Probing Gluon TMD Models with Drell--Yan Structure Functions

This paper employs high-energy factorization to compute Drell-Yan structure functions at 8 TeV using four distinct gluon TMD models, demonstrating that a modified Weizsäcker-Williams model provides the best agreement with ATLAS 2016 data and highlighting the critical role of specific model features in future TMD fits.

The Gist

Imagine you are trying to figure out what's inside a sealed, black box. You can't open it, but you can smash two of these boxes together at incredible speeds and watch the debris fly out. By studying the patterns of the debris, you can deduce what the boxes were made of.

This is essentially what physicists do at the Large Hadron Collider (LHC). They smash protons together to understand the fundamental building blocks of matter. This specific paper is about a new way of looking at the "debris" to see if we are correctly understanding the gluons—the tiny particles that act like super-strong glue holding protons together.

Here is a breakdown of the paper using everyday analogies:

1. The Problem: The "Flat Map" vs. The "3D Terrain"

For a long time, physicists have used a "flat map" to describe protons. This map tells you how much "stuff" (momentum) is moving forward inside the proton. It's like a highway map that only shows traffic moving North or South.

However, we know the traffic isn't just moving North/South. It's also swerving left and right (sideways momentum). The old "flat map" ignores this swerving. When physicists tried to predict what happens when protons collide using this flat map, the predictions didn't quite match the actual data, especially for a specific pattern of debris called the Lam-Tung relation. It was like trying to navigate a mountainous region using a map that only showed a flat plain; you'd get lost.

2. The Solution: The "3D GPS" (TMDs)

The author of this paper, Jan Ferdyan, suggests we need a "3D GPS" instead of a flat map. This is called a Transverse Momentum Dependent (TMD) model. It accounts for the sideways swerving of the gluons.

Think of it this way:

• Old Model (Collinear): Imagine a crowd of people walking in a straight line. You only care about how fast they are walking forward.
• New Model (TMD): Imagine that same crowd, but they are also jostling, bumping into each other, and moving side-to-side. To predict where they will end up, you need to know both their forward speed and their side-to-side wobble.

3. The Experiment: The "Drell-Yan" Crash

The paper focuses on a specific type of crash called the Drell-Yan process.

• The Setup: Two protons collide.
• The Result: They create a pair of particles (a lepton and an antilepton) that fly off in different directions.
• The Clue: The angle at which these particles fly tells us about the internal "swerving" (transverse momentum) of the gluons inside the proton before the crash.

The author calculated what these angles should look like if the gluons were behaving according to four different "3D GPS" theories (models).

4. The Four Competing Theories

The author tested four different ways to describe this "swerving":

1. The Gaussian Model: A simple, bell-curve guess. Like assuming everyone in the crowd swerves randomly but mostly stays close to the center.
2. The Jung-Hautmann (JH) Model: A complex, computer-simulated evolution based on deep quantum rules. Like a sophisticated traffic simulation that accounts for every car's history.
3. The KMR Model: A model that tries to bridge the gap between the old flat map and the new 3D view.
4. The Weizsäcker-Williams (WW) Model: A model based on the idea that gluons are emitted by quarks like light from a flashlight.

5. The "Tweaks" (Phenomenological Adjustments)

The author realized that the raw theories didn't perfectly match the real-world data (from the ATLAS experiment at CERN). So, they made some "tweaks":

• Normalization: They adjusted the overall size of the prediction to fit the total number of crashes observed.
• Rescaling: They realized that because the gluons are swerving sideways, the "forward speed" (momentum fraction) needs to be adjusted. It's like realizing that if a car is driving diagonally, its forward speed is slower than its total speedometer reading. They tested different ways to adjust this math.

6. The Results: Who Won?

The author compared all these predictions against the actual data from the ATLAS experiment.

• The Winner: A modified version of the Weizsäcker-Williams (WW) model, specifically the one where they adjusted the "forward speed" (rescaling), performed the best.
• Why it matters: This model (called WW(3)) didn't just get the total number of crashes right; it perfectly predicted the angles of the debris, including the tricky Lam-Tung relation that the old flat maps failed to explain.

