Equivparticle Model in Dense Quark Matter

• The equivparticle model is a QCD-inspired framework where quarks are treated as quasi-free particles with density-dependent equivalent masses that encode both linear confinement and perturbative effects.
• It employs mean-field approximation and self-consistent rearrangement terms to maintain thermodynamic consistency in systems like strange quark matter, strangelets, and proto-strange stars.
• Applications range from modeling bulk quark matter and compact stars to examining interface physics and quarkyonic matter, providing insights into the quark-hadron transition and the equation of state.

The equivparticle model is a QCD-inspired effective description of dense quark matter in which quarks are treated as quasi-free particles with density-dependent equivalent masses. In the formulation used for strange quark matter, compact stars, and quarkyonic matter, the density dependence is chosen to encode linear confinement at low density and leading-order perturbative interactions at higher density, while thermodynamic consistency is maintained through rearrangement terms induced by derivatives of the effective masses (Xia et al., 2018, Xia et al., 2020, Xia et al., 2023). The framework has been applied to bulk quark matter, strangelets, interface physics, finite-temperature proto-strange stars, in-medium quark condensates, quarkyonic matter, compact multibaryons, and compact dwarfs made of quark nuggets (Xia, 2019, Chen et al., 2021, You et al., 2024, You et al., 27 Jul 2025).

1. Effective-theory definition and mass scaling

The model is formulated as an effective quark theory in which strong interactions are not represented by explicit gluon fields in the medium sector; instead, they are absorbed into an equivalent quark mass. A frequently used Lagrangian is

$$\mathcal{L} = \sum_{i=u,d,s}\bar{\Psi}_i \left[ i \gamma^\mu \partial_\mu - m_i(n_\mathrm{b}) - e q_i \gamma^\mu A_\mu \right]\Psi_i - \frac{1}{4} A_{\mu\nu}A^{\mu\nu},$$

with $q_u=2/3$ and $q_d=q_s=-1/3$, and with baryon density $n_\mathrm{b}=\sum_i n_i/3$ (Xia et al., 2018, You et al., 2024).

In dense-matter applications, the effective quark mass is decomposed as

$m_i = m_{i0}+m_{\mathrm I},$

where $m_{i0}$ is the current quark mass and $m_{\mathrm I}$ is the interaction-induced contribution. Several published implementations write

$$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$

or an equivalent typographically compressed form with the same confinement and perturbative content (You et al., 2024, You et al., 2024). In this interpretation, the $D\,n_{\mathrm b}^{-1/3}$ term represents linear confinement, while the $C\,n_{\mathrm b}^{1/3}$ term represents leading-order perturbative interactions. The sign of $q_u=2/3$0 is used to distinguish interaction character: $q_u=2/3$1 gives a one-gluon-exchange-like attractive behavior, whereas $q_u=2/3$2 mimics a repulsive perturbative contribution (Xia, 2019, You et al., 2024).

The physical interpretation of the parameters is stated explicitly in several applications. $q_u=2/3$3 is related to confinement physics and is connected to string tension, chiral restoration density, and vacuum chiral condensates, while $q_u=2/3$4 is tied to the perturbative interaction strength and linked to $q_u=2/3$5 (Xia et al., 2020, Sun et al., 24 Jan 2026). Some finite-temperature papers use the spelling “equivaparticle model” for the same quasiparticle framework (Chen et al., 2021, Chen et al., 2021).

2. Mean-field realization and thermodynamic consistency

For finite systems such as strangelets and compact multiquark states, the model is typically solved in mean-field approximation with no-sea approximation and, in many cases, spherical symmetry (Xia, 2019, You et al., 27 Jul 2025). The quark spinor is expanded in radial form,

$q_u=2/3$6

leading to a radial Dirac equation for the upper and lower components (Xia et al., 2018, You et al., 2024).

The scalar and vector mean fields are generated by the density dependence of the equivalent mass. In strangelet calculations they are written as

$q_u=2/3$7

while the Coulomb field obeys

$q_u=2/3$8

The quark scalar and number densities are built self-consistently from the radial wave functions, and the solution is obtained iteratively by assuming initial densities, computing mean fields, solving the Dirac equation, filling the lowest single-particle levels, updating densities, and repeating to convergence (Xia, 2019, You et al., 2024).

