Higgs Boson Data Reveal the Gluon’s Hidden Shape Inside the Proton

For decades, physicists have been able to describe the proton with remarkable precision as long as they ignored one crucial detail: the sideways motion of the particles inside it. Now, in a study published in The European Physical Journal C, the MAP (Multi-dimensional Analysis of Partonic distributions) Collaboration has reported the first extraction of the gluon transverse-momentum-dependent parton distribution, or gluon TMD, directly from experimental data. The result, drawn entirely from Higgs boson measurements at the Large Hadron Collider, opens a new window onto the three-dimensional inner life of the proton and marks a milestone in the effort to map how gluons, the carriers of the strong force, are distributed not only in longitudinal momentum but also in intrinsic transverse momentum.

Ordinary parton distribution functions, or PDFs, describe the probability of finding a quark or gluon carrying a given fraction of the proton’s momentum when the proton moves at nearly the speed of light. TMDs extend this picture by also resolving the parton’s transverse momentum, the small sideways kick it carries relative to the proton’s direction of travel. The theoretical machinery that makes this possible is TMD factorisation, which holds when a process produces a colour-singlet final state with a hard scale Q much larger than the transverse momentum q_T of the observed particle. In that regime, the differential cross section can be written as a convolution of TMD distributions with perturbatively calculable hard coefficients, and the TMDs themselves obey evolution equations that resum the large logarithms of Q squared over q_T squared that otherwise wreck the prediction.

Quark TMDs have been studied intensively for two decades, extracted from Drell-Yan scattering and semi-inclusive deep-inelastic scattering data. Gluon TMDs, by contrast, have remained stubbornly under-constrained, largely because the experimental options are so limited. A clean extraction requires a colour-singlet final state, since processes involving coloured particles such as jets or quarkonia suffer from colour entanglement that can spoil factorisation. This is precisely where the Higgs boson shines. At the LHC, the Higgs is produced predominantly by gluon fusion, mediated at leading order by a heavy-quark loop dominated by the virtual top quark, and its electroweak decay modes yield a colour-singlet final state, making inclusive Higgs production arguably the cleanest available probe of the unpolarised gluon TMD.

The new analysis assembles the currently available ATLAS and CMS measurements of the Higgs transverse-momentum distribution at centre-of-mass energies of 8 and 13 TeV, in both the diphoton and four-lepton decay channels. Theoretical predictions are computed at next-to-next-to-next-to-leading logarithmic accuracy, N3LL, matched to collinear PDFs taken from the NNPDF3.1 set. Notably, the calculation includes the linearly polarised gluon TMD, the so-called Boer-Mulders function, which describes how gluons inside an unpolarised proton can be linearly polarised in the transverse plane. Its matching coefficients onto collinear PDFs are known up to next-to-next-to-leading order, allowing both gluon TMDs to be treated consistently at the highest perturbative order considered.

The formalism works in impact-parameter space, Fourier-conjugate to the transverse momentum, where the TMDs can be matched perturbatively onto collinear PDFs at small impact parameter. At large impact parameters, perturbation theory breaks down as the strong coupling approaches the Landau pole, so the analysis follows the classic Collins-Soper-Sterman strategy of separating the perturbative region from a nonperturbative function that must be fitted to data. The team adopted a simple Gaussian parametrisation with a single free parameter g, assumed identical for the unpolarised and linearly polarised TMDs and valid at the single hard scale set by the Higgs mass. Fiducial selection cuts on the final-state photons and leptons were incorporated through a carefully computed phase-space reduction factor, evaluated numerically for both two-body and four-body decay topologies.

Restricting the analysis to the region where TMD factorisation is applicable, the collaboration applied a baseline cut of q_T divided by the Higgs mass below 0.3, retaining a total of 30 experimental data points. The fit, implemented in the NangaParbat framework, uses a bootstrap method with 200 Monte Carlo replicas of the data to propagate uncertainties. The global chi-squared per data point came out at 1.49, corresponding to a p-value of roughly 3 percent, indicating that the theory reproduces both the shape and the normalisation of the measurements reasonably well, with some residual tension traced to statistical fluctuations at relatively large transverse momentum.

