Baryquark Matter: Duality of Baryons & Quarks

• Baryquark matter is a dense phase where baryonic and quark degrees of freedom coexist, blurring the line between confined and deconfined matter.
• It encompasses models from quarkyonic and half-skyrmion phases to momentum-space shell constructions, impacting QCD thermodynamics and nuclear astrophysics.
• Studies leverage effective field theories, holography, and lattice QCD to address chiral symmetry restoration, confinement, and the equation of state in extreme conditions.

Baryquark matter denotes baryon-number-carrying matter in which baryonic and quark descriptions are intertwined, but the term is not standardized across the literature. In heavy-dense QCD it is naturally interpreted as quarkyonic matter, a confined phase whose thermodynamics scale as $p\sim N_c$ immediately beyond baryon onset (Philipsen et al., 2018). In hidden-local-symmetry and Sakai–Sugimoto descriptions it labels a dense hadronic phase with $\langle \bar q q\rangle_{\rm avg}\to 0$ while $F_\pi^*\neq 0$, so chiral symmetry is restored on average although confinement persists (Rho, 2010). In momentum-space quasiparticle constructions it instead means the configuration in which baryons occupy low momenta and quarks populate an outer shell, opposite to the standard quarkyonic arrangement (Poberezhnyuk et al., 2023). In broader astrophysical usage the same label is extended to bulk three-flavor strong matter, quark-cluster matter, and even baryon-number-carrying dark-sector composites (Lai et al., 2022).

1. Terminological range and common conceptual core

Across the cited literature, “baryquark matter” is not a single universally fixed phase name but a family of constructions. The narrowest usage identifies it directly with quarkyonic matter: a low-temperature, high-density state in which confinement survives while bulk thermodynamics is quark-dominated (Philipsen et al., 2018). A second usage, developed in dense hadronic EFT and holography, treats it as a topological reorganization of baryonic matter into half-skyrmion or half-instanton configurations with averaged chiral restoration but no deconfinement (Rho, 2010). A third usage is explicitly kinematic: baryons fill the low-momentum Fermi sea and quarks occupy a shell outside it, so “baryquark” denotes the inversion of the more familiar quarkyonic shell structure (Koch et al., 2022).

The broader astrophysical literature stretches the term further. In the taxonomy of bulk strong matter, it can encompass macroscopic three-flavor matter realized either as deconfined strange quark matter or as localized strangeon matter (Lai et al., 2022). In compressed-baryonic-matter phenomenology it can denote quark-cluster matter, a self-bound phase in which strong coupling localizes quarks into multi-quark clusters rather than leaving them deconfined (Guo et al., 2013). In dark-matter proposals it can even mean baryon-number-carrying antibaryons or quark nuggets that are not “baryonic in the conventional sense” cosmologically, although they are composed of Standard Model quarks (Gorbunov et al., 2013).

The common thread is not a unique microscopic order parameter but the coexistence, duality, or reorganization of baryonic and quark degrees of freedom. This suggests that “baryquark matter” functions less as a sharply delimited phase label than as a descriptor for matter in which baryon number is carried by configurations not cleanly classifiable as either ordinary nuclear matter or asymptotically free quark matter.

2. Large-$N_c$ QCD, heavy quarks, and the quarkyonic interpretation

In the heavy-quark, cold-and-dense regime of lattice QCD, baryquark matter is naturally identified with quarkyonic matter. The starting point is a three-dimensional effective theory obtained from combined strong-coupling and hopping expansions of lattice QCD with Wilson action. Spatial links are integrated out, leaving temporal Wilson lines $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$ and Polyakov loops as effective degrees of freedom, with leading gauge interaction

$$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$

and static fugacity coefficients

$$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$

with $m=-\ln(2\kappa)$ in the static strong-coupling limit (Philipsen et al., 2018).

