Quarkyonic matter and hadron–quark crossover from an ultracold atom perspective

We construct a new isentropic equation of state (EOS) at finite temperature "CRover" on the basis of the hadron-quark crossover at high density. By using the new EOS, we study the structure of hot neutron stars at birth with the typical lepton fraction ($Y_l=0.3-0.4$) and the typical entropy per baryon ($S=1-2$). Due to the gradual appearance of quark degrees of freedom at high density, the temperature T and the baryon density at the center of the hot neutron stars with the hadron-quark crossover are found to be smaller than those without the crossover by a factor of 2 or more. Typical energy release due to the contraction of a hot neutron star to a cold neutron star with 1.4 solarmass is shown to be about 0.04 solarmass with the spin-up rate about 14%.

We discuss the hadron–quark crossover accompanied by the formation of Cooper triples (three-body counterpart of Cooper pairs) by analogy with the Bose–Einstein condensate to Bardeen–Cooper–Schrieffer crossover in two-component fermionic systems. Such a crossover is different from a phase transition, which often involves symmetry breaking. We calculate the in-medium three-body energy from the three-body T-matrix with a phenomenological three-body force characterizing a bound hadronic state in vacuum. With increasing density, the hadronic bound-state pole smoothly undergoes a crossover toward the Cooper triple phase where the in-medium three-body clusters coexist with the quark Fermi sea. The relation to the quarkyonic matter model can also be found in a natural manner.

– This paper constitutes a direct extension of our earlier work on nuclear symmetry energy and neutron star constraints, advancing the discussion from the nuclear saturation region into the supra-nuclear density regime where the internal composition of neutron stars remains deeply contested. We systematically examine three interconnected problems: (i) the hadron–quark phase transition and its imprint on the equation of state (EoS) and neutron star observables; (ii) the hyperon puzzle and its modern proposed resolutions, including momentum-dependent potentials, repulsive hyperon–vector-meson couplings, and the quarkyonic matter framework; and (iii) the speed-of-sound profile cs²(ρ) as a diagnostic observable for the onset of quark degrees of freedom. The non-monotonic behavior of cs² — rising above the conformal limit of 1/3 before declining at higher densities — has emerged as a robust prediction across diverse theoretical frameworks and constitutes indirect evidence for a crossover transition toward quark matter in the cores of massive neutron stars. Combining the Relativistic Mean Field (RMF) framework for hadronic matter, the MIT bag model and Nambu–Jona-Lasinio (NJL) model for the quark phase, and current multimessenger constraints, we present a unified analysis of hybrid star sequences, twin star configurations, and the gravitational-wave signatures of first-order phase transitions. The paper further situates these discussions within the framework of the 2025 lattice QCD constraints on the EoS at isospin-dense conditions and the emerging color-superconducting quark matter picture. Our conclusions consistently favor a smooth hadron–quark crossover near (2–3)ρ₀ as the most observationally compatible scenario, while a strong first-order transition is disfavored by the combined NICER and GW170817 dataset at 90% confidence. Key Words: hybrid stars, hadron-quark phase transition, hyperon puzzle, quarkyonic matter, speed of sound, color superconductivity, MIT bag model, Nambu–Jona-Lasinio model, twin stars, gravitational waves.

We construct a simple model for describing the hadron–quark crossover transition by using lattice QCD (LQCD) data in the [Formula: see text] flavor system, and draw the phase diagram in the [Formula: see text] and [Formula: see text] flavor systems through analyses of the equation of state (EoS) and the susceptibilities. In the present hadron–quark crossover (HQC) model, the entropy density [Formula: see text] is defined by [Formula: see text] with the hadron-production probability [Formula: see text], where [Formula: see text] is calculated by the hadron resonance gas model that is valid in low temperature [Formula: see text] and [Formula: see text] is evaluated by the independent quark model that explains LQCD data on the EoS in the region [Formula: see text] for the [Formula: see text] flavor system and [Formula: see text] for the [Formula: see text] flavor system. The [Formula: see text] is determined from LQCD data on [Formula: see text] and susceptibilities for the baryon-number [Formula: see text], the isospin [Formula: see text] and the hypercharge [Formula: see text] in the [Formula: see text] flavor system. The HQC model is successful in reproducing LQCD data on the EoS and the flavor susceptibilities [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] in the [Formula: see text] flavor system, without changing the [Formula: see text]. We define the hadron–quark transition temperature with [Formula: see text]. For the [Formula: see text] flavor system, the transition line thus obtained is almost identical in [Formula: see text], [Formula: see text], [Formula: see text] planes, when the chemical potentials [Formula: see text] [Formula: see text] are smaller than 250 MeV. This [Formula: see text] approximate equivalence is also seen in the [Formula: see text] flavor system. We plot the phase diagram also in [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] planes in order to investigate flavor dependence of transition lines. In the [Formula: see text] flavor system, [Formula: see text] quark does not affect the [Formula: see text] flavor subsystem composed of [Formula: see text], [Formula: see text], [Formula: see text]. Temperature dependence of the off-diagonal susceptibilities and the [Formula: see text] show that the transition region at [Formula: see text] is [Formula: see text] for both the [Formula: see text] and [Formula: see text] flavor systems.

