Regularization Prescription for the Mixing Between Nonlocal Gluon and Quark Operators

The Gist

Imagine you are trying to understand the inside of a proton (a tiny particle inside an atom) by looking at its building blocks: quarks and gluons. Physicists have two main "languages" to describe how these blocks interact: Coordinate Space (thinking about them as objects at specific distances from each other) and Momentum Space (thinking about them as waves carrying energy and speed).

For a long time, scientists have been able to translate between these two languages for most interactions. However, there was a specific, stubborn translation error when trying to describe how a gluon (the glue holding things together) mixes with a quark (the matter particle) when they get extremely close to each other.

Here is a breakdown of the problem and the solution found in this paper, using simple analogies.

The Problem: The "Infinite" Glitch

Imagine you are trying to measure the distance between two friends holding hands.

• The Gluon is like a heavy backpack (it has a certain "weight" or mass dimension).
• The Quark is like a light shirt (it has a different "weight").

When these two get very close together, the math describing their interaction involves a term that looks like 1 divided by the distance.

• If the distance is 1 meter, the number is 1.
• If the distance is 0.1 meters, the number is 10.
• If the distance is zero (they are touching), the number becomes infinity.

In physics, getting an "infinity" usually means the math has broken down.

The Translation Error:
When scientists tried to translate the "Coordinate Space" result (where the distance is zero) into "Momentum Space" (the wave language), they hit a wall. Because the distance was zero, the math required them to make a guess about how to handle that infinity.

• Some guessed one way, others guessed another.
• This led to ambiguous results: The same physical situation gave different answers depending on which "guess" (or prescription) the scientist used. It was like trying to translate a sentence into another language, but the translator had to invent a word for a concept that didn't exist, leading to confusion.

The Old Fix: Matching Moments

Previously, scientists tried to fix this by looking at "Moments" (think of these as the average weight of the data). They tried to force the "Coordinate Space" average to match the "Momentum Space" average.

• The Paper's Critique: The authors argue this is like trying to fix a broken clock by just setting the hands to match a different clock. It might look right for a few specific points, but it doesn't actually fix the broken gears inside. It leaves the underlying "infinity" problem unsolved and allows for multiple, conflicting answers.

The New Solution: Dimensional Regularization (The "Softening" Tool)

The authors propose a specific mathematical tool called Dimensional Regularization.

The Analogy:
Imagine you are trying to measure the temperature of a flame. If you stick a thermometer directly into the hottest point, it might melt (the "infinity").

• The Old Way: You try to guess what the temperature would have been if the thermometer didn't melt.
• The New Way (Dimensional Regularization): Instead of measuring in our normal 3D world, the math temporarily "softens" the rules of the universe. It treats the space as if it has a tiny bit less than 4 dimensions (like 3.99 dimensions).

In this "softened" space:

1. The "infinity" at zero distance doesn't explode. It becomes a manageable, finite number (a "pole" that can be handled).
2. The math flows smoothly from the "Coordinate" view to the "Momentum" view without needing to make arbitrary guesses.
3. When the math is finished, the scientists "turn the dial back" to our normal 4D world, and the result is clean, consistent, and free of the previous ambiguity.

Why This Matters

• Consistency: This method proves that if you do the math in Coordinate Space and translate it, you get the exact same answer as if you did the math directly in Momentum Space. The "translation error" is gone.
• Lattice QCD: This is crucial for "Lattice QCD," a method where supercomputers simulate the universe on a grid (like a pixelated screen). These simulations naturally produce data in "Coordinate Space." To get real-world predictions (like how a proton behaves in a collider), they must translate to "Momentum Space." This paper provides the official, correct rulebook for that translation, ensuring that simulations of gluon and quark mixing are now accurate and reliable.

Summary

The paper solves a decades-old puzzle where two ways of describing particle physics gave conflicting answers when particles got too close. The authors found that the conflict came from a lack of a proper rule for handling "zero distance." By using a mathematical technique called Dimensional Regularization, they created a consistent rule that works for both descriptions, ensuring that future calculations of how quarks and gluons mix are accurate and unambiguous.

Technical Summary

Technical Summary: Regularization Prescription for the Mixing Between Nonlocal Gluon and Quark Operators

Problem Statement
In the study of parton physics, particularly within the framework of light-front (LF) quantization and lattice QCD (via quasi-LF operators), a persistent ambiguity exists regarding the mixing between flavor-singlet quark operators and gluon operators. This issue arises specifically in the "gluon-in-quark" channel.

