Color Superconductivity and Chiral Symmetry Restoration at Nonzero Baryon Density and Temperature

TL;DR

The authors present a two‑flavor QCD–like model with instanton‑induced four‑fermion interactions, analyzed at finite temperature and baryon density using bosonization and a mean‑field thermodynamic potential $\Omega(\phi,\Delta;\mu,T)$. They show that chiral and color‑superconducting condensates can coexist, with a low‑temperature first‑order chiral transition into a high‑density, color‑superconducting phase and a high‑temperature second‑order (in the chiral limit) or crossover (for $m\neq0$) chiral transition, separated by a tricritical point. In the chiral limit, the tricritical point has mean‑field φ^6 behavior; for nonzero quark mass this point becomes a 3D Ising critical point, suggesting possible long correlation lengths in heavy‑ion collisions traversing the transition. Color superconductivity persists up to tens of MeV in temperature and is sensitive to the form factor and couplings, with a rich mixed‑phase region where chiral restoration and percolation of the dense phase interplay. The study provides a qualitative, universal framework for the interplay between chiral symmetry breaking and color superconductivity at finite density, guiding expectations for heavy‑ion and neutron‑star contexts.

Abstract

We explore the phase diagram of strongly interacting matter as a function of temperature and baryon number density, using a class of models for two-flavor QCD in which the interaction between quarks is modelled by that induced by instantons. Our treatment allows us to investigate the possible simultaneous formation of condensates in the conventional quark--anti-quark channel (breaking chiral symmetry) and in a quark--quark channel leading to color superconductivity: the spontaneous breaking of color symmetry via the formation of quark Cooper pairs. At low temperatures, chiral symmetry restoration occurs via a first order transition between a phase with low (or zero) baryon density and a high density color superconducting phase. We find color superconductivity in the high density phase for temperatures less than of order tens to 100 MeV, and find coexisting $<qq>$ and $<\bar q q>$ condensates in this phase in the presence of a current quark mass. At high temperatures, the chiral phase transition is second order in the chiral limit and is a smooth crossover for nonzero current quark mass. A tricritical point separates the first order transition at high densities from the second order transition at high temperatures. In the presence of a current quark mass this tricritical point becomes a second order phase transition with Ising model exponents, suggesting that a long correlation length may develop in heavy ion collisions in which the phase transition is traversed at the appropriate density.

Figures (5)

• Figure 1: The thermodynamic potential $\Omega$ (in GeV$^4$) as a function of $\phi$ and $\Delta$ at $T=0$ and $\mu=0.292$ GeV. The two degenerate minima have $\phi=\phi_0^{\rm vac}=0.4$ GeV, $\Delta=0$ and $\phi=0$, $\Delta=0.072$ GeV.
• Figure 2: The zero temperature thermodynamic potential $\Omega$ (in GeV$^4$) as a function of $\phi$ at $\Delta=0$ (left panel) and as a function of $\Delta$ at $\phi=0$ (right panel) for several chemical potentials. The curves correspond to (top to bottom) $\mu=0,0.292,0.35$ GeV. The curves at $\mu=\mu_0=0.292$ GeV are sections of Figure 1.
• Figure 3: Left panels show $\phi_0$ and $\Delta_0$ (namely $\phi$ and $\Delta$ at the global minimum of $\Omega$) as functions of $\mu$ for $T=0$ (above) and $T=0.03$ GeV (below). The right panels show the quark number density $n$ for the same parameters. We describe the results at nonzero temperature in the next section.
• Figure 4: Phase diagram as a function of $T$ and $\mu$, and as a function of $T$ and $n^{1/3}$ (all in GeV). This section can be viewed as one long caption for these figures. The solid curves are second order phase transitions; the dashed curves describe the first order transition.
• Figure 5: As for Figure 3 at zero temperature, but with a current quark mass $m=10$ MeV. We observe coexisting $\phi_0 \sim \langle \bar{\psi} \psi \rangle$ and $\Delta_0 \sim \langle \psi \psi \rangle$ condensates in the high density phase.