We investigate many-body properties of equally populated three-component fermions with attractive three-body contact interaction in one dimension. A diagrammatic approach suggests the possible occurrence of Cooper triples at low temperature, which are three-body counterparts of Cooper pairs with a two-body attraction. We develop a minimal framework that bridges the crossover from tightly-bound trimers to Cooper triples with increasing chemical potential and show how the formation of Cooper triples occurs in the grand-canonical phase diagram. Moreover, we argue that this non-trivial crossover is similar to the hadron-quark crossover proposed in dense matter. A coexistence of medium-induced triples and the underlying Fermi sea at positive chemical potential is analogous to quarkyonic matter consisting of baryonic excitations and the underlying quark Fermi sea. The comparison with the existing quantum Monte Carlo results implies that the emergence of these kinds of three-body states can be a microscopic origin of the peak of the sound velocity along the crossover.

Hadron–quark crossover and hybrid stars with quark core

The properties of hybrid stars are studied via the hybrid EOSs that are compatible with astrophysical observables. These hybrid EOSs are constructed by interpolating between hadronic EOS at lower densities and the quark EOS at higher densities. The BSR6 EOS derived from the RMF model is adopted as the hadronic EOS, while the quark EOS is calculated via a quasiparticle model. The maximum masses obtained from the hybrid EOSs are larger than [Formula: see text], and the tidal deformabilities for [Formula: see text] hybrid stars are smaller than 800. The combined tidal deformability [Formula: see text] is a monotonically increasing function of mass ratio [Formula: see text] for both hybrid EOSs and hadronic EOS, and it depends weakly on [Formula: see text]. The results of all hybrid EOSs can strictly satisfy the constraint of [Formula: see text] and the mass and radius constraints from the newest joint analysis of NICER, XMM-Newton and GW170817 data.

On the basis of the percolation picture from the hadronic phase with hyperons to the quark phase with strangeness, we construct a new equation of state (EOS) with the pressure interpolated as a function of the baryon density. The maximum mass of neutron stars can exceed $2M_{\odot}$ if the following two conditions are satisfied; (i) the crossover from the hadronic matter to the quark matter takes place at around three times the normal nuclear matter density, and (ii) the quark matter is strongly interacting in the crossover region and has stiff equation of state. This is in contrast to the conventional approach assuming the first order phase transition in which the EOS becomes always soft due to the presence of the quark matter at high density. Although the choice of the hadronic EOS does not affect the above conclusion on the maximum mass, the three-body force among nucleons and hyperons plays an essential role for the onset of the hyperon mixing and the cooling of neutron stars.

The phase diagram of quantum chromodynamics is conjectured to have a rich structure containing at least three forms of matter: hadronic nuclear matter, quarkyonic matter and quark–gluon plasma. We justify the origin of the quarkyonic phase transition in a chiral-quark model and describe its formulation in terms of Skyrme crystals.

