Confining density functional approach to the QCD phase diagram at low temperatures and thermal twin stars

Abstract: We present a density functional-based equation of state for warm, dense nuclear matter with a transition to deconfined quark matter for applications to simulations of supernova explosions and neutron star mergers, but also for the cosmological evolution of Q-balls. For the quark matter equation of state, we employ a recently developed confining density functional approach while nuclear matter is described within a relativistic density functional model of the DD2 class. The phase transition is obtained by a Maxwell construction at constant entropy per baryon. We discuss the solutions of TOV equations for isentropic hybrid stars for the hybrid equation of state model DDf-SFM (DD2-$χ$CDF) without (with) color superconductivity and find that at finite temperatures above a critical value of entropy per baryon sequences of disconnected third family branches ("thermal twin stars") may appear for the DDf-SFM model, while they are absent for the color superconducting model and at $T=0$. We discuss the relation of this critical entropy per baryon to the Seidov criterion of gravitational instability for $T=0$ and find that it is a good guide. We suggest that the presence of thermal twin stars may be regarded as an indicator for the core-collapse supernova explodability of massive blue supergiant stars and thus serve as a new criterion for the reliability of hybrid equation of state models. By this argument, strong color superconductivity shall be excluded and it remains to be shown whether models with moderate diquark pairing could fulfill the thermal twin constraint. For the case of symmetric matter, we compare the resulting hybrid EOS with the flow constraint by Danielewicz et al. and find a a sensitivity of the onset density for deconfinement on the presence or absence of color superconductivity.

• The paper develops hybrid equations of state using confining density functionals and finds that unpaired DDf-SFM matter produces thermal twin stars above a critical entropy of s/n_B ≈ 1.75, with twins near 1.5 solar masses.
• Strong 2SC color superconductivity in the DD2-χCDF model reverses the temperature dependence of deconfinement, removes thermal twins, and creates a superconducting corridor between hadronic and normal quark matter.
• The paper proposes thermal twin stars as a possible indicator of successful core-collapse supernova explosions, while emphasizing that the connection requires tests across pairing strengths, pasta treatments, and crossover transitions.

This paper develops a hybrid equation of state (EoS) for warm, dense nuclear matter that incorporates a first-order deconfinement transition to quark matter, constructed from a confining density functional (CDF) description of two-flavor quark matter matched to relativistic density functionals for the hadronic phase (2608.18038). The central results are twofold: first, a demonstration that thermal twin stars — disconnected third-family branches appearing only at finite entropy per baryon — arise for the non-color-superconducting DDf-SFM model but are absent when strong color superconductivity is included via the chirally symmetric $\chi$CDF model; second, a proposal that the presence of thermal twin stars serves as an indicator of core-collapse supernova (CCSN) explodability of massive progenitors and hence as a new constraint on hybrid EoS models.

Confining density functional framework

The microscopic input is a Lagrangian density functional for two-flavor quark matter containing four interaction channels: a confining/chiral potential $\mathcal{U}_\chi$, vector-isoscalar repulsion $\mathcal{U}_V$, vector-isovector interaction $\mathcal{U}_I$, and a diquark pairing potential $\mathcal{U}_D$. Confinement is implemented through rapid growth of the scalar mean-field self-energy of quarks at low density, suppressing their thermal excitations. Two parametrizations are considered.

The string-flip model (SFM) variant follows Kaltenborn, Bastian and Blaschke (Kaltenborn et al., 2017), with $\Sigma_\chi \propto \langle\bar{q}q\rangle^{-1/3}$ motivated by string tension scaling with mean interquark separation, and neglects diquark pairing entirely ($\mathcal{U}_D=0$). It operates within the no-sea approximation.

The chirally symmetric $\chi$CDF variant (Ivanytskyi et al., 2022) restores chiral symmetry of the confining potential, includes the zero-point contribution regulated by a Gaussian cutoff, and extends the medium-dependent coupling motivated by nonperturbative one-gluon exchange (Song et al., 2019) to both vector-isoscalar and — a new element here — vector-isovector channels. Diquark pairing leading to 2SC color superconductivity is included self-consistently, with couplings fixed from the vacuum meson spectrum ($M_\pi=140$ MeV, $F_\pi=90$ MeV, $\mathcal{U}_\chi$0 MeV) and a Fierz-motivated ratio $\mathcal{U}_\chi$1. At high density the effective couplings scale as $\mathcal{U}_\chi$2, recovering the conformal limit of QCD.

Hybrid EoSs are built by Maxwell construction: DDf-SFM pairs the soft DDf hadronic EoS with the SFM, while DD2-$\mathcal{U}_\chi$3CDF uses the stiffer DD2 EoS, since no Maxwell crossing exists between the soft DDf EoS and the stiff $\mathcal{U}_\chi$4CDF quark matter. The authors note that pasta-phase calculations at $\mathcal{U}_\chi$5 show only minor deviations ("rounding of edges") from the Maxwell construction for realistic surface tensions (Maslov et al., 2018), and argue this should not materially affect the critical entropy per baryon derived below — though a finite-temperature pasta treatment remains to be demonstrated.