The core difficulty stems from the mismatch in mass dimensions between the gluon LF operator (dimension 4) and the quark LF operator (dimension 3). To compensate, the mixing coefficient must incorporate the lightlike (or spacelike) separation $z_{12}$ of the nonlocal operators, resulting in a term proportional to $1/z_{12}$. When transforming results from coordinate space to momentum space via Fourier transform, this $1/z_{12}$ singularity at $z_{12} \to 0$ leads to an indefinite integration limit.

Conventional approaches attempt to resolve this by:

1. Replacing the coordinate space result with an integral form with an arbitrary lower limit $a$, leading to results dependent on the choice of $a$.
2. Matching Mellin moments between coordinate and momentum space. However, the authors argue this method is insufficient because it implicitly assumes specific integration constants and does not guarantee a unique solution under charge conjugation symmetry.

These ambiguities hinder the consistent extraction of parton distribution functions (PDFs) and generalized parton distributions (GPDs) from lattice QCD calculations, where nonlocal operators are evaluated at finite spacelike separations.

Methodology
The authors propose that the ambiguity is fundamentally due to the lack of a proper regularization prescription for the singularity at zero field separation. To resolve this, they employ Dimensional Regularization (DR) with $d = 4 - 2\epsilon$.

The methodology involves:

1. Coordinate Space Analysis: Examining the Operator Product Expansion (OPE) and factorization formulas for nonlocal correlators. The authors demonstrate that in the limit $z_{12} \to 0$, the factorization formulas contain singular terms (e.g., $\ln z_{12}^2$ or $1/z_{12}$). By applying DR, these singularities are regulated, allowing for a smooth local limit that recovers the expected mixing patterns of local operators.
2. Momentum Space Consistency: The authors perform the Fourier transform of the regularized coordinate space kernels into momentum space. They calculate the matching kernels for quasi-PDFs and quasi-GPDs in $d$ dimensions.
3. Comparison with Alternative Methods:
   • They compare their DR-derived kernels ($C_{gq}^{FT, DR}$) with kernels calculated directly in momentum space ($C_{gq}^{mom}$) and those derived using a Principal-Value (PV) prescription.
   • They analyze the deficiencies of the Mellin moment matching approach, showing it fails to uniquely fix the integration constants required for a consistent Fourier transform.
   • They test the PV prescription, finding that while it agrees with DR at one-loop for forward correlators, it yields inconsistent results (specifically, non-zero local limits where zero is expected) for polarized nonforward correlators.

Key Contributions and Results

• Resolution of Ambiguity: The paper demonstrates that Dimensional Regularization provides a unique and consistent prescription for the $1/z_{12}$ pole. It yields matching kernels that are consistent in both coordinate and momentum space.
• Consistency with Local Limits: Using DR, the local limit ($z_{12} \to 0$) of the nonlocal operator mixing correctly reproduces the known mixing of local operators (e.g., the mixing of gluon average momentum with quark average momentum). Specifically, for the polarized case, the DR prescription ensures the local limit vanishes as required by symmetry, whereas other prescriptions may fail.
• Equivalence of Approaches: The authors show that the matching kernels derived via DR in coordinate space, when Fourier transformed, yield results that are physically equivalent to those calculated directly in momentum space. While superficial differences exist in the functional form of the kernels (attributed to integration-by-parts manipulations in coordinate space), these differences vanish upon convolution with physical parton distributions due to symmetry properties.
• Validation of Mellin Matching: The paper clarifies that while Mellin moment matching can yield equivalent results in specific cases, it is not a rigorous substitute for a proper regularization of the singularity, as it relies on implicit assumptions about integration constants.
• Lattice Compatibility: The DR prescription is shown to be compatible with lattice extractions of parton distributions. Since lattice calculations implicitly include contributions from all orders in $\alpha_s$ (resumming singular terms like $\ln z_{12}^2$), the DR approach, which effectively resums these terms via the renormalization group, aligns with the smooth behavior observed in lattice data near the local limit.

Significance
The authors conclude that their work resolves a long-standing challenge in relating coordinate space and momentum space results for nonlocal operator mixing. By establishing Dimensional Regularization as the correct prescription for handling the singularity at zero field separation, the paper provides a unified framework for:

• Ensuring consistency between evolution equations in coordinate and momentum space.
• Facilitating the systematic study of factorization and mixing for quasi-LF operators.
• Improving the reliability of lattice QCD calculations for gluon PDFs, GPDs, and singlet quark distributions within the Large Momentum Effective Theory (LaMET) framework.

The paper asserts that this prescription eliminates the ambiguity in the gluon-in-quark channel, thereby enabling more precise theoretical predictions and lattice extractions of fundamental parton quantities.