NNLO DGLAP splitting functions from collinear matching of TMDs

This paper presents a complete computation of NNLO helicity and transversity DGLAP splitting functions in both space-like and time-like kinematics, derived from N$^3$LO polarized TMD matching coefficients, to enable high-precision N$^3$LO cross-section calculations and N$^4$LL resummation for future spin physics at the Electron-Ion Collider.

The Gist

Imagine the inside of a proton (the tiny particle at the center of every atom) not as a solid ball, but as a bustling, chaotic city. Inside this city, there are tiny citizens called quarks and gluons zooming around. They are constantly colliding, changing speed, and interacting.

Physicists want to understand exactly how these citizens move and spin. To do this, they use a "map" called a TMD (Transverse-Momentum-Dependent) function. Think of this map as a 3D GPS that doesn't just tell you where a particle is, but also how fast it's moving sideways and which way it's spinning.

However, there's a problem. The rules of the city (Quantum Chromodynamics, or QCD) are incredibly complex. Calculating the exact movements of these particles is like trying to predict the path of a single raindrop in a hurricane while also knowing the wind speed of every other drop. It requires "higher-order" math—essentially, adding more and more layers of detail to the calculation to get it right.

The Big Breakthrough: The "Perfect Map"

This paper, presented by Yu Jiao Zhu, is about creating the most detailed map possible for these spinning particles.

Here is the breakdown using simple analogies:

1. The "Zoom-In" Problem (Collinear Matching)

Imagine you are looking at a high-resolution photo of a crowd. From far away, you just see a blur of colors. As you zoom in, you start to see individual people.

• The TMDs are the high-res, zoomed-in view of the particles.
• The DGLAP Splitting Functions are the rules that tell us how the crowd changes as we zoom out or in. They describe how one particle can split into two, or how two merge, as the energy changes.

For a long time, scientists had a "good enough" map (called NLO or NNLO), but it had some blurry spots. This paper calculates the NNLO (Next-to-Next-to-Leading Order) splitting functions. In our analogy, this is like upgrading from a standard GPS to a satellite-guided, real-time 3D navigation system that accounts for every tiny bump in the road.

2. The Three Types of "Spin"

The paper focuses on three specific ways the particles in the proton city can behave:

• Helicity (The Helicopter): Imagine a particle spinning like a helicopter blade along its path. The paper maps out exactly how this spin changes when particles interact.
• Transversity (The Top): Imagine a particle spinning like a top on a table, sideways to its path. This is harder to see and harder to calculate. The paper finally provides the precise rules for this "sideways spin."
• Linearly Polarized Gluons (The Wobbly Glue): Gluons are the "glue" holding the quarks together. Sometimes, this glue isn't just round; it's squashed or stretched in a specific direction. The paper maps out how this "squashed glue" behaves.

3. The "Future-Proofing" (Small-x and Resummation)

The paper also looks at what happens when particles are moving extremely fast (a regime called "small-x").

• Analogy: Imagine a highway where traffic is so dense that cars are bumper-to-bumper. Standard rules break down.
• The authors developed a way to "resum" (re-summarize) the infinite number of tiny interactions that happen in this dense traffic. They created a formula that predicts the behavior of the city even when it's in a total traffic jam, ensuring the math doesn't crash.

Why Does This Matter? (The "So What?")

You might ask, "Why do we need a map this detailed?"

1. Solving the "Proton Spin Crisis": For decades, scientists thought the spin of a proton came mostly from its quarks. They were wrong! The quarks only account for about 30%. The rest comes from gluons and the orbital motion of the particles. This new, ultra-precise math helps us finally figure out where that missing spin is hiding.
2. The Electron-Ion Collider (EIC): A massive new machine is being built to smash electrons into protons to take "photos" of the inside of the atom. This paper provides the theoretical blueprint for that machine. Without these calculations, the data from the EIC would be like a blurry photo; with them, we get a crystal-clear image.
3. Precision Physics: Just as a watchmaker needs precise gears to make a perfect watch, physicists need these precise "splitting functions" to predict how particles will behave in experiments. If the math is off by even a tiny fraction, the predictions for new discoveries will be wrong.

In a Nutshell

Yu Jiao Zhu and their team have done the heavy lifting of calculating the ultimate rulebook for how spinning particles inside a proton split, merge, and move. They have filled in the missing pages of the book, corrected previous errors, and added a special chapter on how the rules change when traffic gets incredibly dense.

This work ensures that when the next generation of particle accelerators comes online, we will have the perfect map to navigate the mysterious, spinning world inside the atom.

Technical Summary

Here is a detailed technical summary of the paper "NNLO DGLAP splitting functions from collinear matching of TMDs" by Yu Jiao Zhu.

1. Problem Statement

The paper addresses the need for higher-order perturbative precision in the description of Transverse-Momentum-Dependent (TMD) parton distribution functions (PDFs) and fragmentation functions (FFs), specifically for polarized observables.

