Proton isovector helicity PDF at NNLO and the twist-3 moment $\tilde{d}_2$ from lattice QCD at physical quark masses

This paper presents a lattice QCD calculation at physical quark masses within the LaMET framework to determine the proton's isovector helicity parton distribution function at NNLO accuracy and, for the first time, extract the twist-3 moment $\tilde{d}_2$ in the $\overline{\rm MS}$ scheme.

The Gist

The Big Picture: Solving the Proton's "Spin Puzzle"

Imagine the proton (the tiny particle in the center of every atom) as a spinning top. For decades, physicists have been trying to figure out exactly how that top spins.

We know the proton is made of smaller particles called quarks and gluons. The big mystery, known as the "Proton Spin Puzzle," is this: If you add up the spin of all the quarks inside, it only accounts for about 30% of the proton's total spin. Where is the rest coming from? Is it the gluons? Is it the way the quarks are orbiting around?

This paper is a major step in solving that mystery. The authors used a supercomputer to simulate the proton from the ground up and calculated exactly how the quarks are spinning inside it.

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The Challenge: Seeing the Invisible

The Problem:
In the real world, we can't just take a photo of a quark spinning inside a proton. The laws of physics (specifically, the fact that quarks are trapped inside) make it impossible to see them directly in a standard experiment. It's like trying to figure out the shape of a tornado by looking at a single drop of rain; the rain is moving too fast and is too chaotic.

The Solution (LaMET):
The researchers used a clever trick called LaMET (Large Momentum Effective Theory).

• The Analogy: Imagine you are trying to understand the shape of a fast-moving car. If the car is parked, you can walk around it and see every detail. But if it's zooming by at 100 mph, it looks like a blur.
• The Trick: LaMET is like a mathematical "slow-motion camera." It allows the scientists to take a snapshot of the proton while it is moving very fast (boosted to high momentum) on a computer grid. By analyzing this "fast-moving" snapshot, they can mathematically reconstruct what the proton looks like when it's "at rest."

The Two Main Discoveries

The paper presents two major findings, which we can think of as two different ways of looking at the proton's interior.

1. The "Spin Map" (The Helicity PDF)

What it is: This is a map showing how much spin the quarks contribute at different speeds. Think of the proton as a busy highway. Some cars (quarks) are driving fast, some slow. This map tells us: "If a quark is carrying 40% of the proton's speed, how much of the proton's spin does it have?"

The Result:

• The team created a very precise map for the middle section of the highway (where most traffic is).
• They found that in this middle region, the quarks are spinning a bit more than previous global estimates suggested.
• The Catch: The "endpoints" of the highway (very slow or very fast quarks) are hard to see with this method. To fix this, they combined their computer map with a "model" based on how the edges of the highway usually behave. This gave them a complete picture of the proton's spin distribution.

2. The "Twist-3 Moment" (The ˜d2)

What it is: This is a more exotic measurement. While the first measurement looked at the main spin, this one looks at a subtle "wobble" or "twist" caused by the interaction between quarks and gluons.

• The Analogy: Imagine a group of dancers (quarks) spinning in a circle. The "Helicity" measurement tells us how fast they are spinning. The "Twist-3" measurement tells us about the force they feel pushing them sideways as they spin, caused by the invisible strings (gluons) connecting them.
• The Discovery: The researchers measured this "average sideways force" (called the color Lorentz force) for the first time directly from the computer simulation.
• The Surprise: They found this force is almost zero.
  • Why this matters: It means that in this specific type of proton, the quarks aren't being pushed around wildly by the gluons in a way that creates a strong "twist." It's a very quiet, stable dance. This helps confirm that our current theories about how quarks and gluons interact are correct.

How They Did It (The "Lattice" Kitchen)

To get these results, they didn't use a real proton; they built a virtual one on a grid.

• The Grid (Lattice): Imagine a 3D checkerboard. The squares are tiny (about 0.076 femtometers across). The quarks sit on the corners, and the gluons are the strings connecting them.
• The Simulation: They ran a massive simulation on a supercomputer, calculating how these quarks and strings interact. They used "physical quark masses," meaning they didn't cheat by using fake, heavy quarks to make the math easier. They used the real weights, making the result much more accurate.
• The "Renormalization" (Cleaning the Data): The raw data from the computer is messy, like a photo taken with a dirty lens. The authors used advanced mathematical techniques (called "hybrid renormalization" and "resummation") to clean the lens. They removed the "noise" and "blur" to reveal the sharp, true image of the proton's spin.

