First-Order Hadron–Quark Transition

• The first-order hadron–quark phase transition is a non-analytic change between confined hadronic matter and deconfined quark matter, characterized by abrupt jumps in density and energy.
• It is modeled through matched equations of state using Maxwell or Gibbs constructions, with key parameters such as transition density, latent heat, and quark matter stiffness.
• Observational constraints from neutron stars, heavy-ion collisions, and early-Universe cosmology provide practical insights into latent heat limits, quark stiffness, and the apparent absence of twin-star phenomena.

Searching arXiv for recent and relevant papers on first-order hadron–quark phase transitions to ground the article in the cited literature. First-order hadron–quark phase transition (FOPT) denotes a non-analytic change between hadronic matter and deconfined quark matter characterized by coexistence, a discontinuity in density or energy density, and, depending on the thermodynamic ensemble and conserved charges, either a constant-pressure plateau or a non-constant-pressure mixed phase. In neutron-star phenomenology, heavy-ion matter, and early-Universe cosmology, the FOPT is typically formulated through matched hadronic and quark equations of state (EOSs), with the transition specified by quantities such as the onset density, the latent heat or energy-density jump, and the quark-matter stiffness. Recent work has emphasized both direct inversion of neutron-star observables within constant-speed-of-sound parameterizations and more microscopic or phenomenological constructions based on RMF, NJL, Dyson–Schwinger, Friedberg–Lee, and relativistic density-functional frameworks (Zhang et al., 2023, Xie et al., 2020, Bai et al., 2019, Zhou et al., 2020).

1. Definition and thermodynamic structure

A first-order hadron–quark phase transition is conventionally identified by equality of thermodynamic potentials across two phases together with a discontinuous change in density-like observables. In one-component matter, the standard Maxwell construction imposes continuity of pressure and chemical potential at a single transition point, producing a constant-pressure mixed region. In the CSS hybrid-star parameterization, the EOS is written piecewise in energy density,

$$P(\varepsilon) = \begin{cases} P_{\rm HM}(\varepsilon), & \varepsilon<\varepsilon_t,\ P_t + c_{\rm QM}^2(\varepsilon-\varepsilon_t), & \varepsilon\ge \varepsilon_t, \end{cases}$$

with pressure continuity at $\varepsilon_t$ and chemical-potential equality implied by the Maxwell construction when only one conserved charge is considered (Zhang et al., 2023).

The principal parameters in this setting are the transition density $\rho_t$, defined by $\varepsilon_t=\varepsilon(\rho_t)$ on the hadronic branch, the energy-density jump

$$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$

and the quark-matter squared sound speed $c_{\rm QM}^2$ (Zhang et al., 2023). Closely related CSS formulations instead express the quark branch as

$$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$

again making the latent heat and stiffness explicit (Xie et al., 2020).

In systems with two conserved charges, the mixed phase is not generally a constant-pressure plateau. Under Gibbs conditions one requires equality of temperature, baryon chemical potential, electron or charge chemical potential, and pressure, while charge neutrality is imposed globally rather than locally. In this case pressure varies continuously with the quark volume fraction $\chi$, and the mixed phase exhibits intrinsically non-linear structure (Zheng et al., 2014, Shao et al., 2011). This distinction is central in comparing Maxwell-type neutron-star constructions to heavy-ion or multicomponent stellar matter calculations.

The latent heat may be represented either as an energy-density discontinuity $\Delta\varepsilon$ in compact-star EOS models or as a coexistence-region discontinuity in $\varepsilon$, entropy density, or baryon density in finite-temperature QCD-motivated descriptions. In an analytic mean-field Ising mapping to QCD, discontinuities $\varepsilon_t$0 and $\varepsilon_t$1 across the coexistence line imply the Clapeyron relation

$\varepsilon_t$2

making the FOPT part of a broader coexistence and spinodal structure (Karthein et al., 18 Nov 2025).

