Lattice extraction of the Collins-Soper kernel using the auxiliary field representation of the Wilson line

This paper presents a preliminary lattice extraction of the Collins-Soper kernel in Collins' scheme using an auxiliary fermion field representation of the Wilson line, comparing the "ratio" and "double ratio" methods to achieve high statistical precision.

The Gist

Imagine you are trying to understand the internal structure of a proton (a building block of matter). You want to know not just how heavy it is, but how its tiny parts (quarks and gluons) are moving sideways as they zoom around. In physics, this sideways movement is called Transverse Momentum.

To map this out, scientists use a mathematical tool called the Collins-Soper (CS) kernel. Think of the CS kernel as the "instruction manual" or the "weather forecast" for how these particles move. It tells you how the sideways motion changes depending on how fast the particles are moving relative to each other.

However, there's a huge problem: The computer simulations we use to study these particles (Lattice QCD) live in a different "universe" than the real one.

The Problem: The Time Travel Paradox

Real life (Minkowski space) has time flowing forward. Our computer simulations (Euclidean space) treat time like a fourth dimension of space. It's like trying to measure the speed of a car by looking at a frozen photograph; you can see where it is, but you can't see it moving.

Because of this, a specific type of mathematical "divergence" (a number that blows up to infinity) called rapidity divergence appears in the real world but has no direct equivalent in the computer simulation. It's like trying to measure the temperature of a fire using a ruler; the tools just don't match.

The Solution: The "Ghost" Helper

The authors of this paper found a clever workaround. Instead of trying to force the computer to simulate time directly, they introduced a "Ghost" helper (an auxiliary field).

• The Analogy: Imagine you want to know how a heavy box moves through a crowd, but you can't simulate the crowd's movement directly. Instead, you attach a "ghost" string to the box. This string follows a specific path. By studying how the string behaves, you can figure out how the box moves without ever simulating the crowd directly.
• The Magic: In this paper, the "string" is a Wilson line (a path through the quantum field). The authors realized that if they make the "time" part of this string purely imaginary (a mathematical trick), the "ghost" string behaves in the computer simulation exactly like a real-time string would in the physical world.

The Method: The "Double Ratio" Trick

Even with the ghost string, the computer data is messy. It's full of "static" (noise) and "glitches" (mathematical infinities caused by the grid size of the simulation).

To clean this up, the authors use a method called the "Double Ratio."

• The Analogy: Imagine you are trying to measure the exact height of a tree, but your ruler is slightly warped and the ground is uneven.
  1. Single Ratio: You measure the tree against a known pole. This cancels out some of the ruler's warping.
  2. Double Ratio: You measure the tree against the pole, and then you measure a second, slightly different tree against the same pole. By comparing the two results, the errors from the ruler and the ground cancel each other out completely.

In this paper, they compare different "ghost strings" of different lengths and angles. By taking the ratio of these ratios, the messy mathematical noise disappears, leaving behind a clean, precise value for the Collins-Soper kernel.

The Results: A New Map

The team ran these simulations on supercomputers using different grid sizes (like zooming in and out on a map).

1. They proved the theory: They showed mathematically that their "ghost string" method works perfectly, matching the real-world physics equations.
2. They got the data: They extracted the first preliminary numbers for the Collins-Soper kernel using this new method.
3. The Challenge: The numbers are promising, but they still have some "systematic uncertainty." This is like having a very precise map, but you aren't 100% sure where the "North Pole" is on your map yet. They need to refine their matching process to get the final, perfect result.

Why Does This Matter?

If we can perfectly map the Collins-Soper kernel, we can finally understand the 3D structure of protons with incredible precision. This helps us:

• Understand the fundamental forces of nature.
• Interpret data from giant particle colliders (like the Large Hadron Collider) more accurately.
• Potentially discover new physics hidden inside the "noise" of current experiments.

In short: The authors built a mathematical "time machine" using ghost strings and double-ratio math to translate computer simulations into real-world physics, successfully extracting a key piece of the puzzle that describes how matter moves.

Technical Summary

Here is a detailed technical summary of the paper "Lattice extraction of the Collins-Soper kernel using the auxiliary field representation of the Wilson line."

1. Problem Statement

The investigation of Transverse Momentum Dependent (TMD) physics is crucial for mapping the 3D structure of hadrons. However, TMD factorization formulas involve rapidity divergences that must be regularized, giving rise to the Collins-Soper (CS) kernel ($\gamma_q$), which governs the rapidity evolution of TMD observables.

The Core Challenge:

• Rapidity has no direct analogue in Euclidean space, making standard lattice QCD calculations of the CS kernel difficult.
• Previous attempts using time-like Wilson lines in Euclidean space (analogous to Heavy Quark Effective Theory) lead to divergent integrals in perturbative calculations.
• There is a need for a method to compute the soft function and extract the CS kernel directly from Euclidean lattice simulations while maintaining a rigorous connection to Minkowski space physics.

2. Methodology

The authors propose a novel approach combining auxiliary fermion fields with complex directional vectors for Wilson lines.

