A first extraction of gluon TMDs from Higgs data at the LHC

This paper presents the first extraction of the unpolarised gluon transverse-momentum-dependent parton distribution from LHC Higgs-boson production data by fitting ATLAS and CMS measurements within a TMD factorisation framework that includes N$^3$LL accuracy and linearly polarised gluon contributions.

The Gist

The Big Picture: Mapping the Invisible Inside a Proton

Imagine a proton (a tiny particle inside an atom) not as a solid marble, but as a bustling, high-speed city. Inside this city, there are tiny messengers called partons (mostly gluons) zooming around.

For a long time, scientists have had a map of this city that only showed how many messengers were moving in a straight line (forward). This paper is about creating a much more detailed 3D map. It doesn't just tell us how many messengers are there; it tells us how much they are wiggling side-to-side as they zoom forward. This "side-to-side wiggling" is what physicists call transverse momentum.

The authors of this paper have successfully created the first-ever detailed map of this side-to-side motion specifically for gluons (the messengers that hold the proton together) by looking at data from the Large Hadron Collider (LHC).

The Experiment: Catching a Ghost in a Flash

How do you map something you can't see? You have to look at the "footprints" it leaves behind.

1. The Collision: At the LHC, they smash protons together at incredible speeds.
2. The Target: Sometimes, these collisions create a Higgs boson (a heavy, unstable particle). Think of the Higgs as a rare, glowing firework that explodes almost instantly.
3. The Footprints: When the Higgs explodes, it turns into other particles (like two flashes of light or four particles of matter). The scientists measured how much the Higgs was "wiggling" sideways before it exploded.
4. The Clue: The amount of sideways wiggle in the Higgs is directly caused by the sideways wiggles of the gluons inside the protons that created it. By measuring the Higgs, they can reverse-engineer the map of the gluons.

The Challenge: Seeing Through the Fog

The authors faced two main problems, which they solved with clever math:

• The "Fog" of Uncertainty: At very low sideways speeds, the math gets messy because of "quantum fog" (non-perturbative effects). It's like trying to see a car driving through thick fog; you can't see the details clearly. To fix this, the team used a mathematical "lens" (called a Gaussian parametrization) to estimate what the fog looks like. They found that while they could see the general shape of the map, the "fog" was still a bit thick, meaning they couldn't pinpoint the exact details of the wiggles with 100% precision yet.
• The "Zoom" Level: The math works best when you look at the Higgs moving very slowly sideways. If it moves too fast, the rules of the game change. The team had to be very strict, only looking at data where the Higgs was moving slowly enough to fit their "slow-motion" rules. They tested different "slow-motion" limits to make sure their map wasn't biased by the data they threw away.

The Results: A Good First Draft

• The Map: They produced a graph showing how likely gluons are to wiggle at different speeds. They found that the map looks "broad" (the gluons wiggle a lot) and gets wider as the energy of the collision increases.
• The Fit: When they compared their theoretical map to the actual data from the ATLAS and CMS experiments (the giant detectors at the LHC), the shapes matched up very well. The data and the theory agreed on both the shape of the distribution and the number of events.
• The Precision: They tested their math at different levels of complexity (like checking a calculation with a calculator, then a supercomputer, then a quantum computer). They found that once they reached a very high level of complexity (called N3LL), the results stopped changing much. This tells them their math is stable and reliable.

What They Didn't Do (And Why)

The paper is very careful to say what it didn't do:

• They didn't map the "wiggles" of the gluons based on how much energy they carry (the "x" dependence) because the current data isn't detailed enough to show that. Their map is currently driven by the math they used to fill in the gaps, not by the data itself.
• They couldn't separate the "intrinsic wiggles" (how the gluon naturally moves) from the "evolution wiggles" (how the movement changes as energy changes) because all their data came from the same energy level. They need data from different energy levels to untangle these two effects.

The Bottom Line

This paper is a milestone. It is the first time scientists have successfully used Higgs boson data to draw a map of how gluons move sideways inside a proton.

Think of it as taking the first blurry photo of a fast-moving animal. The photo isn't perfectly sharp yet (there is still some uncertainty about the exact details), but it clearly shows the animal's shape, size, and how it moves. This "first photo" provides a solid foundation for future scientists to take sharper, more detailed pictures as they gather more data from the LHC.

Technical Summary

Technical Summary: First Extraction of Unpolarised Gluon TMDs from LHC Higgs Data

Problem Statement
While quark Transverse-Momentum-Dependent (TMD) parton distribution functions have been extensively constrained over the last two decades using Drell-Yan and semi-inclusive deep-inelastic scattering data, gluon TMDs remain poorly constrained. This asymmetry stems from the scarcity of experimental processes with colour-singlet final states, which are required to avoid colour-entanglement issues that compromise TMD factorisation. Inclusive Higgs boson production in proton-proton collisions, dominated by gluon fusion, represents the cleanest probe for unpolarised gluon TMDs ($f_1^g$) and linearly polarised gluon TMDs ($h_1^{\perp g}$). Despite theoretical advancements in calculating the Higgs transverse momentum ($q_T$) distribution up to N$^3$LL accuracy, a direct phenomenological extraction of gluon TMDs from experimental data has remained an open challenge due to the recent availability of precision measurements from the ATLAS and CMS experiments.