ORGANIZING COMMITTEEDr M Ángeles Pérez-García (Área Física Teórica-Universidad de Salamanca & IUFFYM) Dr J A Miralles (Universidad de Alicante) Dr J Pons (Universidad de Alicante) Dr C Albertus (Área Física Nuclear-Universidad de Salamanca & IUFFYM) Dr F Atrio (Área Física Teórica-Universidad de Salamanca & IUFFYM) PREFACEThe second Iberian Nuclear Astrophysics meeting was held at the University of Salamanca, Spain on 22–23 September 2011. This volume contains most of the presentations delivered at this international workshop. This meeting was the second in the series following the previous I Encuentro Ibérico de Compstar, held at the University of Coimbra, Portugal in 2010. The main purpose of this meeting was to strengthen the scientific collaboration between the participants of the Iberian and the rest of the southern European branches of the European Nuclear Astrophysics network, formerly, COMPSTAR. This ESF (European Science Foundation) supported network has been crucial in helping to make a broader audience for the the most interesting and relevant research lines being developed currently in Nuclear Astrophysics, especially related to the physics of neutron stars. It is indeed important to emphasize the need for a collaborative approach to the rest of the scientific communities so that we can reach possible new members in this interdisciplinary area and as outreach for the general public.The program of the meeting was tailored to theoretical descriptions of the physics of neutron stars although some input from experimental observers and other condensed matter and optics areas of interest was also included. The main scientific topics included: Magnetic fields in compact stars Nuclear structure and in-medium effects in nuclear interaction Equation of state: from nuclear matter to quarks Importance of crust in the evolution of neutron stars Computational simulations of collapsing dense objects Observational phenomenology In particular, leading experts from the computational simulation of core-collapse supernovae and the effect of hadron–quark phase transitions developed specialized review talks. Prospects in future observations or a more dilute classification of magnetars were also discussed. The importance of the equation of state, three-body forces, finite nuclei, phenomenological fermionic interaction models, and the microphysics inputs of different many-body approaches to some very important quantities as the symmetry energy were reviewed and discussed from either the non-relativistic to the relativistic framework. The importance of the crust with the existence of a crystallized structure and vortex-crust pinning were some of the important subjects discussed in the context of cooling and field dynamics.Finally, some condensed matter and optics talks presented us the rich insight that Cold Atom Physics can give us on low-density interactions and the new and very intense laser Petawatt beams can test matter under strong external fields, respectively.We would to thank the Faculty of Science and University of Salamanca for hosting the meeting. We also thank for partial financial support the European ERC Network COMPSTAR, The Physics of Neutron Stars under reference 3803 and the Spanish Ministerio de Ciencia e Innovación (MICINN) with project FIS2011-14759 and the local institutions of Instituto de Física Fundamental y Matemáticas (IUFFYM) and Universidad de Salamanca, Spain. Of course we thank those who have contributed to make this meeting a nice occasion to gather and start to develop fruitful collaborations. To them go our grateful acknowledgments.December 2011, Salamanca,Spain M Ángeles Pérez-García, J A Miralles, J Pons, C Albertus, F Atrio Organizing Committee of II Iberian Nuclear Astrophysics Meeting SPONSOR OR FUNDING ACKNOWLEDGMENTSEuropean ERC Network COMPSTAR, The Physics of Neutron Stars under meeting ref. 3803 Spanish Ministerio de Ciencia e Innovación (MICINN) with project FIS2011-14759 Instituto de Física Fundamental y Matemáticas (IUFFYM) Universidad de Salamanca, Spain MULTIDARK Consolider-Ingenio 2010, MICINN ref. CSD2009-00064 PICTURE OF PARTICIPANTS Picture of some of the participants of the II Iberian Nuclear Astrophysics Meeting. LIST OF PARTICIPANTS C Albertus (U. Salamanca, Spain) I Bombaci (U. Pisa, Italy) Rudiney Casali (U. Coimbra, Portugal) Silvia Chiacchiera (U. Coimbra, Portugal) Anthea Fantina (U. Libre Bruxelles, Belgium) Marcio Ferreira (U. Coimbra, Portugal) Miguel Gullón (U. Alicante, Spain) Fabrizio Grill (U. Coimbra, Portugal) Joe Hughto (Indiana University, USA/ U. Alicante, Spain) J M Ibáñez (U. Valencia, Spain) B Juliá Díaz (U. Barcelona/ICFO) D Logoteta (U. Coimbra, Portugal) V Moreno (U. Autónoma de Madrid, Spain) M A Pérez-García (U. Salamanca, Spain) J Pons (U. Alicante, Spain) C Providencia (U. Coimbra, Portugal) Nanda Rea (ICE-CSIC, IEEC, Barcelona, Spain) Xavier Roca-Maza (INFN, Milano, Italy) Luis Roso (CLPU/ U. Salamanca, Spain) Klaas Vantournhout (GSI Darmstadt, Germany) I Vidaña (U. Coimbra, Portugal) Daniele Viganó (U. Alicante, Spain)