Lattice QCD Determination of the Collins-Soper Kernel in the Continuum and Physical Mass Limits

This paper presents a first-principles lattice QCD determination of the Collins-Soper kernel at the continuum and physical mass limits, providing a precise nonperturbative constraint on its long-distance behavior that bridges lattice calculations with phenomenological TMD studies.

The Gist

The Cosmic Speedometer: Mapping the Inner Whirlwind of a Proton

Imagine you are trying to understand how a high-speed Formula 1 car works. You can look at it while it’s parked, or you can watch it zoom past you at 200 mph. If you watch it zoom past, the car becomes a blur. To understand the engine, you can’t just look at a still photo; you have to understand how the parts move, vibrate, and interact while they are in that high-speed "blur."

In the world of particle physics, the proton (the heart of every atom) is that high-speed car. Inside it, tiny particles called quarks and gluons are constantly zooming around. Scientists want to create a 3D map of this internal "whirlwind," but there is a massive problem: the faster the proton moves, the more complicated the "blur" becomes.

This paper describes a breakthrough in how we calculate that blur using supercomputers.

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1. The Problem: The "Rapidity" Blur

When particles move at nearly the speed of light, they undergo something called rapidity evolution.

The Analogy: Imagine you are filming a spinning fan. If the fan spins slowly, you can see the individual blades. If it spins incredibly fast, the blades disappear into a translucent disk. To understand the fan, you need a mathematical rule that tells you: "If the fan goes from 100 RPM to 10,000 RPM, how does that translucent disk change?"

In physics, that "rule" is called the Collins–Soper (CS) kernel. It is the master formula that connects the slow, easy-to-see movements to the ultra-fast, blurry movements. For decades, we had a good formula for the "slow" part, but the "fast/blurry" part (the non-perturbative part) was a mystery. We had to guess, and our guesses weren't very precise.

2. The Solution: The "Digital Universe" (Lattice QCD)

Since we can't easily "see" this inside a real proton without making guesses, scientists use Lattice QCD.

The Analogy: Imagine you want to study how a hurricane forms, but you can't go into the ocean. Instead, you build a "digital ocean" inside a supercomputer. You divide the ocean into a grid (a "lattice") of tiny cubes. By simulating the physics inside these cubes, you can watch the hurricane form from scratch.

The researchers in this paper built a highly advanced digital universe. They simulated quarks and gluons on a grid so fine and so precise that they could finally calculate the "rule" (the CS kernel) from first principles—meaning they didn't have to guess; they let the math do the work.

3. The Breakthrough: The "Continuum" and "Physical" Limits

Until now, digital simulations had two major flaws:

1. The Grid Problem: The "cubes" in the digital ocean were too chunky. To get real results, you have to mathematically shrink those cubes until they are infinitely small (the Continuum Limit).
2. The Weight Problem: Simulating "real" particles is computationally expensive, so scientists often used "heavy" versions of particles to save time. They then had to guess how the results would change if the particles were their actual, lighter weight (the Physical Mass Limit).

The Analogy: It’s like trying to simulate a real human body, but because your computer is slow, you simulate a person made of heavy LEGO bricks. To get the real answer, you have to mathematically prove that as your LEGO bricks get smaller and smaller, and as your "LEGO person" gets lighter and more realistic, you eventually arrive at a real human.

This paper is the first to successfully do both at the same time with extreme precision. They shrunk the grid and lightened the particles simultaneously to reach the "real world" result.

4. Why does this matter?

Why spend years of supercomputer time on a mathematical "rule" for a blur?

Because this rule is the bridge. It connects what we see in massive particle accelerators (like the Large Hadron Collider) to the fundamental theory of how matter is held together. By providing the most precise "speedometer" ever created for the internal whirlwind of the proton, these scientists have given experimentalists a much clearer lens.

In short: They have moved us from looking at a blurry smudge and saying, "I think that's a proton," to saying, "I can see exactly how the engine is spinning."