In this framework, the strong-coupling limit $\beta=0$ exhibits a Silver Blaze structure: at $T\to0$, the quark number density remains zero for $\langle \bar q q\rangle_{\rm avg}\to 0$0 and jumps to lattice saturation for $\langle \bar q q\rangle_{\rm avg}\to 0$1. More importantly, the physically relevant next-to-leading contribution to the pressure above onset scales linearly with color,

$\langle \bar q q\rangle_{\rm avg}\to 0$2

whereas the corresponding below-onset terms are exponentially suppressed by factors $\langle \bar q q\rangle_{\rm avg}\to 0$3 (Philipsen et al., 2018). Since quarkyonic matter is defined by $\langle \bar q q\rangle_{\rm avg}\to 0$4, while the dilute hadronic vacuum scales as $\langle \bar q q\rangle_{\rm avg}\to 0$5 and deconfined quark–gluon plasma as $\langle \bar q q\rangle_{\rm avg}\to 0$6, this result supports the claim that baryonic matter and quarkyonic matter may be thermodynamically indistinguishable at nuclear densities in this regime (Philipsen et al., 2018).

The same study finds that the nuclear liquid–gas transition becomes more strongly first order as $\langle \bar q q\rangle_{\rm avg}\to 0$7 increases, with the critical endpoint moving to higher temperature and potentially connecting to deconfinement in the large-$\langle \bar q q\rangle_{\rm avg}\to 0$8 limit (Philipsen et al., 2018). A related conceptual synthesis argues that dense low-temperature matter should be regarded as quarkyonic because baryon–baryon interaction energy and pressure scale as $\langle \bar q q\rangle_{\rm avg}\to 0$9, and because deconfinement at high $F_\pi^*\neq 0$0 is better viewed as a smooth baryon–quark continuity than as a sharply separated first-order jump (Fukushima, 2014). In that perspective, diquark correlations, color superconductivity, and inhomogeneous chiral order all inhabit the same broad quarkyonic territory (Fukushima, 2014).

Two-color QCD supplies a useful counterexample. There the lightest baryons are bosonic diquarks, and the heavy-quark effective theory yields a continuous onset at $F_\pi^*\neq 0$1 with no binding-energy shift, contrasting directly with the first-order liquid–gas onset of SU(3) nuclear matter (Scior et al., 2015). This emphasizes that baryquark matter, even when defined through mixed baryon–quark language, can inherit very different onset structures depending on color representation and baryon statistics.

3. Topological dense hadronic phases with restored chiral symmetry on average

A different strand of the literature identifies baryquark matter with a topologically reorganized hadronic phase rather than with a quark-dominated Fermi sea. In hidden local symmetry with vector manifestation, and in the holographic Sakai–Sugimoto model, the relevant phase appears above the topology-change density $F_\pi^*\neq 0$2 and below or around the chiral restoration density $F_\pi^*\neq 0$3. Its defining feature is

$F_\pi^*\neq 0$4

while $F_\pi^*\neq 0$5 remains nonzero and hadronic excitations stay confined (Rho, 2010).

In the gauge-theory description this phase is a half-skyrmion crystal. In the gravity dual it is a dyonic “half-instanton salt,” produced when instantons split under holonomy into constituent dyons forming a bcc crystal (Rho, 2010). The onset density is model dependent but typically lies in the range $F_\pi^*\neq 0$6, while the kaon-sector analysis places $F_\pi^*\neq 0$7 (Rho, 2010). Because the averaged condensate vanishes without loss of confinement, this phase is often presented as quarkyonic-like: the symmetry pattern resembles chiral restoration, but the relevant quasiparticles remain color singlets.

The hidden-local-symmetry formulation ties this structure to the vector-manifestation fixed point,

$F_\pi^*\neq 0$8

while the nucleon mass approaches a parity-doublet-like decomposition $F_\pi^*\neq 0$9 rather than vanishing with the vector mass (Rho, 2010). This modifies Brown–Rho scaling across $N_c$0, changes tensor forces, and suppresses short-range many-body repulsion through the density dependence of the $N_c$1-mediated homogeneous Wess–Zumino term once the soft dilaton is included (Rho, 2010).

