Recent update of nucleon axial-vector charge with the PACS10 superfine lattice

This paper presents an ongoing update of the nucleon axial-vector charge using the superfine lattice ensemble of the PACS10 gauge configurations at the physical point, while also examining low-energy relations derived from the PCAC relation to verify the consistency of lattice QCD data with continuum physics.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks. Physicists call these bricks quarks, and when three of them stick together, they form a nucleon (like a proton or neutron), which is the core of every atom in your body.

For a long time, scientists have been trying to measure a specific property of these nucleons called the axial-vector charge (let's call it the "spin strength"). Think of this as the nucleon's "muscle memory" for how it spins and interacts with other particles. We know exactly how strong this muscle is from real-world experiments, but to understand why it is that strong, we need to simulate the universe inside a computer.

This paper is a report card from a team of scientists (the PACS Collaboration) who are running these simulations. Here is the story of what they did, explained simply:

1. The Problem: The "Pixelated" Universe

To simulate the universe, scientists break space and time into a grid, like a giant 3D chessboard. The smaller the squares on the board, the more realistic the simulation.

• Coarse Grid: Big squares. The picture looks blocky and blurry.
• Fine Grid: Smaller squares. Better picture.
• Superfine Grid: Tiny squares. A crystal-clear picture.

In the past, this team used the "Coarse" and "Fine" grids. They got good results, but they wanted to be absolutely sure that their "blocky" grid wasn't hiding any errors. They needed the Superfine grid to prove their math works perfectly when the pixels are tiny enough to look like real, smooth physics.

2. The New Tool: The "Superfine" Lens

The team just finished running their simulations on the PACS10 Superfine Lattice.

• The Scale: Imagine a box of space that is 10 femtometers wide (that's 10 quadrillionths of a meter). Inside this box, they packed a grid so dense it has 256 layers in every direction.
• The Result: This is like zooming in with a microscope so powerful that you can see the individual threads of a spiderweb, whereas before you could only see the web as a blurry smudge.

3. The Test: The "Tug-of-War"

To make sure their Superfine simulation was actually correct, they didn't just look at the spin strength. They played a game of "Tug-of-War" using two different ways to measure the same thing.

• Team A (The Pion Team): They measured a property using simple, two-particle interactions (like two people shaking hands).
• Team B (The Nucleon Team): They measured the same property using complex, three-particle interactions (like a tug-of-war with a third person involved).

The Theory: In a perfect, real-world universe, Team A and Team B should pull with exactly the same force. Their numbers should match perfectly.
The Fear: On a computer grid, sometimes the "pixels" mess things up. If the grid is too blocky, Team A and Team B might pull with different forces, revealing a flaw in the simulation.

The Outcome: The team found that Team A and Team B pulled with the exact same force. Even though they used a complex method that usually gets messed up by grid errors, the results matched perfectly. This proved that their "Superfine" grid was so good that the "pixelation" errors were practically invisible.

4. The Secret Sauce: "Smearing" the Bricks

How did they get such a clean result? They used a technique called Stout Smearing.

• The Analogy: Imagine trying to build a wall with jagged, rough stones. The wall will be wobbly. But if you take a blender and smooth out the edges of every stone before you lay them down, the wall becomes perfectly straight and strong.
• The Science: The team "smoothed out" the quantum fields (the Lego bricks) six times before running the simulation. This removed the "static noise" and high-energy glitches that usually ruin computer simulations, allowing the true physics to shine through.

5. The Conclusion: Why It Matters

The team successfully calculated the "spin strength" of the nucleon on this Superfine grid.

• The Result: Their number matched the real-world experimental value almost perfectly (within about 2% error).
• The Big Picture: This confirms that their computer model is trustworthy. They have proven that they can simulate the subatomic world with such high precision that the artificial "grid" of the computer no longer distorts the reality.

In a nutshell: This paper is like a master chef saying, "I've finally cooked this dish using the finest, most expensive ingredients and the sharpest knives. I've tasted it, and it matches the original recipe perfectly. Now we know our kitchen (the computer simulation) is good enough to cook any dish in the universe without messing up the flavor."

This is a crucial step toward using supercomputers to solve mysteries about the universe that we can't answer with experiments alone.

Technical Summary

1. Problem Statement

The nucleon axial-vector charge ($g_A$) is a fundamental benchmark for understanding nucleon structure and is measured with high precision in experiments. While lattice QCD simulations have successfully calculated $g_A$ at the physical point using "coarse" and "fine" lattice spacings, a comprehensive understanding of discretization uncertainties and a robust continuum limit require data from even finer lattices.

Specifically, this work addresses:

• The need to verify if lattice QCD data correctly reproduces continuum physics within statistical accuracy using the superfine lattice spacing ($a \approx 0.041$ fm).
• The potential systematic errors arising from the use of the local axial-vector current without explicit $O(a)$ improvement. The authors aim to verify if the improvement factor $c_A$ is negligible, which would justify using the simpler unimproved current.
• The validation of the Partially Conserved Axial-Vector Current (PCAC) relation in the context of nucleon three-point functions to ensure consistency with pion two-point functions.

