⚛️ phenomenology

NNLL fit of the transverse-momentum-dependent distribution of unpolarised gluons to LHCb data on $J/\psi$ -pair production

This paper presents the first next-to-next-to-leading-logarithmic accuracy fit of the unpolarised gluon transverse-momentum-dependent distribution to LHCb $J/\psi$ -pair production data, introducing novel methodologies to incorporate perturbative and PDF uncertainties that are crucial for accurately describing low-scale gluon-sensitive measurements.

Original authors: D. Boer, J. Bor, A. C. Serri, M. G. Echevarria, J. P. Lansberg, S. F. Romera, P. Taels

Published 2026-09-02

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CC BY 4.0

Original authors: D. Boer, J. Bor, A. C. Serri, M. G. Echevarria, J. P. Lansberg, S. F. Romera, P. Taels

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside the heart of every atom lies a proton, a tiny, dense bundle of energy that seems simple but hides a chaotic, three-dimensional world within. Protons are not solid marbles; they are seething clouds of even smaller particles called quarks and gluons, held together by the strong force. While scientists have spent decades mapping how these particles move forward and backward inside the proton, a crucial piece of the puzzle has remained elusive: how they move sideways. This sideways motion, known as transverse momentum, is the missing dimension that completes the three-dimensional picture of the proton's interior. Understanding this sideways movement is essential because it reveals how the proton's internal structure changes depending on how hard we probe it, offering a deeper look into the fundamental laws that govern the universe.

A team of researchers has now taken a significant step forward in mapping this hidden landscape. By analyzing data from the Large Hadron Collider, specifically from the LHCb experiment at CERN, they have performed the first fit of the unpolarised gluon transverse-momentum-dependent distribution to extract how gluons—the particles that carry the strong force—move sideways inside a proton. They focused on a specific event where two particles called J/psi mesons are created simultaneously. These mesons are formed when a heavy charm quark and its antimatter partner, the anti-charm, collide and bind together. Because the creation of these pairs is driven almost entirely by the collision of gluons, the sideways motion of the resulting J/psi pair acts as a direct fingerprint of the gluons' own sideways movement before the collision occurred.

The challenge in this work was that the energy scales involved were relatively low, meaning the signals were faint and easily blurred by the messy, unpredictable effects of the quantum world. Previous attempts to measure this sideways motion had struggled to separate the clear signal of the gluons from the background noise and theoretical uncertainties. The researchers found that standard methods used to interpret such data were actually distorting the results, creating artificial wiggles and peaks in the data that did not reflect physical reality. These distortions arose because the mathematical tools used to bridge the gap between the known, predictable laws of physics and the unknown, fuzzy quantum effects were not precise enough for this specific type of low-energy collision.

To solve this, the team developed two new, independent strategies to clean up the data. They introduced a set of strict rules, or constraints, that forced their mathematical models to stay within the boundaries of what is physically possible and theoretically sound. One method checked the consistency of the data against the known uncertainties in the theory itself, while the other ensured that the final results looked smooth and natural, without the unphysical wiggles caused by the mathematical approximations. By applying these new rules, they were able to extract a clear, reliable picture of the gluon's sideways motion for the first time.

The results revealed that the gluons do indeed have a measurable sideways spread, and this spread grows as the energy of the collision increases. This confirms a key prediction of quantum theory: that as we look at particles with higher energy, they appear to have more room to move sideways. The team determined the size of this sideways spread with a specific value, finding that the average squared sideways momentum is roughly 0.17 to 0.36 square GeV, depending on the method used. These numbers are crucial because they provide the first solid experimental anchor for the non-perturbative part of the gluon's behavior—the part that cannot be calculated from first principles and must be measured.

Importantly, the study noted that the contribution from linearly polarized gluons is negligible in this specific analysis, and while the data hints at distinct behavior for gluons compared to quarks, the validity of scaling the gluon behavior to match quark behavior is not confirmed and calls for further investigation. The researchers also found that the uncertainty in their results is heavily influenced by the choice of the underlying model for the proton's structure, suggesting that future measurements will need to account for these variations to be truly precise.

This work marks a turning point in the study of the proton's three-dimensional structure. By successfully isolating the signal of the gluons from the noise and correcting for previous methodological flaws, the team has provided a robust foundation for future experiments. Their findings confirm that the proton is a dynamic, evolving object where the internal motion of its constituents changes with the energy of the probe. While the study does not yet explain every detail of the proton's inner life, it has successfully mapped the first major feature of the gluon's sideways journey, opening the door for even more detailed explorations of the quantum world in the years to come.

Technical Summary: NNLL Fit of the Unpolarised Gluon TMD to LHCb J/ψ-Pair Data

Problem and Context
Transverse Momentum Dependent (TMD) factorisation provides a framework to separate perturbative and non-perturbative dynamics in processes involving a small transverse momentum $q_T$ relative to a hard scale $Q$ . While quark TMDs are well-constrained by Drell-Yan and semi-inclusive deep-inelastic scattering (SIDIS) data, gluon TMDs remain poorly understood because they do not contribute to these processes at leading twist. Quarkonium-pair production, specifically $J/\psi$ -pair production at the LHCb experiment, offers a promising channel dominated by gluon fusion. However, previous attempts to extract the unpolarised gluon TMD ( $f_1^g$ ) from such data using Next-to-Leading Logarithmic (NLL) accuracy and conventional non-perturbative modelling yielded unsatisfactory results. Specifically, the theoretical transverse momentum distributions exhibited peaks at higher $q_T$ than observed, suggesting that the perturbative evolution was "too fast" or that the interplay between perturbative and non-perturbative components was mishandled.