The compact-star implications are substantial. Above $N_c$2, the equation of state is described as stiff but nearly conformal, with $N_c$3 reduced and the sound speed tending toward

$N_c$4

in the range $N_c$5 (Rho, 2010). Related EFT work develops a density sequence NM $N_c$6 KNM $N_c$7 SQM, where a half-skyrmion phase reshapes in-medium mass scaling, stiffens the symmetry energy, and allows a smooth kaon-driven route toward strange quark matter consistent with $N_c$8 stars (Lee et al., 2011). In this usage, baryquark matter is not a deconfined quark phase but a confined, topologically nontrivial dense hadronic medium.

4. Momentum-space shell constructions

Recent quasiparticle models give the term a more literal momentum-space meaning. Standard quarkyonic matter is represented by a quark Fermi sea up to a bulk momentum and a baryonic shell near the Fermi surface. Baryquark matter inverts this arrangement: baryons fill the low-momentum core, while quarks occupy a shell above it. In the excluded-volume model of cold isospin-symmetric matter, the densities of states are taken as

$N_c$9

so nucleons occupy $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$0 and quarks occupy $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$1 (Poberezhnyuk et al., 2023).

The nucleonic sector is described by a quantum van der Waals real-gas model,

$$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$2

with three excluded-volume prescriptions $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$3, $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$4, and $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$5. The parameters $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$6 and $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$7 are fixed to the empirical nuclear ground state at $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$8 with binding energy $$W(\mathbf{x})=\prod_{\tau=1}^{N_\tau}U_0(\mathbf{x},\tau)$$9 MeV per baryon, and all three prescriptions produce a liquid–gas critical point near $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$0 MeV and $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$1 (Poberezhnyuk et al., 2023).

Within this setup, both quarkyonic and baryquark constructions yield a transition to quark-dominated matter at $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$2, but baryquark matter is energetically favored in all excluded-volume schemes considered (Poberezhnyuk et al., 2023). It also avoids the infrared regulator needed in the quarkyonic case to cure acausal behavior of the sound speed at quark onset. A related hard-core repulsion model arrives at the same energetic ordering: when the shell widths are fixed dynamically by minimizing the energy density at fixed baryon number, baryquark matter is always lower in energy than quarkyonic matter, and its $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$3 remains regular without an infrared regulator because quarks first appear at finite momentum rather than at $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$4 (Koch et al., 2022).

The ideal dual Quarkyonic model sharpens the statistical origin of this shell structure. It links the baryon occupation $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$5 and the quark occupation $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$6 through

$$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$7

and shows that once quark occupation saturates at low momentum, the optimal baryon distribution becomes

$$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$8

so the bulk is capped at $$\lambda_1(u,N_\tau)=u^{N_\tau}[1+\cdots],\qquad S_{\rm eff}^{\rm gauge,LO}=\lambda_1\sum_{\langle x,y\rangle}(L_xL_y^*+L_x^*L_y),$$9 while a baryonic shell survives near the Fermi surface (Kojo, 31 May 2026). In this formulation quark saturation at a few times $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$0 drives rapid stiffening and a peak in $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$1, and the same quark-level Pauli constraints shift hyperon thresholds upward, mitigating the hyperon-softening problem (Fujimoto et al., 2024).

Not all dynamical shell models favor early quark admixture, however. In a parity-doublet model with chiral dynamics, a self-consistent minimization of the momentum-space shell finds baryquark matter lower in energy than quarkyonic matter at fixed quark fraction, yet purely hadronic matter remains the true minimum up to densities well beyond nuclear saturation; only near $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$2 does a shallow baryquark minimum with $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$3 appear for the physical constituent-quark choice $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$4 (Gao et al., 3 Sep 2025). This demonstrates that chiral restoration and quark onset need not coincide.