7. The Takeaway

This paper is a victory lap for the "3D GPS" approach. It shows that:

1. Ignoring the sideways swerving of gluons leads to wrong answers.
2. The specific way gluons "swerve" (the TMD shape) is crucial.
3. The Weizsäcker-Williams model, with a few smart adjustments, seems to be the best description of how gluons behave inside a proton right now.

In simple terms: The author tried four different recipes to describe the "jiggling" inside a proton. By comparing the results to real-life crash data, they found that one specific recipe (the tweaked Weizsäcker-Williams one) cooks up the most accurate picture of reality. This helps physicists build better models for the future, potentially leading to a deeper understanding of the universe's fundamental structure.

Technical Summary

1. Problem Statement

The Drell–Yan (DY) process is a primary probe for understanding the internal structure of hadrons. While standard collinear perturbative QCD (pQCD) describes many observables accurately, it faces challenges in describing the Lam–Tung relation ($A_0 - A_2 = 0$) at higher transverse momenta ($q_T$) and fails to fully capture the angular distribution of dileptons at $q_T \sim M$ (where $M$ is the dilepton invariant mass).

The discrepancy between Next-to-Next-to-Leading Order (NNLO) collinear predictions and experimental data (specifically the Lam–Tung combination $A_{LT}$) suggests that parton transverse momentum ($k_T$) effects are significant and must be incorporated from the outset. The paper aims to test various Transverse Momentum Dependent (TMD) gluon distribution models against high-precision Drell–Yan structure function data from the ATLAS collaboration to identify which TMD frameworks best describe the internal dynamics of the proton.

2. Methodology

Formalism: High-Energy ($k_T$) Factorization

The authors utilize the high-energy factorization (or $k_T$-factorization) framework. Unlike collinear factorization, this approach employs off-shell initial partons and resums large logarithms of $1/x$.

• Process: Proton-Proton collisions at $\sqrt{S} = 8$ TeV producing a virtual electroweak boson ($V^* = \gamma^*, Z^0$) which decays into a dilepton pair.
• Channels: The calculation includes two tree-level scattering channels:
  1. $q_{val}g^* \to qV^*$: Scattering of a collinear valence quark and an off-shell gluon.
  2. $g^*g^* \to qqV^*$: Scattering of two off-shell gluons.
• Observables: The authors compute the differential cross-section decomposed into helicity structure functions ($A_0$ through $A_9$). Special attention is given to:
  • Parity-conserving functions: $A_0, A_1, A_2$.
  • The Lam–Tung combination: $A_{LT} = A_0 - A_2$.
  • Parity-violating functions: $A_3, A_4$.

Gluon TMD Models Tested

Four distinct classes of gluon TMD models were evaluated, along with phenomenological modifications:

1. Gaussian (Gauss.): A quasi-collinear model inspired by the Golec-Biernat–Wüsthoff saturation model, featuring a narrow $k_T$ dependence.
2. Jung–Hautmann (JH): Derived from the CCFM evolution equation, which interpolates between BFKL (small $x$) and DGLAP evolution. Three grid sets were tested (JH2013, PB2018, PB2023).
3. Kimber–Martin–Ryskin (KMR): Based on DGLAP evolution with the last splitting treated as non-integrated, including a Sudakov form factor.
4. Weizsäcker–Williams (WW): A phenomenological model where gluon TMDs are driven by gluon emission from valence quarks.
   • Modifications: The authors introduced modified WW (WW') and modified KMR models. These modifications replace the intrinsic $x$-dependence of the TMD with that of the collinear gluon PDF and apply $x$-rescaling ($x \to x/2, x/4$) to account for the kinematic differences between collinear and $k_T$-factorization processes.

Data and Analysis

• Data Source: ATLAS 2016 data at $\sqrt{S} = 8$ TeV, focusing on the $Z^0$ resonance region ($M = M_Z$).
• Normalization: Since many models underestimated the total cross-section, models were normalized to experimental data.
• Statistical Metric: The goodness of fit was quantified using the reduced chi-squared ($\chi^2$) per degree of freedom for individual structure functions and a global $\chi^2$ average.