• Context: While unpolarized TMDs have seen significant progress (reaching N3LO matching), the polarized sector (helicity, transversity, and linearly polarized gluons) has lagged behind.
• Gap: A complete set of Next-to-Next-to-Next-to-Leading Order (N3LO) twist-2 matching coefficients for polarized TMDs was missing. Consequently, the associated NNLO DGLAP splitting functions in both space-like and time-like kinematics were not fully derived from this specific framework.
• Goal: To compute these missing higher-order ingredients to enable N3LO differential cross-section calculations and N4LL resummation for spin-dependent observables, particularly for the upcoming Electron-Ion Collider (EIC).

2. Methodology

The author employs Soft-Collinear Effective Theory (SCET) to perform the calculations. Key methodological features include:

• Operator Definitions: TMDs are defined using gauge-invariant collinear fields ($\chi_n$ for quarks, $A_{n\perp}$ for gluons) within the SCET framework. The definitions include Wilson lines to ensure gauge invariance.
• Rapidity Regulation: The calculation utilizes an exponential rapidity regulator. This is crucial for systematically handling rapidity divergences that arise in TMD factorization. The scheme preserves non-abelian exponentiation and ensures the zero-bin soft function is identical to the TMD soft function.
• Collinear Matching: The core calculation involves matching the TMD operators (defined at a low scale $\mu \sim b_T^{-1}$) onto the standard collinear PDFs/FFs (defined at a high scale $\mu$). This matching is performed at N3LO (order $\alpha_s^3$) for the twist-2 operators.
• Extraction of Splitting Functions: The DGLAP splitting functions are extracted from the renormalization group evolution of the matching coefficients. The calculation covers both space-like (PDFs) and time-like (FFs) kinematics.
• Small-$x$/$z$ Analysis: The paper investigates the small-$x$ (for PDFs) and small-$z$ (for FFs) behavior of the matching coefficients and performs resummation of these logarithms.

3. Key Contributions

The paper makes several distinct theoretical contributions:

1. Complete N3LO Twist-2 Matching: It provides the first complete computation of N3LO matching coefficients for:
   • Quark Helicity ($\Delta q$)
   • Quark Transversity ($\delta q$)
   • Linearly Polarized Gluons ($h_1^{\perp g}$)
   • Both distribution and fragmentation functions.
2. NNLO DGLAP Splitting Functions: From the matching results, the author derives the complete set of NNLO DGLAP splitting functions for the polarized sectors in both space-like and time-like regimes.
3. Resolution of Discrepancies: The author compares the new results with existing literature and identifies specific discrepancies:
   • Quark-Gluon Splitting: A discrepancy is found in the non-diagonal quark-gluon splitting function ($P_{qg}$) at NNLO compared to reference [136]. The paper provides an explicit formula for the difference $\Delta P_{qg}^{(2)}$.
   • Helicity Splitting: A discrepancy is identified in the cubic color structure ($d_{abc}^2$) term for the "sea" quark difference in the helicity splitting function compared to reference [139].
   • Transversity: The results for NNLO transversity splitting functions are confirmed to be consistent with existing literature [137–139].
4. Small-$z$ Resummation: The paper performs small-$z$ resummation for unpolarized and linearly polarized gluon TMD FFs, demonstrating that resummation effects are significant even at N3LO for $z < 10^{-2}$.

4. Key Results

• Splitting Functions: The paper presents the full analytical expressions for the NNLO polarized splitting functions. These are essential for evolving TMDs from the non-perturbative scale to the hard scattering scale.
• Matching Coefficients: The N3LO matching coefficients for polarized TMDs are now available, filling the gap that previously prevented N3LO precision predictions for spin-dependent processes.
• Discrepancy Formulas:
  • The difference in the quark-gluon splitting function is given by:
    $$\Delta P_{qg}^{T,(2)}(x) = \frac{\pi^2}{3}(C_F - C_A)\beta_0 [-4 + 8x + x^2 + 6(1-2x+2x^2)\ln x]$$
  • The difference in the helicity splitting function involving the $d_{abc}^2$ color structure is explicitly detailed in Eq. (20) of the paper.
• Resummation Impact: Figures 1–3 illustrate that for gluon TMD FFs, adding higher-order resummation contributions (truncated at $\alpha_s^{15}$) significantly alters the fixed-order predictions in the small-$z$ region.

5. Significance and Impact

• Precision Spin Physics: These results provide the necessary perturbative ingredients to compute N3LO differential cross-sections below the resolution scale $q_T^{cut}$ and enable N4LL resummation of $q_T$ observables in the Sudakov region for polarized processes.
• Electron-Ion Collider (EIC): The work establishes a consistent theoretical framework for precision spin measurements at the EIC, where understanding the proton spin structure (gluon helicity, orbital angular momentum) is a primary goal.
• Unified Evolution: The paper highlights the importance of a joint treatment of small-$x$ evolution (BFKL) and transverse-momentum resummation (Sudakov). The derived small-$x$ structure of the matching coefficients provides data for future efforts to unify these two resummation regimes, which is critical for high-energy phenomenology where both types of logarithms become relevant simultaneously.
• Standard Model Benchmark: By improving the theoretical control over polarized TMDs, the results enhance the precision of Standard Model benchmarks for processes like Drell-Yan, SIDIS, and Higgs production.

In summary, this work represents a major step forward in the perturbative QCD description of spin-dependent TMDs, resolving previous inconsistencies and providing the high-order tools required for the next generation of collider experiments.