A. Theoretical Framework

• Complex Wilson Lines: Instead of time-like vectors, the authors use Wilson lines with purely imaginary time components in Euclidean space: $\tilde{n} = (i n_0, \vec{0}_\perp, n_3)$.
• Mapping to Minkowski Space: These complex vectors map directly to space-like Wilson lines in Minkowski space (Collins' scheme). The authors verified this mapping to one-loop order in perturbation theory.
• Auxiliary Field Representation: The Wilson line operator is represented as a path integral over 1D auxiliary fermion fields ($\psi, \bar{\psi}$). The propagator of this field satisfies a Green's function equation with a directional vector. This allows the Wilson line to be solved iteratively in Euclidean time.
• Butterfly Loop: The computation focuses on the "butterfly loop" diagram (Fig. 1), which represents the TMD soft function.

B. Perturbative Analysis

• Infinite vs. Finite Length: The authors performed a one-loop calculation for both infinite and finite length Wilson lines.
• Convergence Conditions: They derived that for the Euclidean integral to converge, the directional parameters $r_a, r_b$ must satisfy $|r| > 1$. This condition is naturally met by space-like vectors but fails for time-like vectors, explaining why the latter approach yields divergent integrals.
• Renormalization: Lattice regularization breaks O(4) symmetry, requiring the renormalization of the directional vectors (rapidity).

C. Numerical Strategy: The Double Ratio

To handle lattice artifacts (linear divergences in $1/a$and$L/a$) and extract the CS kernel, the authors employ a Double Ratio method:

1. Single Ratio ($R_{single}$): Compares soft functions at different transverse separations ($b_\perp$) to cancel UV cutoff effects introduced by the auxiliary propagator.
2. Double Ratio ($S_{double}$): Combines ratios at different rapidities ($r$) and different $b_\perp$ values.
   $$S_{double} \propto \exp\left[ (\gamma_q(b_{\perp,1}) - \gamma_q(b_{\perp,2})) \cdot 2(y_1 - y_2) \right]$$
   This construction eliminates linear divergences ($b_\perp/a$ and $L/a$) and isolates the difference in the CS kernel.

D. Extraction Procedure

1. Compute the double ratio on the lattice for large Euclidean time ($\tau$) to reach a plateau.
2. Determine the renormalized rapidity ($y_{ren}$) by matching the lattice double ratio to perturbative QCD results in a specific "perturbative window" ($3a \lesssim b_\perp \lesssim 0.2$ fm).
3. Use the renormalized rapidity difference to extract the CS kernel difference $\gamma_q(b_\perp) - \gamma_q(b'_{\perp})$.
4. Match to perturbative results to obtain the absolute value of the CS kernel.

3. Key Contributions

• Validation of Space-like Mapping: Demonstrated that using complex directional vectors in Euclidean space corresponds to space-like Wilson lines in Minkowski space, providing a convergent alternative to time-like approaches.
• Auxiliary Field Implementation: Successfully implemented the auxiliary field representation for Wilson lines on the lattice, solving the propagator iteratively with improved stability.
• Double Ratio Technique: Developed a robust double-ratio method that cancels linear divergences and operator renormalization factors, allowing for high statistical precision in extracting the CS kernel.
• One-Loop Verification: Provided a rigorous one-loop perturbative proof that the Euclidean auxiliary field formulation correctly reproduces the Minkowski space soft function in the space-like regime.

4. Results

• Preliminary Extraction: The authors present a preliminary extraction of the CS kernel using the double ratio method on quenched lattice configurations (Table 1).
• Plateau Behavior: Figure 3 shows that the single ratio $R_{single}$ exhibits a stable plateau in Euclidean time, confirming that UV cutoff effects are successfully cancelled as predicted.
• Kernel Values: Figure 4 displays the extracted CS kernel ($\gamma_q$) for various $b_\perp$ values within the perturbative window. The results are consistent with N3LL (Next-to-Next-to-Next-to-Leading Log) perturbative predictions.
• Systematic Uncertainty: The current uncertainty is dominated by systematic effects related to the perturbative matching window and the choice of renormalization scale. Figure 5 illustrates a conservative estimate of these errors.
• Data Status: The results are based on configurations with lattice spacings $a = 0.041$ fm and $0.048$fm. The authors are currently processing finer lattices ($a=0.03$ fm) to better control the perturbative window.

5. Significance

• Bridging Euclidean and Minkowski: This work provides a concrete pathway to calculate non-perturbative TMD quantities (specifically the CS kernel) on the lattice, which are essential for precision phenomenology in high-energy physics (e.g., Drell-Yan and Higgs production).
• Solving the Rapidity Problem: By utilizing space-like Wilson lines via complex vectors, the authors circumvent the fundamental difficulty of defining rapidity in Euclidean space.
• Precision Tool: The double ratio method offers a high degree of statistical precision, making it a viable tool for future high-precision TMD studies.
• Future Outlook: The methodology lays the groundwork for future calculations with dynamical fermions and finer lattice spacings, potentially leading to the first ab initio determination of the Collins-Soper kernel from lattice QCD.