5. Unified effective and holographic coexistence models

Several frameworks attempt to describe baryonic matter, quark matter, and their interpolating or coexisting regimes within one formalism. In V-QCD, two implementations of homogeneous baryons were analyzed. The phenomenologically viable one, based on a homogeneous bulk gauge field, produces at $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$5 a vacuum phase, a confining chirally broken baryonic phase, and a deconfined chirally symmetric quark phase, separated by strongly first-order transitions as $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$6 increases (Ishii et al., 2019). The baryonic equation of state is stiff, with $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$7 in two intervals before the transition to quark matter, and the vacuum-to-baryon onset occurs near $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$8 MeV with $$h_1=(2\kappa e^{a\mu})^{N_\tau}=e^{(\mu-m)/T},\qquad \bar h_1=(2\kappa e^{-a\mu})^{N_\tau}=e^{-(\mu+m)/T},$$9 (Ishii et al., 2019).

The Witten–Sakai–Sugimoto model realizes quarkyonic matter more literally through coexistence of pointlike baryons and fundamental quarks on flavor branes. Baryons are represented by instanton layers at $m=-\ln(2\kappa)$0, quarks by strings or horizon-reaching gauge fields, and the grand potential takes the on-shell form

$m=-\ln(2\kappa)$1

with coexistence determined by stationarity conditions for the baryon and quark sources (Kovensky et al., 2020). In this construction holographic quarkyonic matter is chirally symmetric in the chiral limit, favored over pure nuclear and pure quark matter at large chemical potential, and connected to the nuclear phase by a first-order transition at small or physical pion mass, while sufficiently heavy pions allow a quark–hadron continuity through the quarkyonic phase (Kovensky et al., 2020).

An extended SU(3) NJL model takes yet another route by treating baryons as three-quark clusters with density-dependent structural function

$m=-\ln(2\kappa)$2

and adding a Pauli-blocking contribution $m=-\ln(2\kappa)$3 to baryon masses so that baryons become unbound in a quark Fermi sea (Xia, 2024). In this setup quarks can emerge as quasi-free particles and coexist with baryons, which is explicitly identified as quarkyonic matter, while deconfinement is modeled as a Mott transition when baryons disappear (Xia, 2024). Depending on $m=-\ln(2\kappa)$4 and the quark–vector couplings, the quarkyonic, chiral, and deconfinement transitions may be first order or continuous, and some parameter sets produce compact-star sequences compatible with NICER radii, GW170817 tidal deformabilities, and the $m=-\ln(2\kappa)$5 mass constraint (Xia, 2024).

Taken together, these models do not agree on a universal onset density or transition order. What they do share is the replacement of a sharp baryon-versus-quark dichotomy by coexistence, interpolation, or sequential dissolution.

6. Bulk strong matter, quark clusters, and dark-matter extensions

Outside the dense-QCD phase-diagram literature, “baryquark matter” is often generalized to macroscopic baryon-number-carrying quark phases. One review defines bulk strong matter as macroscopic, self-bound matter dominated by the strong interaction and argues that above a critical baryon number

$m=-\ln(2\kappa)$6

weak equilibrium and electron energetics favor three-flavor symmetry. In this taxonomy, baryquark matter includes both deconfined strange quark matter and localized strangeon matter (Lai et al., 2022). Strangeon matter is a nonperturbative quark-cluster phase, potentially solid and self-bound, with a stiff equation of state capable of supporting $m=-\ln(2\kappa)$7 and even $m=-\ln(2\kappa)$8 in the scenarios reviewed (Lai et al., 2022).

A related astrophysical treatment of compressed baryonic matter argues that at a few times nuclear density the relevant energy scale is $m=-\ln(2\kappa)$9 MeV, large enough to make strangeness energetically relevant, while strong coupling $\beta=0$0 favors localization into quark clusters rather than deconfined quasifree quarks (Guo et al., 2013). The same work treats quark-cluster stars as self-bound compact stars with stiff equations of state and distinctive phenomenology: strong surface binding, featureless thermal spectra, possible global solidity, and large maximum masses (Guo et al., 2013).