The fit returned a nonperturbative parameter g of 15.6 plus or minus 5.1 GeV squared, a relative uncertainty of about 35 percent. That number tells an important story: the current Higgs dataset has only moderate sensitivity to the nonperturbative content of gluon TMDs, which is expected because the hard scale of Higgs production is so large that the perturbative contribution dominates. Splitting the parameter into an intrinsic component and a nonperturbative evolution component, by rescaling a quark-sector estimate of the Collins-Soper kernel by the ratio of colour factors, the team estimated an intrinsic value of 14.4 plus or minus 5.1 GeV squared, suggesting that nonperturbative evolution effects are small within the current kinematic coverage.

The extracted gluon TMD itself grows markedly in size as the momentum fraction x decreases, mirroring the rapid rise of the collinear gluon PDF at small x, and becomes broader in transverse momentum at higher energy scales as a consequence of TMD evolution. Tests of perturbative convergence showed that the agreement between data and theory improves substantially from NLL-prime to NNLL-prime accuracy and then changes only mildly at N3LL, indicating that the perturbative series has effectively stabilised. Varying the q_T cut to more conservative values of 0.25 and 0.2 reduced the chi-squared per point to as low as 1.08 while keeping the extracted parameters compatible within one-sigma uncertainties, confirming that the baseline choice does not significantly bias the result. The linearly polarised gluon contribution, affecting the spectrum at only the few-percent level, remains below the roughly 50-percent experimental uncertainties and could not be independently constrained.

The authors emphasise that this analysis is a baseline rather than a final word. Because all data sit at essentially one hard scale, the Higgs mass, the fit cannot disentangle the nonperturbative part of the gluon Collins-Soper kernel from the intrinsic transverse-momentum structure, and the x dependence of the extracted TMD is currently driven entirely by perturbative matching onto collinear PDFs. Future extractions combining Higgs measurements with other gluon-sensitive processes at very different scales, such as quarkonium production or measurements at future fixed-target experiments, could separately determine the nonperturbative evolution kernel. Meanwhile, the increased statistical precision expected from Run III and the High-Luminosity LHC should tighten the nonperturbative parameters, expose genuine x dependence, and eventually allow more flexible parametrisations that treat the unpolarised and linearly polarised gluon TMDs on their own terms.

What makes the result resonate beyond the specialist community is what it represents: the first time the diffuse, swirling cloud of gluons inside the proton has been quantitatively imaged in transverse momentum using real collider data. The proton’s mass, its spin, and virtually all the physics of the strong force at collider energies hinge on gluon dynamics, yet gluons have remained the least mapped constituents. By squeezing a coherent picture of the gluon TMD out of the tiny, precious sample of measured Higgs transverse-momentum spectra, the MAP Collaboration has turned a long-standing theoretical aspiration into a working extraction tool, one that will only sharpen as the LHC delivers more Higgs bosons and as theorists and experimentalists combine forces across processes, energies, and decay channels in the years ahead.

Subject of Research: First phenomenological extraction of the unpolarised gluon transverse-momentum-dependent distribution from LHC Higgs boson transverse-momentum data within TMD factorisation.

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

Article References: Anedda, S., Bertone, V., Bozzi, G., & Cerutti, M. (2026). A first extraction of gluon TMDs from Higgs data at the LHC. The European Physical Journal C, 86(8), Article 1024. https://doi.org/10.1140/epjc/s10052-026-16288-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16288-6

Keywords: gluon TMD, Higgs boson, LHC, TMD factorisation, proton structure, ATLAS, CMS, perturbative QCD, N3LL resummation, nonperturbative QCD, Boer-Mulders function, Collins-Soper kernel