2. EOS constructions and phase-equilibrium prescriptions

A large fraction of the literature models the FOPT through two-phase EOS matching. In compact-star applications, hadronic matter has been described by RMF or meta-modeling approaches, while quark matter is represented by CSS, Dyson–Schwinger, NJL, string-flip or confinement density-dependent mass models (Xie et al., 2020, Bai et al., 2019, Bastian, 2020, Li et al., 2015). The transition is then implemented by Maxwell or Gibbs conditions depending on the conserved-charge content.

In CSS-based neutron-star studies, the hadronic EOS is fixed or sampled, then matched at a transition pressure $\varepsilon_t$3 to a quark branch with constant $\varepsilon_t$4. The three independent FOPT parameters are commonly taken as $\varepsilon_t$5, $\varepsilon_t$6 or $\varepsilon_t$7, and $\varepsilon_t$8 (Xie et al., 2020, Zhang et al., 2023, Miao et al., 2020). In one implementation, the priors are

$\varepsilon_t$9

with Maxwell matching through continuity of $\rho_t$0 and $\rho_t$1 (Xie et al., 2020).

More microscopic quark-sector descriptions alter both the coexistence region and the existence of critical endpoints. In the RMF + lattice-motivated QGP treatment, the hadronic sector includes the full baryon octet, pions, and kaons interacting through $\rho_t$2, while the quark–gluon sector is an MIT-bag model with $\rho_t$3 corrections. Solving $\rho_t$4 yields a first-order line ending at a critical endpoint near $\rho_t$5 (Uddin et al., 2012).

In asymmetric matter relevant to heavy-ion collisions, the two-EOS RMF + NJL approach gives Gibbs equilibrium under fixed global asymmetry. The mixed phase is characterized by total densities

$\rho_t$6

and the coexistence region terminates around

$\rho_t$7

with the FOPT occurring for $\rho_t$8 MeV and $\rho_t$9 (Shao et al., 2011).

Alternative constructions replace explicit mixed-phase microphysics with interpolation. In the Dyson–Schwinger/RMF framework, the sound-speed interpolation scheme parametrizes

$\varepsilon_t=\varepsilon(\rho_t)$0

and reconstructs the coexistence-window EOS through

$\varepsilon_t=\varepsilon(\rho_t)$1

matched to hadronic and quark branches at $\varepsilon_t=\varepsilon(\rho_t)$2 (Bai et al., 2019, Bai et al., 2019).

Finite-size and surface-tension effects motivate mixed-phase smoothing. In a twin-star study based on the ACB4 piecewise-polytropic EOS, the sharp Maxwell plateau is replaced by a quadratic interpolation in chemical potential,

$\varepsilon_t=\varepsilon(\rho_t)$3

where $\varepsilon_t=\varepsilon(\rho_t)$4 controls the degree of pasta-induced smoothing; $\varepsilon_t=\varepsilon(\rho_t)$5 reproduces the sharp transition (Pradhan et al., 2023).

3. Neutron-star constraints and inversion of transition parameters

Recent neutron-star studies have shifted from forward modeling to inverse constraints on FOPT parameters. A CSS inversion in a three-dimensional $\varepsilon_t=\varepsilon(\rho_t)$6 parameter space fixes all hadronic EOS parameters at their best-known values, constructs the hybrid EOS for each parameter triplet, solves the TOV equations, and computes $\varepsilon_t=\varepsilon(\rho_t)$7, $\varepsilon_t=\varepsilon(\rho_t)$8, and $\varepsilon_t=\varepsilon(\rho_t)$9. Imposing

$$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$0

delimits an allowed “tube” in CSS parameter space (Zhang et al., 2023).

The most striking result of that inversion is a lower limit on quark-matter stiffness above the conformal bound: $$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$1

$$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$2

No meaningful upper limit on $$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$3 is obtained from current data (Zhang et al., 2023). This indicates that, within that framework, quark matter in neutron stars must be stiffer than a conformal gas.

A Bayesian analysis with nine EOS parameters—six hadronic meta-model parameters and three CSS parameters—finds posterior peaks at

$$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$4

at $$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$5 confidence level (Xie et al., 2020). The same study reports that the total probability of $$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$6 is very small, and that the posterior distributions of hadronic and quark EOS parameters are very weakly correlated.