Phase diagrams and isentropes

For the DDf-SFM model, increasing temperature lowers the baryon chemical potential and onset density of deconfinement, because unpaired quark matter responds more strongly to thermal excitations than hadronic matter. Isentropes crossing the transition cool down, reflecting the increase in thermal degrees of freedom. A counter-intuitive finding is that the phase diagrams of isospin-symmetric matter and neutron star matter are nearly identical; this follows from the imposed continuity of the symmetry energy across the phase border in the SFM parametrization. Without that assumption, as in the $\mathcal{U}_\chi$6CDF case, deconfinement in neutron star matter occurs at lower chemical potential and density than in symmetric matter. The soft DDf hadronic sector ensures agreement with the Danielewicz proton flow constraint [nucl-th/0208016], while isospin asymmetry stiffens the mixed phase by roughly $\mathcal{U}_\chi$7 in the cold limit.

The DD2-$\mathcal{U}_\chi$8CDF model exhibits qualitatively opposite behavior: the phase boundary bends rightward, so the critical density for deconfinement rises with temperature. This is a direct consequence of 2SC pairing — flavor-color entanglement reduces the effective number of degrees of freedom in the quark phase, so thermal pressure grows more slowly than in the hadronic phase. Correspondingly, isentropes heat up across the transition rather than cooling.

Thermal twin stars and the Seidov criterion

Solving the TOV equations along constant-entropy-per-baryon sequences reveals the key distinction. For DDf-SFM, the density jump at deconfinement grows with $\mathcal{U}_\chi$9, and above a critical value the mass-radius relation loses uniqueness: for $\mathcal{U}_V$0, twin configurations near $\mathcal{U}_V$1 appear, one purely hadronic and one with a quark core. Applying the Seidov gravitational instability criterion in its density-jump form,

$\mathcal{U}_V$2

the authors interpolate between $\mathcal{U}_V$3 (subcritical, $\mathcal{U}_V$4) and $\mathcal{U}_V$5 (supercritical, $\mathcal{U}_V$6), obtaining a critical value $\mathcal{U}_V$7. Notably, this zero-temperature criterion proves a good guide even at finite entropy, consistent with the value $\mathcal{U}_V$8 found in a nonlocal chiral quark model study (Carlomagno et al., 2024).

By contrast, the DD2-$\mathcal{U}_V$9CDF sequences form an "onion-shaped" family with no instability: increasing $\mathcal{U}_I$0 stiffens the EoS, raising both masses and radii monotonically. Thermal twins are absent. Since prior simulations showed that quark deconfinement can trigger explosions of massive blue supergiants of $\mathcal{U}_I$1–$\mathcal{U}_I$2 that would otherwise collapse to black holes (D'Adderio et al., 2017, Fischer, 2021), and since those exploding scenarios involved thermal twin formation while failing scenarios did not (Hempel et al., 2015), the authors advance the conjecture that thermal twin stars indicate CCSN explodability. Under this criterion, the strongly color-superconducting $\mathcal{U}_I$3CDF parametrization would predict failed supernovae — a tension with observed massive-star explosions that constrains the model.

Color superconductivity as a phase-diagram corridor

A distinctive structural result concerns the topology of the QCD phase diagram in the $\mathcal{U}_I$4CDF model. Solving the curvature criterion $\mathcal{U}_I$5 yields two zeroes at $\mathcal{U}_I$6: appearance of 2SC at $\mathcal{U}_I$7 MeV and disappearance at $\mathcal{U}_I$8 MeV. Combined with the dramatic drop of the dynamical quark mass at $\mathcal{U}_I$9 MeV (partial chiral restoration inducing Mott dissociation of hadrons), this implies that the hadronic phase has no direct border with normal quark matter anywhere in the diagram — even at vanishing baryon density it borders instead a corridor of 2SC color superconducting matter. The authors state that neither the quark mass nor the gap obeys the BCS relation between pseudocritical temperature and zero-temperature gap. Such a phase structure, they report, has not been seen in local or nonlocal NJL-based studies (Sabatucci et al., 19 Mar 2026, Carlomagno et al., 2023), and they attribute it to the combination of sufficiently strong diquark coupling with confinement-induced enhancement of chiral symmetry breaking.

Limitations and open questions

Several caveats bear directly on these conclusions. The mean-field treatment neglects beyond-mean-field mesonic and diquark correlations, which are subdominant at compact-star conditions but essential for connecting to the low-energy hadron spectrum. The color chemical potential $\mathcal{U}_D$0 required for exact color neutrality is neglected on grounds of smallness. The Maxwell construction omits finite-temperature pasta effects, whose impact on the critical $\mathcal{U}_D$1 is asserted by analogy to the $\mathcal{U}_D$2 result rather than demonstrated. Most consequentially, the explodability conjecture rests on the comparison of two specific models — a bag-model-like EoS that cannot reach $\mathcal{U}_D$3 and a confining NJL-type model without confinement — so the link between thermal twins and supernova success is suggestive rather than established. The authors explicitly leave open whether models with moderate diquark pairing could satisfy the thermal twin constraint, and call for a systematic investigation of the diquark coupling strength including its medium dependence. They also note that even in a crossover scenario thermal twins are not excluded, leaving the relationship between transition order and twin formation unresolved.

Conclusion

The paper demonstrates within a thermodynamically consistent confining density functional approach that the presence or absence of strong color superconductivity decisively controls the low-temperature QCD phase structure: it reverses the temperature dependence of the deconfinement onset density, eliminates thermal twin stars, and inserts a 2SC corridor between the hadronic and normal quark phases. The proposed thermal-twin criterion for CCSN explodability offers a falsifiable link between the microphysics of dense QCD and stellar phenomenology, but its validity for moderate pairing strengths and for crossover-type transitions remains an open question for future work.