A finite temperature framework for quark matter with color-superconducting phases

TL;DR

The paper addresses how to obtain finite-temperature equations of state for dense quark matter with color-superconducting phases, which are relevant for neutron-star mergers. It introduces a finite-$T$ framework built on the cold, three-flavor NJL EoS: a Taylor expansion for unpaired quarks plus an analytic quasiparticle contribution for paired quarks, with a stability-preserving extrapolation, validated against a renormalization-group-consistent mean-field calculation. The authors demonstrate that the reconstructed EoS matches exact NJL results to within a few percent up to $T\sim 50$ MeV and remains reasonable up to $T\sim100$ MeV across densities of interest, including 2SC and CFL phases. This framework enables efficient finite-temperature modeling suitable for numerical relativity simulations of merger scenarios and provides insight into the thermal signatures of color-superconducting quark matter beyond the cold EoS.

Abstract

Current observations of neutron stars and measurements of gravitational waves only provide constraints on the zero temperature ($T=0$) equation of state (EoS) of dense matter. The detection of the post-merger gravitational-wave signal from a binary neutron star merger would additionally provide access to finite-temperature properties of the EoS which contain more information about the composition and the interactions of dense matter than the cold EoS alone. In particular deconfined quark matter may be probed by its characteristic finite temperature effects. This is especially the case for color-superconducting phases, in which the quasiparticle contribution to the thermal pressure is exponentially suppressed at low temperatures. Here we develop a new finite $T$ framework to model the thermal EoS for dense quark matter based on the cold quark matter EoS which is useful for numerical relativity simulations. We test the validity of the framework against a three-flavor NJL mean-field calculation, both with and without diquark pairing. We find that even for the complicated phase diagram of the NJL model including multiple different phases the framework is accurate to the few percent level for temperatures up to $T\sim 50\,$MeV.

Figures (12)

• Figure 1: Phase diagrams of the RG-consistent NJL model (minimal scheme) without a diquark coupling ($G_D=0$, top panels), and with color superconductivity ($G_D=G_S$, bottom panels). Phase boundaries are shown as solid (first order), dashed (second order) and gray, dashed (crossover) lines. A heat plot with contours (white lines) of constant quark number fractions $Y_u=Y_d$ (left) and $Y_s$ (right) of up and down and strange quarks, respectively, is overlaid. The acronyms for the phases denote chiral symmetry breaking ($\chi{\text{SB}}$), normal quark matter (NQM), two-flavor color-superconducting quark matter (2SC) and color-flavor locked matter (CFL), see \ref{['tab:phasesofmatter']}.
• Figure 2: The pressure versus the baryon chemical potential at different temperatures $T=1,50,75,100\,$MeV for the NJL model without diquark pairing (top) and with diquark pairing (bottom), respectively.
• Figure 3: Derivatives $c_2=\partial s/ \partial T$ (top), $c_3=\partial^2 s/ \partial T^2$ and $c_4=\partial^3 s/ \partial T^3$ (symmetric-log plot, middle and bottom) which are coefficients for the $T^2$ term, the $T^3$ term and the $T^4$ term in the Taylor expansion, respectively. For easier numerics, the derivatives were calculated at $T=1\,$MeV. Both the model without diquark interactions ($G_D=0$) and with diquark interactions ($G_D=G_S$) are shown.
• Figure 4: From top to bottom: pressure versus temperature at fixed baryon chemical potentials $\mu_B=1200, 1400, 1500\,$MeV (left panels) and pressure versus baryon chemical potential at fixed temperatures $T=50,75,100\,$MeV (right panels) of the model without diquark interactions ($G_D=0$). The exact model calculation (black solid line) is compared with the Taylor expansion to order $T^2$ and orders $T^3$ and $T^4$ (only left panels).
• Figure 5: Absolute value of the relative error in the Taylor series expansion up to order $\mathcal{O}(T^2)$ compared to the exact NJL solution. The results are overlaid on tope of the phase diagram in the plane of baryon chemical potential and temperature of the model without diquark interactions.