Within the quark-mean-field + CSS framework, systematic exploration of $$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$7 together with the symmetry-energy slope $$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$8 shows that increasing either $$\Delta\varepsilon \equiv \varepsilon_t^+ - \varepsilon_t^-,$$9 or $c_{\rm QM}^2$0 softens the core and reduces $c_{\rm QM}^2$1, while a stiffer quark phase raises $c_{\rm QM}^2$2 and yields more compact stars (Miao et al., 2020). That analysis quotes robust limits $c_{\rm QM}^2$3 and $c_{\rm QM}^2$4 km, and finds that a too-weak transition,

$c_{\rm QM}^2$5

taking place at low densities $c_{\rm QM}^2$6 is strongly disfavored (Miao et al., 2020).

Sound-speed interpolation combined with astronomical calibration yields a complementary description in terms of transition chemical potentials rather than direct CSS parameters. In one such analysis, the most probable values are

$c_{\rm QM}^2$7

corresponding to $c_{\rm QM}^2$8 and $c_{\rm QM}^2$9 (Bai et al., 2019). A closely related study using extensive sampling finds peak values near

$$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$0

with modest sensitivity to interpolation order and hyperon content (Bai et al., 2019).

These results collectively favor early onset and moderate latent heat together with a very stiff quark branch. A plausible implication is that present multimessenger data are more compatible with moderate first-order transitions than with either vanishingly weak crossovers in the same parameterizations or very strong latent-heat jumps that destabilize the stellar sequence.

4. Correlations, critical criteria, and the twin-star problem

The structure of the FOPT in neutron stars is strongly constrained by stability criteria. In CSS-type models, a necessary condition for a connected stable hybrid branch is the Seidov inequality

$$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$1

with larger jumps allowing a disconnected third family (Miao et al., 2020). The Bayesian study of canonical stars used the Seidov condition in its default likelihood filter, although removing it changed the posterior only slightly (Xie et al., 2020).

An important empirical finding is the approximately linear anti-correlation between transition density and latent heat at the causality boundary $$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$2. The upper envelope of allowed points in $$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$3 is fitted by

$$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$4

for $$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$5 km, and

$$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$6

for $$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$7 km, both with Pearson $$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$8 (Zhang et al., 2023). This suggests a narrow tradeoff: later transitions permit smaller latent heat if the EOS is to satisfy current mass-radius-tidal constraints.

Twin-star scenarios remain a focal controversy. A commonly cited heuristic is that twins require a strong first-order transition, typically $$\varepsilon(p)= \begin{cases} \varepsilon_{\rm HM}(p),&p<p_t,\ \varepsilon_{\rm HM}(p_t)+\Delta\varepsilon+\dfrac{p-p_t}{c_{\rm QM}^2},&p>p_t, \end{cases}$$9 (Zhang et al., 2023). Yet the inversion study finds upper limits

$\chi$0

for $\chi$1 km and

$\chi$2

for $\chi$3 km; a survey of 137 hybrid EOS mass–radius curves in the allowed region showed no twin-star branches (Zhang et al., 2023).

By contrast, models purposely engineered for large discontinuities or explicit mixed-phase control can generate third-family branches. In the ACB4 plus pasta interpolation framework, $\chi$4 produces a vertical jump in $\chi$5 and $\chi$6 at branch onset, whereas increasing $\chi$7 up to approximately $\chi$8 smooths the transition and eventually removes the third family (Pradhan et al., 2023). In the excluded-volume/chiral hybrid model, an abrupt first-order-like jump to pure quark matter at $\chi$9 produces a plateau in $\Delta\varepsilon$0 at $\Delta\varepsilon$1 and a latent heat $\Delta\varepsilon$2, which may produce a “mass twin” phenomenon if the latent heat is large enough (Kouno et al., 2023).

The apparent tension is model-dependent rather than contradictory. Present observational inversion in constrained CSS spaces disfavors twins (Zhang et al., 2023), while broader phenomenological constructions can still realize them if the transition is sufficiently abrupt or if mixed-phase smoothing is weak (Pradhan et al., 2023, Kouno et al., 2023).

