematic effects, including topological freezing (at high $\beta$) and light quark mass dependence, showing them to be negligible for the flow observables studied.

4. Key Results

Gradient Flow Scales (Physical Units):
Using a weighted average of determinations from bottomonium splittings, decay constants, and the $\phi$ mass, the authors obtained:

• $\sqrt{t_0} = 0.14428(48)$ fm
• $w_0 = 0.17391(52)$ fm
• $r_1 = 0.3112(24)$ fm

Comparisons and Consistency:

• The values of $\sqrt{t_0}$ and $w_0$ are consistent with the FLAG 2024 average for $N_f=2+1$ but are significantly larger than recent $N_f=2+1+1$ determinations (e.g., by BMW and Fermilab-MILC). This confirms that the dynamical charm quark affects the scale setting.
• The ratio $\sqrt{t_0}/w_0 = 0.8291(8)(6)$ is consistent with $N_f=2+1+1$ results, suggesting the ratio is less sensitive to flavor changes than the absolute scales.
• The $r_1$ scale determined here agrees with previous $N_f=2+1$ results (e.g., MILC) but differs from the $N_f=2+1+1$ TUMQCD result, further supporting the flavor-dependence hypothesis.

Strong Coupling and $\Lambda_{\overline{MS}}$:

• The gradient flow coupling $g^2_{\text{flow}}$ was successfully extrapolated to the continuum.
• Perturbative predictions match the lattice data up to $\sqrt{8\tau_F} = 0.15$ fm when a 3-loop term ($k_3 \alpha^3_{MS}$) is included in the conversion to the $\overline{MS}$ scheme.
• The extracted QCD scale parameter is:
  • $\Lambda^{N_f=3}_{\overline{MS}} = 309.1^{+0.7}_{-0.7} \pm 5.0 \pm 33.9$ MeV (stat, continuum, perturbative uncertainty).
  • This yields $\alpha^{(5)}_{\overline{MS}}(M_Z) = 0.1161^{+0.0027}_{-0.0016}$, consistent with the FLAG 2024 average.

5. Significance

• Benchmark for High-Temperature QCD: The precise determination of $t_0$ and $w_0$ provides a robust alternative to the $r_1$ scale for setting the lattice spacing in HotQCD's high-temperature calculations, reducing systematic uncertainties in the study of the QCD phase transition.
• Flavor Physics Insight: The clear distinction between $N_f=2+1$ and $N_f=2+1+1$ scales highlights the necessity of including the charm quark explicitly in high-precision lattice simulations aiming for sub-percent accuracy.
• Perturbative Validation: The successful matching of the gradient flow coupling to perturbation theory up to relatively large flow times ($0.15$ fm) strengthens the utility of gradient flow as a tool for defining non-perturbative renormalization schemes and extracting fundamental QCD parameters.
• Methodological Rigor: The use of multiple physical inputs, improved discretization schemes, and extensive checks for topological freezing and mass dependence sets a high standard for future scale-setting studies.