Why This Matters

1. First Principles: This isn't just an educated guess based on other experiments. This is a calculation starting from the fundamental laws of quantum physics (QCD) with almost no assumptions.
2. Precision: They achieved a level of detail (NNLO) that is the "gold standard" in physics, meaning their math is incredibly precise.
3. New Tool: They successfully used a new method to measure the "twist-3" moment. This is like inventing a new type of microscope that can see things other microscopes miss.

The Bottom Line

This paper is like a high-definition, slow-motion video of the inside of a proton. It tells us that the quarks inside are spinning in a specific, predictable pattern, and that the "twisting" forces between them are surprisingly small. It brings us one step closer to answering the ultimate question: What makes the proton spin?

Technical Summary

1. Problem Statement

The internal spin structure of the proton is a fundamental question in Quantum Chromodynamics (QCD). While the contribution of quark spin to the proton spin is known to be roughly 30%, a precise, first-principles determination of the helicity Parton Distribution Function (PDF), $g_1(x)$, and its moments remains challenging.

• Leading Twist: Standard lattice QCD cannot directly compute light-cone PDFs because they are defined by non-local operators on the light-cone, which are inaccessible in Euclidean spacetime.
• Twist-3: Beyond leading twist, the transverse spin structure function $g_T(x) = g_1(x) + g_2(x)$ contains twist-3 contributions representing quark-gluon correlations. A key benchmark is the reduced moment $\tilde{d}_2$, which is proportional to the average color Lorentz force experienced by a quark in the proton. Extracting this is difficult due to operator mixing with lower-dimensional operators and power-divergent subtractions.
• Current Limitations: Previous lattice studies often used unphysical quark masses, coarse lattice spacings, or lacked higher-order perturbative matching (NNLO) and resummation techniques necessary for precision.

2. Methodology

The authors employ Large-Momentum Effective Theory (LaMET) combined with Short-Distance Factorization (SDF) to extract both leading-twist and twist-3 observables.

A. Lattice Setup

• Ensemble: Generated using the $N_f = 2+1$ Highly Improved Staggered Quark (HISQ) action by the HotQCD collaboration.
• Parameters: Physical light and strange quark masses ($m_\pi \approx 140$ MeV), lattice spacing $a = 0.076$ fm, and volume $64^3 \times 64$.
• Boosts: Proton states were boosted to momenta $P_z = \{0, 0.25, 1.02, 1.53\}$ GeV.
• Techniques:
  • All-Mode Averaging (AMA): Used to improve signal-to-noise ratios by combining "exact" and "sloppy" propagator solves.
  • Momentum Smearing: Coulomb-gauge smearing applied to quark fields to enhance boosted hadron signals.
  • Excited State Control: Two-state fits to three-point correlation functions to isolate ground-state matrix elements.

B. Extraction of Moments (OPE Approach)

• Renormalization-Group (RG) Invariant Ratios: To cancel UV divergences, the authors construct ratios of matrix elements at different momenta.
• Operator Product Expansion (OPE): These ratios are expanded in terms of Wilson coefficients and Mellin moments $\langle x^n \rangle$.
• Twist-3 Extraction: A specific ratio $M_T$ is constructed for the transverse correlator. By subtracting the leading-twist (Wandzura-Wilczek) contribution at Next-to-Leading Order (NLO), the residual dependence on the momentum $\lambda = P_z z$ isolates the genuine twist-3 moment $\tilde{d}_2$.