A defining technical point is thermodynamic self-consistency. Because the equivalent masses depend on density, derivatives of the mass must enter the Dirac equation, chemical potentials, pressure, and, in finite-temperature formulations, entropy. The corresponding rearrangement term is identified in the vector potential and in bulk thermodynamics; without it, the model gives inconsistent thermodynamics (Xia et al., 2018, Chen et al., 2021, Jin et al., 2022). In cold quark matter, the thermodynamic chemical potential is written as a free-gas contribution plus an interaction correction induced by the density dependence of the mass, and the pressure is then obtained from the standard identity $q_u=2/3$9 using the full chemical potentials (Jin et al., 2022).

This formal structure makes the model neither a bag-model pressure shift nor a direct microscopic QCD calculation. It is an equivalent quasiparticle theory whose interaction content is concentrated in the state dependence of the effective masses and in the corresponding rearrangement contributions (Xia et al., 2020, Xia et al., 2023).

3. Interfaces, strangelets, and finite-size energetics

One of the most developed uses of the equivparticle model is the study of interface effects in strange quark matter. In mean-field calculations of strangelets, confinement generated by the density-dependent mass makes the baryon density and mean fields vanish smoothly at the quark-vacuum boundary, rather than abruptly as in the MIT bag model. The surface layer is reported to be about $q_d=q_s=-1/3$0–$q_d=q_s=-1/3$1 fm thick (Xia, 2019, Xia et al., 2018).

Interface quantities are extracted from the finite-size dependence of the strangelet energy per baryon through a liquid-drop expansion,

$q_d=q_s=-1/3$2

with

$q_d=q_s=-1/3$3

Here $q_d=q_s=-1/3$4 is the surface tension and $q_d=q_s=-1/3$5 is the curvature term (Xia, 2019, Xia et al., 2018).

A central result is that, at zero external pressure, both $q_d=q_s=-1/3$6 and $q_d=q_s=-1/3$7 increase monotonically with the strange-quark-matter density $q_d=q_s=-1/3$8. The interface papers summarize this trend approximately as

$q_d=q_s=-1/3$9

with units consistent with the original table (Xia, 2019). For the parameter set $n_\mathrm{b}=\sum_i n_i/3$0, constrained according to the $n_\mathrm{b}=\sum_i n_i/3$1 strange-star requirement tied to PSR J0348+0432, the reported values are $n_\mathrm{b}=\sum_i n_i/3$2, $n_\mathrm{b}=\sum_i n_i/3$3, $n_\mathrm{b}=\sum_i n_i/3$4, $n_\mathrm{b}=\sum_i n_i/3$5, $n_\mathrm{b}=\sum_i n_i/3$6, and $n_\mathrm{b}=\sum_i n_i/3$7 (Xia, 2019). Other tested parameter sets give larger surface tensions, including $n_\mathrm{b}=\sum_i n_i/3$8 for $n_\mathrm{b}=\sum_i n_i/3$9, $m_i = m_{i0}+m_{\mathrm I},$0 for $m_i = m_{i0}+m_{\mathrm I},$1, $m_i = m_{i0}+m_{\mathrm I},$2 for $m_i = m_{i0}+m_{\mathrm I},$3, and $m_i = m_{i0}+m_{\mathrm I},$4 for $m_i = m_{i0}+m_{\mathrm I},$5 (Xia, 2019).

The same studies compare full mean-field results with the multiple reflection expansion (MRE). In this framework, raw MRE is found to overestimate the surface tension and underestimate the curvature term. The stated reason is that standard MRE was designed for bag-like hard-wall confinement, whereas the equivparticle model produces a smooth interface with finite thickness. To reproduce the mean-field results, the MRE density of states should be modified by damping factors, with a reported better match around $m_i = m_{i0}+m_{\mathrm I},$6 and $m_i = m_{i0}+m_{\mathrm I},$7, especially for $m_i = m_{i0}+m_{\mathrm I},$8 to $m_i = m_{i0}+m_{\mathrm I},$9 (Xia et al., 2018).

These interface results are used to distinguish stable and unstable strange quark matter. For absolutely stable strange quark matter, surface and curvature energies raise the energy per baryon of small strangelets and can destabilize them, while shell effects may make specific baryon numbers especially stable. For unstable strange quark matter, the interface tension influences whether quark-hadron matter forms structured mixed phases or approaches Maxwell-construction behavior (Xia, 2019).

4. Finite-temperature formulations

Finite-temperature versions of the model extend the equivalent mass to depend on both density and temperature. In the thermodynamically self-consistent “equivaparticle” treatment of strange quark matter and proto-strange stars, the interaction mass is taken as

$m_{i0}$0

so that the model reduces to the zero-temperature density-dependent form at $m_{i0}$1 and the interaction mass tends to zero rather than becoming negative at high $m_{i0}$2 (Chen et al., 2021, Chen et al., 2021).