3. Key Contributions

1. Systematic Comparison of TMD Models: The paper provides the first comprehensive comparison of four distinct gluon TMD families (Gaussian, JH, KMR, WW) specifically against the full set of DY structure functions, rather than just the total cross-section.
2. Phenomenological Rescaling: The introduction of $x$-rescaling in the KMR and WW models to bridge the gap between collinear PDF kinematics and $k_T$-factorization kinematics. This was crucial for improving the description of the total cross-section and specific structure functions.
3. Channel Decomposition Analysis: The authors explicitly analyzed the contribution of the $q_{val}g^*$ vs. $g^*g^*$ channels to the structure functions. They demonstrated that the $q_{val}g^*$ channel dominates the parity-violating functions ($A_3, A_4$) and significantly influences the shape of $A_1$ and $A_4$, while the $g^*g^*$ channel drives the total cross-section magnitude.
4. Identification of Model Sensitivities: The study isolates which features of a TMD model (e.g., $k_T$ fall-off vs. $x$-dependence) are most critical for describing specific observables.

4. Results

• Best Overall Model: The modified Weizsäcker–Williams model with $x/4$ rescaling (WW(3)) provided the best global description of the data, yielding the minimum global $\chi^2$ (normalized to 1.00 in Table 3).
• Structure Function Specifics:
  • $A_0$ and $A_2$: All models described $A_0$ well. However, $A_2$ at high $q_T$ clearly distinguished models; WW-based models matched data best, while Gaussian and JH models significantly overestimated it.
  • Lam–Tung ($A_{LT}$): The WW(3) model provided the best description of the Lam–Tung combination, closely matching the data trend where $A_{LT}$ rises with $q_T$. The Gaussian model performed poorly here.
  • $A_1$: The WW-based models (especially WW(3)) significantly outperformed others, accurately capturing the local maximum position and value. KMR and JH models overestimated $A_1$ at low $q_T$.
  • Parity-Violating ($A_3, A_4$): These functions showed the largest discrepancies. The $A_4$ function was particularly sensitive to the TMD model choice, with the WW model family showing a convexity opposite to the data in some cases.
• Impact of $x$-Rescaling: Rescaling the $x$ variable (e.g., $x \to x/4$) generally improved the description of the total cross-section and the $A_1$ function, suggesting that the effective $x$ in $k_T$-factorization is smaller than in the collinear limit.
• Channel Dominance: The analysis revealed that models with a dominant $g^*g^*$ contribution (like JH2013 and KMR(3)) tended to describe $A_3$ better, while models with a stronger $q_{val}g^*$ component (like WW) were better for $A_1$ and $A_4$.

5. Significance

• Validation of TMD Frameworks: The study confirms that Drell–Yan structure functions are highly sensitive probes for gluon TMDs, capable of distinguishing between models that might yield similar total cross-sections.
• Guidance for Future Fits: The results suggest that future global fits of gluon TMDs should prioritize:
  1. A Weizsäcker–Williams-like $k_T$ dependence (slow fall-off).
  2. A distinct $x$-dependence that differs from standard collinear PDFs, likely requiring rescaling to account for transverse momentum effects.
• Theoretical Implications: The success of the WW(3) model implies that the simple phenomenological ansatz of gluon emission from valence quarks, combined with appropriate kinematic rescaling, captures the essential physics of the proton's transverse structure better than complex evolution-based grids (like CCFM) in this specific kinematic regime.
• Future Directions: The authors propose using this formalism for dedicated TMD fits and suggest that including Next-to-Leading Order (NLO) corrections and off-shell initial quark channels could further refine the description, particularly for the parity-violating functions $A_3$ and $A_4$.

In conclusion, the paper demonstrates that the interplay between transverse momentum and longitudinal momentum fraction ($x$) is critical for describing Drell–Yan data, and that modified phenomenological TMD models can outperform rigorous evolution-based models in specific kinematic regimes.