C. Extraction of $x$-dependent PDFs (LaMET Approach)

• Hybrid Renormalization: The bare matrix elements are renormalized using a hybrid scheme. At small $z$, a ratio scheme is used; at large $z$, linear divergences are subtracted using a mass counterterm $\delta m$.
• Renormalon Resummation: To handle the infrared ambiguity of the linear divergence (renormalon), the authors apply Leading Renormalon Resummation (LRR) in the coordinate space, ensuring the perturbative matching is consistent.
• Asymptotic Extrapolation: Since lattice data is limited in $z$, the long-tail behavior of the matrix elements is extrapolated using two functional forms (inspired by Regge behavior and rigorous large-$z$ derivations) before Fourier transforming to momentum space.
• Perturbative Matching: The resulting quasi-PDFs are matched to the $\overline{\text{MS}}$ light-cone PDFs. The matching kernel is calculated at Next-to-Next-to-Leading Order (NNLO).
• Resummation: To control large logarithms in the small-$x$ (soft) and large-$x$ (threshold) regions, Renormalization Group (RG) and Threshold Resummations are applied.

D. Endpoint Modeling (SDF)

Since LaMET fails at the endpoints $x \to 0, 1$, the authors use Short-Distance Factorization (SDF) to model the endpoint behavior ($x^a(1-x)^b$) and fit these parameters to the coordinate-space matrix elements, combining them with the LaMET mid-$x$ results.

3. Key Contributions

1. First Direct Lattice Extraction of $\tilde{d}_2$ in $\overline{\text{MS}}$: The paper provides the first calculation of the twist-3 reduced moment $\tilde{d}_2$ directly in the $\overline{\text{MS}}$ scheme using non-local correlators, bypassing the complications of local operator mixing.
2. NNLO Precision: The helicity PDF is determined with NNLO matching accuracy, including RG and threshold resummations, significantly reducing theoretical uncertainties compared to previous LO/NLO studies.
3. Physical Point Calculation: The calculation is performed at physical quark masses on a fine lattice ($a=0.076$ fm), eliminating chiral and continuum extrapolation uncertainties.
4. Systematic Error Control: The study rigorously addresses systematic errors, including excited-state contamination, renormalon ambiguities, and asymptotic extrapolation uncertainties.

4. Results

A. Mellin Moments

The lowest few moments of the isovector helicity PDF at $\mu = 2$ GeV were extracted:

• $\langle x \rangle^{u-d} = 0.291(7)$
• $\langle x^2 \rangle^{u-d} = 0.101(10)$
  These values are slightly higher than those obtained from global fits (e.g., JAM22), suggesting a harder distribution in the mid-$x$ region.

B. Twist-3 Moment $\tilde{d}_2$

The extracted value for the isovector twist-3 moment is:
$$\tilde{d}_2^{u-d}(2 \text{ GeV}) = 0.0024(46)$$

• Interpretation: The result is statistically consistent with zero. This implies that the genuine quark-gluon correlation contribution to the twist-3 moment is strongly suppressed at this scale, supporting the Wandzura-Wilczek approximation and phenomenological constraints.

C. Helicity PDF $g_1(x)$

• Mid-$x$ Region ($x \in [0.25, 0.75]$): The lattice result shows a more elevated distribution compared to global fits (JAM22, BDSSV24).
• Endpoint Regions: The fitted parameters for the endpoint behavior ($a \approx -0.38, b \approx 1.68$ for the valence distribution) were constrained using SDF.
• Comparison: The results show good agreement with recent Coulomb-gauge lattice calculations for $x < 0.6$, but discrepancies appear for $x > 0.6$, likely due to power corrections at the current momentum ($P_z \approx 1.5$ GeV).

5. Significance

• Validation of LaMET: This work demonstrates the maturity of the LaMET framework, capable of producing high-precision, physical-point results for both leading-twist and twist-3 observables.
• Proton Spin Puzzle: By providing a first-principles constraint on the quark helicity distribution and its moments, it helps quantify the quark spin contribution more accurately.
• Quark-Gluon Dynamics: The near-zero result for $\tilde{d}_2$ offers crucial insight into the nature of quark-gluon correlations, suggesting that the average color Lorentz force is small in the isovector channel, a finding that complements experimental efforts to measure $g_2(x)$.
• Future Directions: The paper outlines the path forward: increasing hadron momentum ($P_z$) to reduce power corrections and improve the $x$-coverage, and performing simulations at multiple lattice spacings to control continuum limits.

In summary, this paper represents a state-of-the-art lattice QCD calculation that successfully combines advanced renormalization, high-order perturbative matching, and effective field theory techniques to deliver precise constraints on the proton's spin structure.