In that formulation, the free energy density is taken as the fundamental thermodynamic potential, and the chemical potentials, entropy, and pressure acquire additional terms through the temperature and density derivatives of the equivalent masses. The papers emphasize a direct numerical consistency check: at the minimum of the free energy per baryon, the pressure vanishes. They also report that the energy per baryon increases with temperature, while the free energy per baryon decreases and can eventually become negative; the entropy per baryon tends to zero as $m_{i0}$3; and the sound velocity approaches the extreme relativistic limit $m_{i0}$4 as density increases, although larger $m_{i0}$5 and $m_{i0}$6 slightly weaken that tendency (Chen et al., 2021).

The finite-temperature strangelet extension combines this thermodynamics with MRE finite-size corrections and Coulomb energy. In that study, $m_{i0}$7 and the charge-to-mass ratio $m_{i0}$8 decrease with baryon number $m_{i0}$9, while the mechanically stable radius $m_{\mathrm I}$0 and the strangeness per baryon $m_{\mathrm I}$1 increase. At fixed $m_{\mathrm I}$2, increasing $m_{\mathrm I}$3 raises $m_{\mathrm I}$4, $m_{\mathrm I}$5, and $m_{\mathrm I}$6, while lowering $m_{\mathrm I}$7 (Chen et al., 2021).

A later microscopic strangelet study adopts mean-field Dirac equations with Fermi-Dirac occupations and neglects explicit temperature dependence of the quark masses below the deconfinement temperature because the sub-$m_{\mathrm I}$8 dependence is described as weak. In that treatment, temperature weakens shell effects through shell dampening, and the free energy per baryon is fitted by a liquid-drop formula after subtracting the Coulomb term. The striking result is that the surface tension does not decrease with temperature but instead rises until it peaks at $m_{\mathrm I}$9–$$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$0 MeV, whereas the curvature term decreases with $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$1 (You et al., 2024). The same work computes neutron and proton emission widths from equilibrium external nucleon gas densities and finds that the emission rates generally increase with $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$2 for stable strangelets but decrease for strangelets that are unstable to nucleon emission at $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$3 (You et al., 2024).

For stellar applications, the finite-temperature scaling allows massive proto-strange stars with maximum mass above $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$4 at $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$5 MeV for some parameter choices, a feature stated to be absent in earlier quark-mass scalings (Chen et al., 2021).

5. Compact stars, mixed phases, and chiral order parameters

In compact-star studies, the equivparticle model is used both as a quark-matter EOS and as a microscopic estimate of quark-hadron interface tension. A systematic mixed-phase analysis adopts the surface-tension prescription

$$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$6

where $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$7 replaces the quark-vacuum density $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$8 to include the hadronic contribution roughly (Xia et al., 2020). Because $$m_i(n_{\mathrm b})=m_{i0}+D\,n_{\mathrm b}^{-1/3}+C\,n_{\mathrm b}^{1/3},$$9 increases with density, its derivative enters the mixed-phase thermodynamics and stiffens the mixed-phase EOS. In that work, the equivparticle and damped-MRE surface tensions are generally well below the critical $D\,n_{\mathrm b}^{-1/3}$0, favoring structured pasta phases rather than bulk-separated Maxwell behavior, and the authors report that in almost all cases allowed by pulsar observations the quark phase persists inside the most massive compact stars (Xia et al., 2020).

A distinct compact-star program combines a nuclear-matter EOS generated by Taylor expansion with quark matter predicted by the equivparticle model and matches the two by Maxwell construction. In this setting, the phase transition is treated as first order with local charge neutrality in each phase. The reported astrophysically allowed hybrid EOSs place quark deconfinement at $D\,n_{\mathrm b}^{-1/3}$1 to $D\,n_{\mathrm b}^{-1/3}$2, with central pure-quark densities $D\,n_{\mathrm b}^{-1/3}$3 to $D\,n_{\mathrm b}^{-1/3}$4 in the most massive hybrid stars; quark cores are described as rather small and do not emerge for compact stars with $D\,n_{\mathrm b}^{-1/3}$5 (Jin et al., 2022).

The same framework is used to extract the in-medium quark condensate. The equivalent Hamiltonian is defined so that the medium-vacuum energy difference matches that of QCD, leading to

$D\,n_{\mathrm b}^{-1/3}$6

and, for the averaged two-flavor condensate,

$D\,n_{\mathrm b}^{-1/3}$7

In neutron-star matter constrained by nuclear and astrophysical data, the condensate decreases nonlinearly with density but does not vanish inside stable neutron stars, so chiral symmetry is stated to be only partially restored (Jin et al., 2021). In the hybrid-star Maxwell-construction study, the condensate remains large in the pure-quark phase and does not necessarily decrease with density, which is interpreted as evidence for significant nonperturbative contributions in the density range covered by compact stars (Jin et al., 2022).