Methodology
The authors present the first extraction of the unpolarised gluon TMD ($f_1^g$) within the TMD factorisation framework, utilizing the complete set of available Higgs $q_T$ distribution measurements from ATLAS and CMS at center-of-mass energies of $\sqrt{s} = 8$ and $13$ TeV. The analysis focuses on the diphoton ($H \to \gamma\gamma$) and four-lepton ($H \to 4\ell$) decay channels, restricted to the small-$q_T$ region ($q_T \ll M_H$) where TMD factorisation is valid.

Key methodological components include:

• Theoretical Framework: The differential cross-section is computed up to N$^3$LL accuracy, incorporating the contribution of the linearly polarised gluon TMD ($h_1^{\perp g}$) with its NNLO matching coefficient. The calculation employs the $b$-space formalism, matching TMDs to collinear PDFs (NNPDF3.1) at small $b_T$ and introducing a nonperturbative function $f_{NP}$ for large $b_T$.
• Nonperturbative Parametrisation: A Gaussian parametrisation, $f_{NP} = \exp[-\frac{1}{2}g b_T^2]$, is adopted for the nonperturbative content. The parameter $g$ is treated as a free parameter to be fitted. The analysis assumes the same nonperturbative function for both unpolarised and linearly polarised gluons due to current data limitations.
• Experimental Data Handling: The fit utilizes the NangaParbat framework (MAP Collaboration). Fiducial selection cuts specific to each experiment and decay channel are consistently incorporated via a phase-space reduction factor $P$.
• Fit Strategy: A global $\chi^2$ fit is performed using a bootstrap method with 200 Monte Carlo replicas of the experimental data. The analysis is restricted to data points satisfying $q_T/M_H < 0.3$ (baseline), with variations to $0.25$ and $0.2$ to assess stability.

Key Results

• Fit Quality: The baseline fit (N$^3$LL, $q_T/M_H < 0.3$) yields a global $\chi^2/N_{dat} = 1.49$ (p-value $\approx 3\%$), indicating a reasonable reproduction of both the shape and normalisation of the experimental data. While some tension exists, particularly in the CMS Run II combined dataset and the tail of the ATLAS Run II distribution, this is attributed to statistical fluctuations at relatively large $q_T$.
• Perturbative Convergence: The study demonstrates that the agreement between data and theory improves significantly when moving from NLL' to NNLL'. The transition from NNLL' to N$^3$LL results in only mild changes, suggesting the perturbative series has stabilized. The authors conclude that NNLL' accuracy or higher is necessary for a reliable extraction.
• Extracted Parameter: The fit determines the nonperturbative parameter $g = 15.6 \pm 5.1 \text{ GeV}^2$. After decomposing $g$ into an intrinsic component ($g_1$) and an evolution component (estimated via rescaling quark results), the intrinsic parameter is found to be $g_1 = 14.4 \pm 5.1 \text{ GeV}^2$. The relative uncertainty of approximately 35% reflects the moderate sensitivity of current Higgs data to the nonperturbative content, a consequence of the large hard scale ($M_H$) where perturbative contributions dominate.
• Stability: Varying the $q_T$ cut from $0.3$ to $0.2$ reduces the $\chi^2/N_{dat}$ to $1.08$ but significantly reduces the dataset size. The extracted values of $g_1$ across different cuts are compatible within one-sigma uncertainties, indicating the baseline choice does not introduce significant bias.

Significance and Claims
The paper claims to provide the first extraction of the unpolarised gluon TMD from LHC Higgs data. The analysis establishes a baseline for future extractions that will combine Higgs measurements with other gluon-sensitive processes spanning a broader range of hard scales.

The authors explicitly state that the current dataset offers only moderate sensitivity to the nonperturbative content of gluon TMDs. Consequently, the extraction of the $x$-dependence of the gluon TMD is currently driven entirely by the perturbative matching onto collinear PDFs, rather than by the data itself. The narrow-width approximation and the single hard scale of the dataset prevent the separate determination of the nonperturbative Collins-Soper kernel; this component was estimated by rescaling quark coefficients.

The work serves as a proof of concept, demonstrating that Higgs $q_T$ distributions can be used to constrain gluon TMDs, while highlighting the need for future data from different scales (e.g., quarkonium production) and higher luminosity (Run III and HL-LHC) to disentangle intrinsic nonperturbative effects from evolution and to constrain the $x$-dependence directly.