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

That hidden feature is the gluon condensate, a nonperturbative property of the quantum chromodynamics (QCD) vacuum that measures the expectation value of the gluon field strength squared. In plain terms, even empty space seethes with gluonic fluctuations, and the intensity of this background seething shapes how strongly quarks bind into hadrons. In a paper published in The European Physical Journal C, Fei Wang and Zi-qiang Zhang of China University of Geosciences in Wuhan have computed, for the first time within their framework, how varying the strength of the gluon condensate reshapes the thermal spectral functions of charmonium and bottomonium — the bound states of charm and bottom quark pairs. Their conclusion is striking: a stronger gluon condensate systematically narrows the spectral peaks of these states and boosts their spectral weight, meaning the bound states live longer and survive more readily in the plasma. The condensate, far from being a spectator, actively hinders thermal dissociation.

The technical engine behind this result is holography, the remarkable dictionary arising from the AdS/CFT correspondence that translates intractable strongly coupled gauge theories into tractable classical gravity problems in a higher-dimensional spacetime. In this picture, a four-dimensional quantum field theory without a known analytical solution is recast as a five-dimensional gravitational system, with the extra dimension playing the role of an energy scale. Quarkonia — heavy vector mesons such as the J/psi and the Upsilon — correspond to waves of a five-dimensional gauge field propagating in this curved bulk geometry. The authors employed an improved soft-wall AdS/QCD model, in which a background scalar field with three independent energy scales encodes the nonperturbative physics responsible for the masses and decay constants of heavy mesons. Unlike earlier hard-wall and original soft-wall constructions, this refined profile reproduces both the vacuum mass spectrum and the experimentally observed monotonic decrease of decay constants as one climbs the ladder of radial excitations.

Spectral functions are the central objects in this analysis. At zero temperature, the two-point correlator of the heavy quark vector current decomposes into an infinite sum of infinitely sharp delta-function peaks, one for each meson state, with heights set by the decay constants. At finite temperature, these razor-sharp resonances broaden into structures with finite width, and the position, width, and height of each peak encode the quasiparticle mass, lifetime, and dissociation behavior of the corresponding bound state inside the hot medium. A peak that broadens and sinks is a meson dissolving; a peak that stays tall and narrow is a meson holding together. Tracking these peaks as functions of temperature and condensate strength therefore offers a direct, dynamical window into quarkonium melting that bulk thermodynamic measures cannot provide.

To introduce the gluon condensate, the authors extended the setup to finite temperature using a dilaton black hole geometry. In holographic language, temperature appears as a black hole in the fifth dimension, and the gluon condensate enters through a second scalar field — the dilaton — whose back-reaction warps the black hole metric itself. Crucially, the construction keeps two scalar sectors physically distinct: the soft-wall scalar fixes the vacuum masses and decay constants of the mesons, while the dilaton, whose strength parameter is proportional to the expectation value of the gluon field strength squared, deforms only the finite-temperature geometry. This separation has a clean consequence visible in the numerical results: increasing the condensate changes the widths and heights of the spectral resonances but leaves their central positions untouched, because the real part of the quasiparticle energy is governed by the vacuum sector alone.

The numerical machinery relies on the membrane paradigm, a technique that reformulates the computation of retarded Green’s functions as a flow equation for a frequency-dependent conductivity that is integrated from the black hole horizon out to the boundary of spacetime. Regularity at the horizon supplies the boundary condition, and the real part of the resulting boundary conductivity, multiplied by frequency, yields the spectral function. Because no closed-form solution exists, the flow equations must be integrated numerically, using parameter sets fitted in earlier work: for charmonium a soft-wall scale of 1.2 GeV with a string-tension scale of 0.55 GeV, and for bottomonium 2.45 GeV and 1.55 GeV respectively, along with mass-scale parameters tied to non-hadronic decay channels.