Technical Summary

Technical Summary: Lattice QCD Determination of the Collins–Soper Kernel in the Continuum and Physical Mass Limits

1. The Problem: The Rapidity Evolution Challenge

In the study of the three-dimensional structure of the nucleon, Transverse-Momentum-Dependent parton distribution functions (TMD PDFs) are essential for describing processes like semi-inclusive deep-inelastic scattering (SIDIS) and Drell–Yan production. A fundamental theoretical requirement for TMD physics is rapidity evolution, which connects TMDs across different energy scales. This evolution is governed by the Collins–Soper (CS) kernel, $K(b_\perp, \mu)$.

While the CS kernel is perturbatively calculable at small transverse separations ($b_\perp$), its behavior at large $b_\perp$ is intrinsically non-perturbative. Current knowledge of this long-distance regime relies heavily on phenomenological extractions from experimental data, which are subject to significant model dependencies and parametrization uncertainties. There is an urgent need for a first-principles determination of the CS kernel using Lattice QCD.

2. Methodology: LaMET and Quasi-TMD Wave Functions

Direct calculation of light-cone TMDs is impossible in Euclidean lattice space due to the presence of lightlike Wilson lines. To circumvent this, the authors employ Large-Momentum Effective Theory (LaMET).

• Quasi-TMD Wave Functions (Quasi-TMDWFs): The authors simulate nonlocal equal-time correlation functions of hadrons boosted to large longitudinal momentum ($P_z$). These "quasi-distributions" are related to their light-cone counterparts through a perturbative matching kernel.
• Extraction Strategy: The CS kernel is extracted by investigating how the quasi-TMDWFs evolve with respect to the boost momentum $P_z$. Specifically, the variation of the quasi-TMDWFs with $P_z$ encodes the same rapidity evolution that governs the physical TMDs.
• Lattice Setup: The study uses $(2+1)$-flavor gauge configurations with multiple lattice spacings ($a \in [0.052, 0.105]$ fm) and multiple pion masses, including the physical pion mass ($m_\pi \approx 136$ MeV).
• Technical Improvements:
  • HYP Smearing: Applied to improve the signal-to-noise ratio of the Wilson links.
  • Wilson Loop Renormalization: Used to remove linear divergences from the staple-shaped Wilson links.
  • $b_\perp$-unexpanded Matching: Instead of a small-$b_\perp$ expansion, the authors use the full $b_\perp$-dependent perturbative matching kernel to improve reliability across a broader range of transverse separations.
  • Extrapolations: The authors perform a systematic "chained" extrapolation: first to the infinite-momentum limit ($P_z \to \infty$), then to the continuum limit ($a \to 0$), and finally to the chiral limit (physical pion mass).

3. Key Contributions

• First-Principles Determination: This is the first lattice QCD determination of the CS kernel that achieves both the continuum limit and the physical pion mass without relying on an explicit $b_\perp$-dependent parametrization.
• Extended Reach: The determination is provided up to transverse separations of $b_\perp \simeq 1$ fm, covering the crucial non-perturbative long-distance regime.
• Robust Error Budget: The paper provides a comprehensive analysis of systematic uncertainties, including those from the imaginary part of the kernel, $\lambda$-extrapolation, chiral/continuum limits, and power corrections ($1/P_z^2$).

4. Results

• Consistency: The extracted CS kernel shows remarkable agreement with perturbative QCD at small $b_\perp$ and is highly consistent with existing global phenomenological extractions at larger $b_\perp$.
• Precision: The results provide the most precise non-perturbative constraint on the long-distance behavior of the CS kernel from Lattice QCD to date.
• Phenomenological Impact: A global one-parameter fit demonstrated that incorporating these lattice results into phenomenological analyses significantly reduces the statistical uncertainty of the non-perturbative parameters (e.g., $g_2$ in various models) compared to using experimental data alone.

5. Significance

This work represents a major milestone in hadronic physics. By bridging the gap between Lattice QCD, perturbation theory, and experimental nucleon structure data, it provides a model-independent input for future global TMD analyses. It establishes a high-precision foundation for understanding the 3D momentum structure of the nucleon and enhances the predictive power of QCD factorization in high-energy scattering processes.