2. Methodology

Lattice Configuration

The study utilizes the third ensemble of the PACS10 gauge configurations, generated by the PACS Collaboration.

• Action: Six stout-smeared $O(a)$-improved Wilson-clover quark action and Iwasaki gauge action.
• Parameters:
  • Lattice Volume: $(10 \text{ fm})^4$ (negligible finite-volume effects).
  • Lattice Spacing ($a$): 0.041 fm (superfine), corresponding to $\beta = 2.20$.
  • Pion Mass: $m_\pi \approx 142$ MeV (close to the physical mass).
  • Comparison: Results are compared against previous ensembles with $a=0.085$ fm (coarse) and $a=0.063$ fm (fine).

Computational Techniques

• Interpolating Operators: Exponentially smeared quark fields with Coulomb gauge fixing are used to construct nucleon interpolating operators, effectively suppressing excited-state contamination.
• Measurement: The All-Mode-Averaging (AMA) technique is employed to reduce computational costs while achieving high statistical precision.
• Renormalization: The renormalization factor $Z_A$ is determined using the Schrödinger functional method.

Key Observables and Relations

1. Axial-Vector Charge ($g_A$): Calculated using the standard plateau method. The ratio of the nucleon three-point function to the two-point function is analyzed as a function of the operator insertion time ($t$) to extract the ground-state matrix element.
2. PCAC Quark Mass Comparison: To test discretization effects, the authors compare two definitions of the PCAC quark mass:
   • $m_{\text{PCAC}}^{\text{pion}}$: Derived from pion two-point functions.
   • $m_{\text{PCAC}}^{\text{nucl}}$: Derived from nucleon three-point functions using the axial Ward-Takahashi identity.
   • Theoretical Basis: In the continuum limit, these should be identical. On the lattice, discrepancies arise from $O(a)$ discretization errors. If the local (unimproved) current is used, $m_{\text{PCAC}}^{\text{nucl}}$ should theoretically differ from $m_{\text{PCAC}}^{\text{pion}}$ by terms proportional to $a c_A q^2$. Agreement between them implies $c_A \approx 0$.

3. Key Contributions

• First Results on Superfine Lattice: This paper presents the first updated results for $g_A$ and PCAC relations using the PACS10 superfine ensemble ($a=0.041$ fm).
• Validation of Unimproved Current: The study provides strong evidence that the local axial-vector current (without explicit $O(a)$ improvement terms) is sufficient for calculating nucleon matrix elements at this lattice spacing. This is attributed to the effectiveness of the six stout-smeared action in restoring chiral properties and suppressing UV fluctuations.
• Systematic Error Control: The work demonstrates that excited-state contamination is well-controlled even at large source-sink separations ($t_{\text{sep}} \approx 1.2$ fm) due to optimized smearing parameters.

4. Results

Nucleon Axial-Vector Charge ($g_A$)

• Plateau Formation: Clear plateaus were observed for both source-sink separations ($t_{\text{sep}}/a = 20$ and $29$).
• Precision:
  • For $t_{\text{sep}}/a = 20$ ($\approx 0.8$ fm), the relative error is < 2-3%.
  • For $t_{\text{sep}}/a = 29$ ($\approx 1.2$ fm), the relative error is larger ($\sim 5\%$) due to insufficient statistics, but the central value remains consistent.
• Continuum Consistency: The renormalized $g_A$ values from the superfine lattice agree with experimental values ($1.2754(13)$) and previous results from coarser lattices, confirming that discretization effects on $g_A$ are small.

PCAC Relation Verification

• Consistency: The PCAC quark mass calculated from nucleon three-point functions ($m_{\text{PCAC}}^{\text{nucl}}$) coincides with the value from pion two-point functions ($m_{\text{PCAC}}^{\text{pion}}$) within statistical errors.
• Momentum Independence: $m_{\text{PCAC}}^{\text{nucl}}$ shows no significant dependence on the momentum transfer ($q^2$) across seven different momentum modes.
• Implication: The agreement implies that the $O(a)$ correction term (proportional to $c_A$) is negligible. This confirms that the improvement factor $c_A$ is effectively zero for this specific action, validating the use of the unimproved local current.

5. Significance

• Continuum Limit Preparation: These results are a critical step toward a full continuum extrapolation of nucleon structure functions. The superfine lattice data confirms that the discretization errors are under control, allowing for a reliable combination with coarse and fine lattice data.
• Efficiency in Lattice QCD: The finding that $c_A \approx 0$ for the stout-smeared action simplifies future calculations. Researchers can use the computationally cheaper local axial-vector current without needing to calculate the improvement coefficient $c_A$ or include the derivative term in the current, provided the same smearing and action are used.
• Benchmarking: The study reinforces the reliability of the PACS10 ensembles for high-precision physics at the physical point, serving as a robust benchmark for other lattice QCD collaborations.

In conclusion, the paper successfully updates the nucleon axial-vector charge using the superfine PACS10 lattice, demonstrating excellent agreement with experiment and validating the theoretical framework used to handle discretization effects in modern lattice QCD simulations.