Taken together, these compact-star applications establish two recurring uses of the model: the generation of a density-dependent quark EOS and the extraction of interface or condensate observables that are difficult to fix from constant-parameter phenomenology alone (Xia et al., 2020, Jin et al., 2022).

6. Extended uses: quarkyonic matter, compact multibaryons, and quark-nugget dwarfs

Later work embeds the equivparticle model in broader many-body constructions.

Domain	Use of the model	Representative papers
Quarkyonic matter	Quark Fermi sea with baryonic Fermi surface in extended RMF	(Xia et al., 2023, Sun et al., 24 Jan 2026)
Compact multibaryons	Spherical color-singlet $D\,n_{\mathrm b}^{-1/3}$8-quark states in MFA	(You et al., 27 Jul 2025)
Compact dwarfs	Strangelets or $D\,n_{\mathrm b}^{-1/3}$9 quark-matter nuggets in bcc lattices	(You et al., 2024)

In quarkyonic-matter studies, the quark sector of an extended RMF model is supplied by the equivparticle mass scaling, while baryons are described by density-dependent covariant density functionals such as TW99, PKDD, and DD-ME2. The construction follows the picture of “a quark Fermi Sea” plus “a baryonic Fermi surface,” with the transition fixed by matching single-particle energies at the interface, for example

$C\,n_{\mathrm b}^{1/3}$0

In the strangeness extension, analogous matching is imposed for $C\,n_{\mathrm b}^{1/3}$1 hyperons, and hyperons $C\,n_{\mathrm b}^{1/3}$2 as well as strange quarks are included in a unified framework (Xia et al., 2023, Sun et al., 24 Jan 2026). These studies report that quarkyonic transition softens overly stiff RMF EOSs and can improve agreement with heavy-ion and astrophysical constraints. When hyperons and quark-hadron transition effects are both included, the EOS exhibits additional softening and the maximum sound velocity is reported as $C\,n_{\mathrm b}^{1/3}$3, close to the ultrarelativistic limit of $C\,n_{\mathrm b}^{1/3}$4 (Sun et al., 24 Jan 2026).

A different extension treats compact multibaryon states as self-consistent mean-field solutions of the equivparticle model. In that work, color-singlet $C\,n_{\mathrm b}^{1/3}$5-quark configurations with $C\,n_{\mathrm b}^{1/3}$6 are assumed to be spherically symmetric with quarks in the $C\,n_{\mathrm b}^{1/3}$7 state, and at fixed $C\,n_{\mathrm b}^{1/3}$8 the states form a single irreducible SU(6) representation (You et al., 27 Jul 2025). The model parameters are constrained by Bayesian inference using eight baryons and $C\,n_{\mathrm b}^{1/3}$9, after which the study predicts bound H-dibaryon, $q_u=2/3$00, and a dibaryon with $q_u=2/3$01 relative to $q_u=2/3$02, $q_u=2/3$03, and $q_u=2/3$04 thresholds, while heavier compact multibaryons are described as unlikely to be stable (You et al., 27 Jul 2025).

The compact-dwarf application uses the equivparticle model to compute strangelets and nonstrange $q_u=2/3$05 quark-matter nuggets at fixed baryon and charge numbers from self-consistent Dirac equations in mean-field approximation. Those nuggets are then arranged in body-centered cubic lattices embedded in a uniform electron background, leading to compact-dwarf EOSs softer than ordinary white-dwarf matter because the nugget charge-to-mass ratios are smaller (You et al., 2024). In that model, strangelets become absolutely stable for $q_u=2/3$06, while $q_u=2/3$07 nuggets become more stable than nuclei for $q_u=2/3$08 and absolutely stable against decay to nuclei for $q_u=2/3$09 (You et al., 2024). Radial-oscillation calculations indicate that these compact dwarfs are dynamically stable and typically have higher oscillation frequencies than traditional white dwarfs (You et al., 2024).

A consistent theme across these extensions is that the density-dependent equivalent mass is used as the organizing device that connects confinement physics, perturbative corrections, and thermodynamic consistency. The quantitative outcomes, however, depend strongly on how that mass scaling is implemented, how interface physics is modeled, and whether finite-size, shell, or quark-baryon matching effects are treated microscopically.