Dark-matter models broaden the term still further. One proposal treats dark matter as antibaryons built from hypothetical fourth-generation antiquarks, with the lightest neutral state $\beta=0$1 carrying $\beta=0$2, vanishing $\beta=0$3-boson couplings, and a target mass $\beta=0$4 GeV, leading to proton-decay-like annihilation signatures with $\beta=0$5 GeV visible energy release (Gorbunov et al., 2013). Another class considers macroscopic quark and antiquark nuggets formed during the QCD transition, with baryon number $\beta=0$6 up to about $\beta=0$7 and flux at Earth scaling as

$\beta=0$8

so that anti-nuggets could generate muon-rich atmospheric showers (Lawson, 2010). Six-flavor quark-matter nuggets provide a more specific cosmological realization: with electroweak supercooling, a first-order six-flavor QCD transition yields nuggets of typical mass $\beta=0$9 g and radius $T\to0$0 cm whose relic abundance can be comparable to the observed dark-matter density (Bai et al., 2018).

These usages are conceptually much broader than the dense-matter definitions. Here baryquark matter no longer denotes an intermediate phase between nuclear and quark matter inside neutron stars alone; it denotes any stable or metastable quark configuration that carries baryon number in bulk or composite form.

7. Open questions, fault lines, and recurring controversies

The principal controversy is semantic and physical at once: whether baryquark matter names a specific QCD phase or merely a recurrent motif linking baryonic and quark descriptions. The cited literature supports the latter reading. Depending on framework, baryquark matter may mean quarkyonic matter, a half-skyrmion phase, a baryon-core/quark-shell configuration, a strangeon medium, or a dark-matter composite.

Within the quarkyonic interpretation, the leading unresolved issue is robustness beyond the strong-coupling or model limit. The heavy-quark effective theory finds $T\to0$1 immediately beyond onset, but explicitly notes that stability under gauge corrections, enforcement of the ’t Hooft limit $T\to0$2, and possible noncommutativity of strong-coupling and large-$T\to0$3 limits remain open (Philipsen et al., 2018). Momentum-space shell models likewise disagree on onset: some find baryquark matter energetically favored over quarkyonic matter and phenomenologically attractive because it avoids infrared pathologies (Koch et al., 2022), whereas chiral parity-doublet dynamics can suppress any relevant quark admixture up to densities well beyond saturation (Gao et al., 3 Sep 2025).

Another fault line concerns the relation between chiral restoration and quark appearance. Half-skyrmion and holographic approaches allow $T\to0$4 with confinement intact (Rho, 2010). The parity-doublet shell analysis makes the non-equivalence sharper by showing that chiral symmetry may become restored while purely hadronic matter still minimizes the free energy (Gao et al., 3 Sep 2025). A plausible implication is that “quark onset,” “deconfinement,” and “chiral restoration” cannot be treated as interchangeable markers of the same transition.

Hyperons provide a further diagnostic. Conventional hadronic models often suffer strong hyperon softening, but quark-level Pauli blocking in ideal dual Quarkyonic models shifts the $T\to0$5 neutral hyperon onset from $T\to0$6 to

$T\to0$7

and makes the ensuing softening mild because little low-energy phase space remains available (Fujimoto et al., 2024). Whether this mechanism survives more realistic interactions, charge-neutral $T\to0$8-equilibrated matter, and finite-temperature transport remains unsettled.

The cumulative picture is therefore not a settled taxonomy but a structured ambiguity. “Baryquark matter” consistently signals matter in which baryons and quarks are not cleanly separable descriptions, yet the microscopic realization, symmetry content, thermodynamic scaling, and observational consequences depend strongly on whether the underlying framework is large-$T\to0$9 QCD, topological EFT, holography, quasiparticle shell modeling, or astrophysical composite-matter phenomenology.