The temperature scan with the condensate switched off establishes the baseline physics. At 195 MeV, just above the pseudo-critical deconfinement temperature, the charmonium spectrum still displays a sharp J/psi peak, indicating that the ground state remains largely intact. The bottomonium spectrum at the same temperature is even more robust, showing a dominant Upsilon(1S) resonance accompanied by visible peaks from the 2S and 3S excited states — a reflection of the bottom quark’s larger mass and correspondingly stronger binding. As the temperature climbs to 330, 465, and 600 MeV, the peaks for both systems systematically broaden and lose height, the unmistakable signature of medium-induced dissociation: thermal fluctuations increasingly tear the heavy quark pairs apart, draining spectral weight from the quasiparticle resonances.

The condensate scan reverses this narrative. Holding the temperature fixed at 280 MeV for charmonium and 300 MeV for bottomonium — both comfortably above the deconfinement crossover — the authors varied the condensate parameter through the values 0.2, 0.5, 0.8, and 0.9. Even at these temperatures, the 1S resonances remain clearly distinguishable, while the excited-state peaks, broader and weaker as expected from their larger spatial extent and looser binding, respond most dramatically. As the condensate strengthens, the peaks progressively narrow and grow taller, and the effect builds gradually rather than switching on abruptly. Physically, the interpretation is that a stronger nonperturbative gluonic background reinforces the gluon-mediated interaction between the heavy quark and antiquark, making it harder for thermal fluctuations to pry the pair apart. The result is a longer-lived bound state — a narrower resonance — and a larger occupation probability — a taller peak.

Perhaps the most novel observation is a mass-dependent asymmetry between the two quarkonium families. For the same increment in condensate strength, the narrowing and amplification of the bottomonium 2S peak is more pronounced than for its charmonium counterpart. The authors attribute this to the fact that, at the temperatures considered, the binding energies of bottomonium excited states are larger relative to the thermal energy than those of charmonium excited states, making them more sensitive to the condensate-induced changes in the effective interaction. This differential response, they note, had not been emphasized in previous separate analyses of the two systems, and it offers a potentially testable fingerprint: if the gluon condensate in the plasma is large, excited bottomonium states should show disproportionately enhanced survival compared with excited charmonium.

The findings dovetail with two independent lines of holographic evidence. Earlier calculations of the imaginary part of the interquark potential and of entropic forces in condensate-bearing backgrounds both concluded that gluon condensation impedes quarkonium melting. The new spectral-function results also mesh with the authors’ own prior work using configurational entropy, an information-theoretic stability measure which showed that larger condensates reduce microscopic disorder in the plasma. Together, these converging approaches suggest that condensate-driven stabilization of quarkonia is a robust nonperturbative effect rather than an artifact of any single formalism. The stakes are high for experiment: quarkonium suppression patterns measured at the LHC and at RHIC are among the primary diagnostics of plasma formation, and any condensate-induced stabilization directly alters how those patterns should be interpreted. The authors point to comparisons with lattice QCD and potential models, and to data from LHC Run 4 and the RHIC Beam Energy Scan II, as the next steps in constraining the magnitude of the gluon condensate in hot QCD matter — a quantity that, if these calculations hold, may be quietly deciding which of the plasma’s heaviest messengers survive the fire.

Subject of Research: Holographic study of gluon condensate effects on heavy quarkonium spectral functions and thermal dissociation in the quark–gluon plasma

Article Title: Gluon condensate effects on heavy quarkonium spectral functions and thermal dissociation

Article References: Wang, F., & Zhang, Z.-Q. (2026). Gluon condensate effects on heavy quarkonium spectral functions and thermal dissociation. The European Physical Journal C, 86(9), Article 1129. https://doi.org/10.1140/epjc/s10052-026-16323-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16323-6

Keywords: quark–gluon plasma, quarkonium, gluon condensate, spectral functions, holographic QCD, AdS/QCD, charmonium, bottomonium, thermal dissociation, heavy-ion collisions, dilaton black hole, nonperturbative QCD