Methodology
This work presents the first fit of the unpolarised gluon TMD $f_1^g$ to LHCb double-differential data on inclusive $J/\psi$ -pair production at $\sqrt{s}=13$ TeV, performed at Next-to-Next-to-Leading Logarithmic (NNLL) accuracy. The analysis focuses on the self-normalised differential cross section in bins of the pair's invariant mass $Q$ (6.6, 7.9, and 11 GeV) and transverse momentum $q_T$ (restricted to $q_T < Q/2$ ).

The authors identify critical limitations in conventional TMD fitting approaches, particularly regarding the $b_T$ -space prescription used to separate perturbative and non-perturbative regions. The standard $b^*_T$ prescription, combined with a non-perturbative Sudakov factor $S_{NP}$ , was found to artificially distort the perturbative calculation within the perturbative region ( $b_T < b_{T,max}$ ), leading to unphysical oscillations in the $q_T$ spectrum.

To address this, the paper proposes and implements two novel fitting strategies that incorporate theory constraints:

$b_T$ -Space Constraint: This method retains the conventional smooth $b^*_T$ prescription ( $n=2$ ) but modifies the $\chi^2$ minimisation. A penalty term is added to penalise non-perturbative parameters ( $A$ and $B$ ) that cause the theoretical prediction in $b_T$ space to deviate from the perturbative uncertainty band (estimated via 9-point scale variations). This ensures the non-perturbative contribution does not arbitrarily alter the perturbative input.
$q_T$ -Space Constraint: This method employs a sharper $b^*_T$ prescription ( $n=10$ ) to minimise the influence of non-perturbative parameters on the perturbative region. To counteract the unphysical oscillations that arise in $q_T$ space with this sharper prescription, a penalty is applied to the curvature of the $q_T$ distribution, specifically suppressing negative curvature (oscillations) at low $q_T$ .

Both methods account for uncertainties from perturbative scale variations and Parton Distribution Function (PDF) sets (using Hessian replicas). The fit parameters are the non-perturbative coefficients $A$ and $B$ in the Sudakov factor $S_{NP} \propto (A \ln(Q/Q_{NP}) + B)b_T^2$ .

Key Results

Conventional Fit Failure: A standard fit without theory constraints yields a low $\chi^2/\text{dof} = 0.30$ and $A = 0.23 \pm 0.07 \text{ GeV}^2$ . However, this result corresponds to a non-perturbative modification of the perturbative calculation exceeding 100% in the perturbative region and produces an unphysical $q_T$ distribution with a pronounced peak at very low $q_T$ and oscillatory tails.
Constrained Fits: Both novel strategies successfully eliminate unphysical oscillations and yield compatible results, though with different uncertainties:
- $b_T$ -constrained fit: $\chi^2/\text{dof} = 1.18$ , $A = 0.37 \pm 0.20 \text{ GeV}^2$ , $B = 1.04 \pm 0.68 \text{ GeV}^2$ .
- $q_T$ -constrained fit: $\chi^2/\text{dof} = 1.03$ , $A = 0.17 \pm 0.06 \text{ GeV}^2$ , $B = 1.03 \pm 0.23 \text{ GeV}^2$ .
TMD Evolution: The extracted gluon TMDs exhibit the expected evolution: as the hard scale $Q$ increases, the distribution broadens in transverse momentum, and the peak is suppressed.
Collins-Soper Kernel: By applying Casimir scaling ( $C_F/C_A$ ), the authors extract the non-perturbative part of the Collins-Soper kernel for quarks. The results are compatible with recent extractions (MAPNN, ART) but differ from older fits (Ref. [91]), highlighting the sensitivity of the kernel to the choice of $b_{T,max}$ and the fitting methodology.
PDF Sensitivity: The study demonstrates that results from different NLO PDF sets (MSHT20, CT18, NNPDF40) are only mutually consistent when PDF uncertainties are explicitly included in the fit. Using only central eigensets is discouraged.
Perturbative Accuracy: Comparisons between NLL and NNLL fits show that while the central values of non-perturbative parameters are stable, the NNLL accuracy provides a significantly better description of the data and tighter constraints on the parameter space.

Significance and Claims
The paper claims to provide the first extraction of the unpolarised gluon TMD $f_1^g$ and the non-perturbative contributions to the Collins-Soper kernel from $J/\psi$ -pair production data at NNLL accuracy. The authors argue that the consideration of perturbative and PDF uncertainties is crucial for meaningful fits to low-scale gluon-sensitive data.

The work highlights that conventional modelling of non-perturbative effects (specifically the $b^*_T$ prescription) can introduce uncontrolled distortions in the perturbative region for gluon-initiated processes. The proposed fitting strategies, which enforce consistency with perturbative uncertainties or physical $q_T$ behaviour, are presented as necessary improvements to obtain reliable extractions. The authors note that the linearly polarised gluon TMD ( $h_1^{\perp g}$ ) contribution is negligible in this analysis due to kinematic suppression and the smallness of the associated hard coefficient.

The paper concludes that while the current data allows for the extraction of the $Q$ -dependence of the non-perturbative parameters, the lack of sensitivity to the rapidity dependence prevents a constraint on the $x$ -dependence of the non-perturbative factor. Future measurements at different energies and with higher precision are identified as necessary to disentangle these dependencies further.

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