5. Bubble nucleation, spinodals, and finite-temperature dynamics

Finite-temperature FOPT dynamics are often formulated in terms of nucleation and metastability. In the Friedberg–Lee model, the tree-level potential

$\Delta\varepsilon$3

has two minima when $\Delta\varepsilon$4, and the cubic term $\Delta\varepsilon$5 generates a barrier even at $\Delta\varepsilon$6, producing a strong FOPT (Zhou et al., 2020). Including one-loop thermal and chemical-potential corrections yields an effective potential $\Delta\varepsilon$7 with degenerate minima at the coexistence point.

The critical bubble is the $\Delta\varepsilon$8-symmetric bounce solving

$\Delta\varepsilon$9

with $\varepsilon$0 and $\varepsilon$1 (Zhou et al., 2020). In the thin-wall limit, the bubble free energy is

$\varepsilon$2

leading to the critical radius

$\varepsilon$3

The surface tension and bubble free-energy shift are

$\varepsilon$4

$\varepsilon$5

and the thermal nucleation rate is approximated by

$\varepsilon$6

For the strong transition considered there, $\varepsilon$7 does not go to zero at low $\varepsilon$8, $\varepsilon$9 remains finite as $\varepsilon_t$00, and $\varepsilon_t$01 is non-monotonic, unlike in weak FOPT scenarios approaching spinodal decomposition (Zhou et al., 2020).

Spinodal physics has re-emerged in analytic QCD EOS constructions. A mean-field Ising mapping to the QCD phase diagram reconstructs both coexistence and spinodal regions by solving the cubic Landau equation

$\varepsilon_t$02

and locating stability loss through

$\varepsilon_t$03

This yields explicit spinodal curves $\varepsilon_t$04, providing access to super-heated hadronic matter and super-cooled quark–gluon plasma (Karthein et al., 18 Nov 2025). Such formulations are intended for hydrodynamic simulations at low collision energy where metastable trajectories matter.

This finite-temperature perspective clarifies a frequent misconception: the coexistence line does not exhaust FOPT physics. Metastable and unstable domains, nucleation barriers, and spinodal instabilities are essential for dynamical realizations in heavy-ion collisions and, by analogy, in astrophysical transient environments (Zhou et al., 2020, Karthein et al., 18 Nov 2025).

6. Physical settings and observational signatures

The FOPT has been investigated in three principal arenas: neutron stars, heavy-ion collisions, and the early Universe. In neutron stars, the main observables are maximum mass, radii, tidal deformabilities, and oscillation spectra (Zhang et al., 2023, Xie et al., 2020, Pradhan et al., 2023). In heavy-ion matter, emphasis falls on coexistence regions, spinodals, isospin distillation, and critical endpoints (Shao et al., 2011, Karthein et al., 18 Nov 2025). In cosmology, the focus is on transition timing, bubble growth, and the modification of thermodynamic evolution by non-standard gravity (Haddad et al., 2014, Saaidi et al., 2012).

Gravitational-wave asteroseismology has been proposed as a probe of the transition character. In the full general relativistic $\varepsilon_t$05-mode analysis of sharp and smooth transitions, the sharp-Maxwell case produces a gap in the $\varepsilon_t$06 curve around onset mass: two stars of the same mass but with $\varepsilon_t$07 km have $\varepsilon_t$08 differing by $\varepsilon_t$09. With Einstein Telescope-level errors of $\varepsilon_t$10 and $\varepsilon_t$11, twin branches could be distinguished if $\varepsilon_t$12 km at $\varepsilon_t$13; smooth transitions with $\varepsilon_t$14 erase these jumps (Pradhan et al., 2023).

Heavy-ion studies of asymmetric matter predict isospin-sensitive signals of the mixed phase, including enhanced $\varepsilon_t$15 and $\varepsilon_t$16 ratios, increased $\varepsilon_t$17 and hyperon yields, and event-by-event neutron/proton fluctuations due to isospin distillation (Shao et al., 2011). In the analytic Ising-based EOS, the spinodal region permits super-heating and super-cooling and may seed nucleation and cavitation in hydrodynamic evolution at $\varepsilon_t$18 GeV (Karthein et al., 18 Nov 2025).

