Cold quarks in medium: An equation of state

We derive a compact, semialgebraic expression for the cold-quark-matter equation of state (EoS) in a covariant model that exhibits coincident deconfinement and chiral symmetry restoring transitions in medium. In doing so we obtain algebraic expressions for the number- and scalar-density distributions in both the confining Nambu and the deconfined Wigner phases and the vacuum-pressure difference between these phases, which defines a bag constant. Our qualitative study illustrates that a confining interaction can materially alter distribution functions from those of a Fermi gas and impact significantly on a system’s thermodynamic properties, possibilities that are apparent in the EoS.

Figure 1

Wigner-phase quark number density distribution, Eqs. (10) and (13). Top: $\mu =0.5$, 1, 2, 4, 8, and 16 GeV; the Fermi gas result is recovered with increasing $\mu$. Bottom: $f_1^W(0;\mu )$ increases with increasing $\mu$ for ${\mu }^2<2{\eta }^2$, illustrated at equidistant $\mu \in [0.2,2.0]$ GeV. At fixed ${\mu }^2>2{\eta }^2$, $f_1$ drops from 1 to 0 in the domain ${\mu }^2-2{\eta }^2<{\mathrm{p⃗}}^2<{\mu }^2$.

Figure 2

Nambu-phase quark number density distribution, Eq. (17). Top to bottom: $\mu =0.2$, 0.4, 0.6, 0.8, and 1.0 GeV. The density decreases with increasing $\mu$, as it must: the confined, DCSB phase is destabilized by increasing $\mu$ and hence the dressed-quark component of the pressure, measured with respect to that at $\mu =0$, must decrease.

Figure 3

Nambu-phase single-quark scalar density distribution $f_2^N(|\mathrm{p⃗}|;\mu )$ in Eq. (19). Dashed curve: $\mu =0$. Solid curves: $\mu =0.2$, 0.4, 0.6, 0.8, and 1.0 GeV.

Figure 4

Dotted curve: difference between Nambu-Goldstone and Wigner-Weyl vacuum pressures, which vanishes at ${\mu }_{\mathrm{cr}}=300$ MeV. Dashed curve: thermodynamic pressure in the Nambu phase. Solid curve: thermodynamic pressure in the Wigner phase, which is the ground state for $\mu >{\mu }_{\mathrm{cr}}$.

Figure 5

Solid curve, thermodynamic pressure associated with the Wigner-Weyl phase of our model; dashed curve, pressure of a noninteracting Fermi gas. Evidently, $P_T^W\propto \mu P_T^{\mathrm{Fg}}\propto {\mu }^5$ for $\mu \lesssim \eta$ and the chemical potential overwhelms the interaction for $\mu \gtrsim 4\hspace{0.5em}\eta$.