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