Astrophysical table-based EOS models with explicit first-order transitions have also been deployed in core-collapse supernovae and binary neutron-star mergers. In the phenomenological quark–hadron EOS, the FOPT can trigger a second shock wave in core-collapse supernova simulations and produce a distinct neutrino burst when the quark core forms; in binary mergers it can shift the post-merger gravitational-wave frequency $\varepsilon_t$19 and alter collapse outcomes (Bastian, 2020).

In early-Universe applications, first-order quark–hadron transitions have been studied in DGP + Brans–Dicke and chameleon Brans–Dicke brane cosmologies. In the DGP + BD model, using quark and hadron EOSs matched at $\varepsilon_t$20 MeV for $\varepsilon_t$21 MeV and $\varepsilon_t$22 MeV, the transition occurs on microsecond scales, with faster completion on the self-accelerating branch (Haddad et al., 2014). In the chameleon Brans–Dicke brane model, the modified Friedmann law $\varepsilon_t$23 pushes the QCD transition into the nanosecond regime, with $\varepsilon_t$24 ns and completion around $\varepsilon_t$25 ns (Saaidi et al., 2012). These cosmological results are model-specific and should not be conflated with standard FRW timing.

7. Open issues and recurrent points of contention

Several unresolved issues organize current debate. The first concerns the stiffness of deconfined matter. Multiple neutron-star inferences place the favored $\varepsilon_t$26 well above the conformal limit $\varepsilon_t$27, with lower bounds $\varepsilon_t$28 or $\varepsilon_t$29 in one inversion and a posterior peak at $\varepsilon_t$30 in another (Zhang et al., 2023, Xie et al., 2020). This does not establish a unique microscopic mechanism, but it does constrain viable quark-matter descriptions within the adopted hybrid-star frameworks.

A second issue is whether the mixed phase should be sharp or smooth. Maxwell constructions enforce abrupt jumps at a single pressure, while Gibbs constructions and pasta-inspired interpolations yield extended non-constant-pressure coexistence. Surface tension, finite-size effects, and the number of conserved charges determine which description is appropriate in a given setting (Zheng et al., 2014, Pradhan et al., 2023). Therefore, statements about “the” hadron–quark transition must be qualified by thermodynamic context.

A third issue concerns the critical endpoint. Some models locate a CEP near $\varepsilon_t$31 (Uddin et al., 2012), while NJL-based asymmetric-matter calculations place it near $\varepsilon_t$32 MeV and $\varepsilon_t$33 (Shao et al., 2011). A plausible implication is that CEP coordinates are highly model-dependent because they arise in a regime where chiral symmetry breaking, deconfinement, and dense-matter interactions compete strongly.

A fourth issue is whether twin stars are generic signatures of FOPT. Current observational inversion under tightly constrained CSS assumptions makes them improbable (Zhang et al., 2023), whereas broader EOS classes still permit them under sufficiently large latent heat or sufficiently sharp transitions (Pradhan et al., 2023, Kouno et al., 2023). The more precise statement is that twin stars are possible in some FOPT realizations but not generically implied by the existence of a hadron–quark transition.

Finally, there is a methodological divide between direct microphysical constructions and inference-driven phenomenology. Microphysical models specify the origin of the barrier, the role of chiral restoration, or the behavior of metastable branches (Zhou et al., 2020, Tsue et al., 2010, Kouno et al., 2023). Inference approaches instead ask which transition parameters are compatible with current astrophysical data (Zhang et al., 2023, Xie et al., 2020, Bai et al., 2019). Together they indicate that, while the existence of a first-order hadron–quark transition remains unconfirmed, a consistent research program now links coexistence thermodynamics, compact-star observables, heavy-ion signatures, and finite-temperature nucleation dynamics within a shared parameter language of onset, latent heat, stiffness, and phase stability.