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

The new work, published in The European Physical Journal C by Kazem Bitaghsir Fadafan of Shahrood University of Technology, Jesús Cruz Rojas of the National Autonomous University of Mexico, and Jonas Mager of the Vienna University of Technology, applies holography, the correspondence that links strongly coupled quantum field theories to weakly coupled gravitational theories in higher dimensions. The central obstacle in describing quark matter inside neutron stars is that the relevant regime of quantum chromodynamics, or QCD, is strongly coupled. First-principles lattice simulations fail there because of the notorious fermion sign problem at large densities, while traditional phenomenological approaches such as nucleon effective field theories struggle under the extreme conditions found in stellar cores. Holographic models offer a way around this impasse: instead of computing the equation of state of dense quark matter directly, one solves a comparatively tractable problem in a higher-dimensional gravitational dual and reads off the properties of the quantum field theory from the geometry.

The researchers built their model on the well-studied D3/D7-brane configuration of type IIB string theory, a bottom-up construction in which D7-branes embedded in the curved background of D3-branes describe quarks moving in a gauge theory. The key innovation lies in the dilaton, a scalar field that controls the running of the gauge coupling in the dual theory. Rather than keeping the dilaton constant, as in the simplest version of the model, the team adopted a phenomenological profile that interpolates smoothly and monotonically from its ultraviolet value to a finite, regular value in the infrared. Three parameters, denoted A, lambda and kappa, govern the precise shape of this profile, controlling the infrared value of the dilaton, the scale at which conformal symmetry is broken, and the steepness of the transition. This tailored profile captures essential features of QCD such as chiral symmetry breaking and produces a deconfined yet massive quark phase at finite density.

In the holographic setup, the embedding function of the D7-brane encodes the quark mass and condensate, while a gauge field living on the brane describes the quark chemical potential and density. Two distinct phases emerge: a Minkowski, or vacuum, phase with broken chiral symmetry and zero baryon density, and a quark phase in which chiral symmetry is restored and the density is nonzero. By computing the on-shell action of the brane, which the holographic dictionary identifies with the grand canonical potential, the researchers obtained the pressure as a function of chemical potential and hence the equation of state of the quark phase. At asymptotically large densities the model correctly reproduces the perturbative QCD behavior, with the pressure scaling as the fourth power of the chemical potential.

The team also attempted to describe the baryonic phase within the same holographic framework, modeling nucleons as D5-branes wrapped on a five-sphere. The nontrivial dilaton profile stabilizes these wrapped branes against collapse, which is itself a notable achievement. However, in the homogeneous, smeared approximation used here, the resulting nuclear equation of state turned out to be unrealistic, predicting pressures in sharp conflict with phenomenology just above the onset of the baryonic phase. The authors showed analytically that near the transition the chemical potential scales as the one-third power of the density, forcing a fourth-order transition that cannot reproduce the first-order behavior expected of isospin-symmetric QCD. They therefore discarded the holographic baryon phase and instead adopted the phenomenological equations of state of Hebeler and collaborators, derived from chiral effective field theory and constrained by nuclear physics and observation.

With the baryonic phase fixed phenomenologically and the quark phase supplied by holography, the researchers scanned the parameter space of the model, varying the AdS radius and the t’Hooft coupling. They found a region of parameters, with the AdS radius between roughly 0.015 and 0.02 inverse megaelectronvolts and the t’Hooft coupling between 1.9 and 3, that produces rather stiff equations of state with only weakly first-order transitions between baryonic and quark matter. This softness of the transition is crucial: it allows a smooth conversion from nuclear matter to quark matter inside a star rather than a violent discontinuity that would destabilize the configuration. When combined with the stiff Hebeler equation of state, three parameter choices produced equations of state supporting stable quark cores, with phase transitions occurring at chemical potentials around 400 megaelectronvolts.

To connect the microscopic equation of state to observable stellar properties, the team solved the Tolman-Oppenheimer-Volkoff equations of general relativity, which determine the structure of non-rotating compact stars. The resulting mass-radius curves show stars that begin their lives as ordinary nucleonic objects and develop quark cores as the central density increases. Stability requires that the mass increase with central energy density, and the stars with quark cores satisfy this criterion all the way up to the maximum of the mass-radius curve, beyond which a radial mode becomes unstable and the star collapses into a black hole. The maximum masses obtained for the three quark-star-supporting parameter choices are 2.17, 2.13 and 1.90 solar masses, the largest of which is broadly consistent with the heaviest precisely measured neutron star mass of about 2.35 solar masses, albeit somewhat favoring lower values.

The authors also computed the tidal deformability, the quantity that measures how easily a star is deformed by the gravitational field of a companion and that leaves a characteristic imprint on gravitational wave signals from binary neutron star inspirals. Once a star becomes heavy enough to develop a quark core, the tidal deformability drops rapidly, falling by roughly 80 units between the onset of the quark phase and the maximum allowed mass. This rapid decrease is a potentially observable signature of quark matter formation. However, the very stiffness of the baryonic phase required to obtain stable quark cores causes the model to overshoot the tidal deformability constraint from the GW170817 event at 1.4 solar masses, predicting a value near 950 compared with the observational bound of about 190 with large uncertainties. The authors note that a baryonic equation of state intermediate between the stiff and medium phenomenological cases would likely ease this tension while still permitting stable quark cores.

Several other results add nuance to the picture. The speed of sound in the quark phase remains below the speed of light, although the stiff baryonic phase violates a recently derived transport bound. More strikingly, the polytropic index of the holographic quark phase exceeds the value of 1.75 that has been proposed in the literature as a criterion for the onset of quark matter, reaching values as high as 2.5 just after the transition. This suggests that the criterion may be too restrictive, and it contrasts with predictions from the V-QCD holographic model, which places transitions at much larger energy densities with smaller polytropic indices. The authors emphasize that their model cannot definitively establish whether stable quark stars exist in nature, since the answer depends on parameter choices, but it demonstrates that such objects can in principle emerge from holographic models, opening a computational window onto their phenomenology. Future work will explore localized D5-brane configurations, which may improve the low-density baryonic description, along with neutrino transport and rotating stars with quark cores, as multi-messenger observations continue to tighten the constraints on the densest matter in the universe.

Subject of Research: Holographic modeling of stable massive quark stars and the equation of state of dense quark matter in compact star cores

Article Title: Properties of stable massive quark stars in holography

Article References: Bitaghsir Fadafan, K., Cruz Rojas, J., & Mager, J. (2026). Properties of stable massive quark stars in holography. The European Physical Journal C, 86(9), Article 1094. https://doi.org/10.1140/epjc/s10052-026-16338-z

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16338-z

Keywords: quark stars, holographic QCD, neutron stars, D3/D7-branes, equation of state, chiral symmetry breaking, tidal deformability, compact stars, gauge-gravity duality, quark matter, TOV equations, dense nuclear matter