Vacuum-pressure treatment and chemical-potential-dependent dynamical mass in the CJT/Dyson-Schwinger quark-matter equation of state for compact stars

The quark-matter equation of state (EOS) at zero temperature and finite quark chemical potential constructed by Zong and Sun from a model-independent CJT (Cornwall-Jackiw-Tomboulis) / Dyson-Schwinger derivation contains a single parameter $m_D$ setting the scale of dynamical chiral symmetry breaking, plus an additive vacuum-pressure term $P(\mu=0)$ . When applied to quark stars through Tolman-Oppenheimer-Volkoff integration (Li, Luo, Jiang, Zong, 2011), two internal questions were left open: (i) the CJT-derived $P(\mu=0)$ is positive, unlike the phenomenological negative bag constant $-B$ , and its inclusion produces a negative energy density at small chemical potential, so the correct field-theoretic definition and inclusion criterion for the vacuum term in stellar EOSs is unsettled; (ii) the dynamical-mass scale $m_D$ should in principle depend on $\mu$ , but was held $\mu$ -independent, leaving the impact of a self-consistent $m_D(\mu)$ on the EOS and stellar sequences unexplored. Both questions remain open: later work by the same school and others replaced the vacuum term with competing prescriptions (the pressure difference between the chirally broken Nambu and chirally restored Wigner solutions of the quark gap equation, or the pressure at the chiral-critical-point chemical potential) without an accepted criterion — the issue is still under active study in 2026 — and no implementation of a $\mu$ -dependent $m_D$ in this framework exists, while the observational discrimination of the resulting self-bound stellar sequences against neutron-star alternatives is ongoing.

Background

A quark star is a hypothetical compact star made of self-bound deconfined quark matter, i.e. matter whose energy per baryon at zero pressure is lower than that of iron-56 or of nuclear matter, so it is stable without being gravitationally bound. Whether such objects exist is an old question (Bodmer, Witten) that observations have neither confirmed nor excluded. The interior structure of any relativistic compact star follows from the Tolman-Oppenheimer-Volkoff (TOV) equations, which require as input the equation of state $P(\varepsilon)$ relating pressure to energy density. For deconfined cold quark matter, controlled first-principles computations are unavailable: perturbation theory fails at the relevant chemical potentials $\mu$ , and lattice QCD is blocked at finite baryon density by the sign problem (the fermion determinant becomes complex, defeating Monte Carlo sampling). Practical work therefore uses models such as the MIT bag model, in which nearly free quarks are confined by a vacuum pressure represented by a bag constant $B$ appearing as a negative additive term $-B$ in the pressure, or the Nambu-Jona-Lasinio (NJL) model, a four-fermion chiral model, or Dyson-Schwinger equations (DSEs), which are the infinite tower of exact integral equations for the Green functions of QCD, tractable only after truncation, e.g. the rainbow-ladder truncation, which closes the quark gap equation (the equation governing the dressed quark propagator) by pairing a dressed gluon propagator with a matching vertex. Zong and Sun (Int. J. Mod. Phys. A 23, 3591 (2008)) derived a semi-analytic EOS at $T=0$ and finite $\mu$ from a model quark propagator within the Cornwall-Jackiw-Tomboulis (CJT) effective-action formalism — a variational generating functional for the equations of motion of Green functions. Their EOS contains one adjustable parameter $m_D$ , connected to the dynamically generated quark mass — the large constituent-like mass that the strong interaction produces through spontaneous chiral symmetry breaking — and an additive vacuum term $P(\mu=0)$ computed as $2N_cN_f\,m_D^4/(3^{3/2}\,16^{1/3}\pi)$ , which is positive. Below a critical chemical potential $\mu_0=4^{1/3}m_D/2$ the quark-number density vanishes identically and the pressure equals $P(\mu=0)$ , a robust qualitative feature of zero-temperature QCD at finite density. Li, Luo, Jiang and Zong (Phys. Rev. D 83, 025012 (2011)) integrated this EOS in the TOV equations to obtain quark-star mass-radius sequences, computed both with and without the $P(\mu=0)$ term, and compared with other models and with the then-recent two-solar-mass pulsar mass measurement of Demorest et al. (2010). The two prescriptions give very different stellar structures, so the treatment of the vacuum term is not a small ambiguity. Yet the CJT vacuum pressure is positive whereas the MIT bag constant enters as a negative contribution $-B$ , and keeping it makes the energy density negative at small chemical potential — a behavior the authors themselves flag as hard to understand physically. This exposes a general problem for DSE/NJL-derived stellar EOSs: in non-renormalizable effective models the vacuum contribution is regulator- and prescription-dependent, and there is no agreed field-theoretic criterion for which vacuum to subtract or how to include the remainder when matching to a stellar EOS. Separately, the single parameter $m_D$ stands for the dynamical chiral-symmetry-breaking scale; since the dynamically generated quark mass depends on the medium, $m_D$ should in principle be a function $m_D(\mu)$ , but the approximation of Zong and Sun holds it constant, leaving the effect of a self-consistent $m_D(\mu)$ on the EOS and on the stellar sequences unquantified. The predecessor paper Li, Luo, Zong (Phys. Rev. D 82, 065017 (2010)) had instead identified $P(\mu=0)$ with the phenomenological $-B$ , illustrating that the school itself changed prescription from paper to paper.

Problem Statement

Within the CJT/Dyson-Schwinger quark-matter equation of state at zero temperature and finite chemical potential of Zong and Sun, in which the only parameter $m_D$ sets the scale of dynamical chiral symmetry breaking and an additive vacuum term $P(\mu=0)$ appears, establish a physically and field-theoretically well-defined equation of state suitable for quark-star structure calculations, and determine its stellar consequences. Concretely: (i) resolve the status of the vacuum-pressure term — the CJT-derived $P(\mu=0)$ is positive, whereas the phenomenological bag constant enters as $-B$ , and keeping the CJT term produces a negative energy density at small chemical potential; a complete answer must either give a principled derivation of the correct vacuum contribution (which vacuum state is to be subtracted, with which regularization, so that the pressure and energy density are physically sensible and thermodynamically consistent at all relevant $\mu$ ) or a demonstrated criterion for excluding the term; and (ii) implement the dynamically generated mass scale as a self-consistent function $m_D(\mu)$ within this framework, and quantify how the resulting EOS and the Tolman-Oppenheimer-Volkoff stellar sequences (maximum mass, radius, mass-energy-density relations, the sensitivity to $m_D$ , and the differences between the with-term and without-term sequences) change relative to the $\mu$ -independent approximation. The two parts are linked requirements of one objective: a complete answer must satisfy both, resolving (i) in one of the two forms stated above and satisfying (ii) by reporting each of the five quantities listed there (that list is a required minimum, not a set of examples), and it must use the same vacuum-term treatment and the same mass function it adopts in (i)-(ii) to construct the EOS that is integrated through the Tolman-Oppenheimer-Volkoff equations, so that the EOS and its stellar predictions follow from them.

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