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

GPDs are the mathematical objects that go beyond the familiar parton distribution functions measured in inclusive deep inelastic scattering. While ordinary PDFs tell you the probability of finding a quark carrying a given fraction of the proton’s momentum, GPDs also encode what happens when the proton recoils, absorbing a momentum transfer that shifts it sideways. That extra information, accessed through processes like deeply virtual Compton scattering, where an electron knocks a virtual photon out of the proton and a real photon emerges, is what allows physicists to reconstruct genuinely three-dimensional portraits of nucleon structure, including the elusive orbital angular momentum carried by quarks and gluons.

The central difficulty is that experiments do not measure GPDs directly. What they measure are Compton form factors, convolutions of the GPDs with perturbatively calculable kernels that are sharply singular at particular points. Recovering the underlying GPDs from these convolutions is an inverse problem that is, in principle, intractable without additional input. The Virginia team’s strategy is to narrow the space of possible solutions by building the parametrization directly on the symmetries of quantum chromodynamics: crossing symmetry under the exchange of quark and antiquark momenta, polynomiality of the Mellin moments, and the area rules dictated by baryon number and momentum conservation.

The mathematical backbone of UVA2 is the reggeized spectator model, in which the proton is pictured as a struck parton bound to a remnant whose invariant mass is allowed to fluctuate. This yields compact analytic expressions for the two leading vector GPDs, H and E, for each quark flavor, including the valence u and d quarks, the antiquark sea, and the gluons. In a first for this framework, the authors provide fully analytic closed forms in all kinematic variables, eliminating the need for numerically expensive transverse momentum integrations and making the parametrization immediately usable in regression analyses, neural network training, and Monte Carlo event generators.

Constraining such a multidimensional function requires an enormous amount of input. In the forward limit, where the GPD H reduces to ordinary PDFs, the team fitted their parameters to reproduce the NNPDF21 parton distributions, checking the quality of the match with Kolmogorov-Smirnov tests. The t-dependence, describing how the distributions fall off with transverse momentum transfer, was then pinned down by fitting the flavor-separated Dirac and Pauli form factors extracted from lepton-nucleon scattering, together with the second Mellin moments recently computed in lattice QCD. The gluon sector was anchored to lattice determinations of the gluon form factors evaluated at a pion mass closer to the physical value than in previous versions.

All of these ingredients are defined at a common initial evolution scale of 0.58 square gigaelectronvolts, an improvement over earlier versions in which quarks and gluons started from inconsistent scales. From there, the distributions are evolved to higher energy scales using the leading-order off-forward DGLAP and ERBL evolution equations, which the authors solved with the Adams predictor-corrector method rather than the conventional Runge-Kutta approach. Because the right-hand side of the evolution equations involves convolution integrals that are notoriously expensive to evaluate, this choice of integrator delivers a substantial gain in computational speed, particularly in the low-momentum-fraction region that will dominate at future colliders.

The practical payoff comes in the form of predictions for the Compton form factors across a vast kinematic range, from the fixed-target regime of Jefferson Lab, where valence quarks dominate, all the way to the low Bjorken-x frontier of the upcoming Electron-Ion Collider, where sea quarks and gluons take over. The team’s calculations show that at EIC kinematics the Compton form factors are expected to be several orders of magnitude larger than at fixed-target facilities, and that the valence contribution is strongly suppressed in the collider setting. Crucially, because the flavor-singlet quark combination couples to gluons in the evolution equations, the gluon GPDs can in principle be extracted directly from DVCS data at the EIC, turning a machine designed to map gluon structure into a precision instrument for proton tomography.

The authors are careful about uncertainties. The error budget on their GPDs originates entirely from the fits to the t-dependence of the electromagnetic form factors, the lattice gluon form factors, and the second moments of the quark distributions, while the forward-limit parameters, obtained by matching to PDF parametrizations rather than raw data, carry no statistical errors and are instead validated through distribution-to-distribution comparisons. They flag this Hessian-based treatment as a first step, with maximum-likelihood and Markov-chain Monte Carlo analyses, along with neural network approaches, planned to properly capture correlations among the seven to eight parameters that define each flavor sector.

Everything is openly available. The UVA2 parametrization, including analytic forms at the initial scale and precomputed grids spanning momentum fractions from ten thousandths up to nearly one, momentum transfers up to four square gigaelectronvolts, and evolution scales up to a thousand square gigaelectronvolts, can be downloaded from a public GitHub repository in easily readable CSV formats. This openness matters, because GPD parametrizations are becoming shared infrastructure: they feed event generators, guide experimental analyses of exclusive processes, and provide the benchmark curves against which lattice QCD results and future DVCS measurements will be judged.

As the Electron-Ion Collider moves from blueprint to beamline, tools like UVA2 will determine how much physics can be squeezed out of the first data. By fusing elastic scattering, inclusive deep inelastic data, and lattice QCD into a single symmetry-respecting framework, the Virginia collaboration has delivered what amounts to a flexible, evolving atlas of the proton’s interior, one that promises to turn the coming decade of scattering experiments into a genuine three-dimensional census of the most abundant visible matter in the universe.

Subject of Research: Global parametrization of generalized parton distributions describing the three-dimensional quark and gluon structure of the proton

Article Title: Updated flexible global parametrization of generalized parton distributions from elastic and deep inelastic inclusive scattering data

Article References: Panjsheeri, Z., Adams, D. Q., Khawaja, A., Pandey, S., Tezgin, K., & Liuti, S. (2026). Updated flexible global parametrization of generalized parton distributions from elastic and deep inelastic inclusive scattering data. The European Physical Journal C, 86(9), Article 1078. https://doi.org/10.1140/epjc/s10052-026-15872-0

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-15872-0

Keywords: generalized parton distributions, proton structure, quantum chromodynamics, deeply virtual Compton scattering, lattice QCD, parton distribution functions, Electron-Ion Collider, QCD evolution, elastic form factors, Compton form factors, nucleon tomography, reggeized spectator model