Color-superconducting quarkyonic matter

TL;DR

The paper develops a color-superconducting quarkyonic (CSQY) model to describe dense QCD matter in neutron stars, incorporating a momentum-dependent color superconducting gap Δ_{f k} within a two-flavor 2SC framework and a hadron shell in momentum space to reflect confinement. Using a flexible meta-modeling approach for the nuclear EoS and enforcing strong-equilibrium, color neutrality, β-equilibrium, and electric neutrality, the authors obtain an EoS that stiffens at CS onset and approaches the conformal limit $c_S^2 o1/3$ as density increases, aided by a nucleon-shell thickness that vanishes asymptotically. They explore NSs with CSQY cores for different onset densities, κ controlling the shell width, and Δ the pairing gap, finding that early onset and larger Δ lead to more massive and larger NSs while remaining consistent with χEFT, pQCD, and GW170817 constraints. The framework offers a consistent tool for studying smooth quark–hadron transitions in NSs and motivates further work on chiral restoration and vector repulsion effects.

Abstract

We explore the role of color superconductivity in quarkyonic matter under the conditions of color and electric neutrality at $β$- and strong equilibrium, as relevant for neutron stars. By explicitly incorporating the color-superconducting pairing gap into the phenomenological model of a smooth transition from hadron to quark matter, we extend the known quarkyonic framework to include this essential aspect relevant at high densities. The momentum dependence of the pairing gap, motivated by the running of the QCD coupling and introduced similarly to chiral quark models with nonlocal interaction, is a novel element of the model that is crucial for enabling the simultaneous onset of all color-flavor quark states in the presence of color superconductivity. While asymptotically conformal behavior of the present model is ensured by construction, we demonstrate that reaching the conformal limit in agreement with the predictions of perturbative QCD is provided by the proper momentum dependence of the thickness of the hadron shell in momentum space. We employ the flexible meta-modeling approach to nuclear matter, analyzing the structure of the hadron shell in momentum space and focusing on the effects of color superconductivity in quarkyonic matter. Similar to the effects induced by the onset of the quarkyonic phase, color superconductivity leads to stiffening of the equation of state of the NS matter. This causes a significant impact on observable properties of neutron stars, which are analyzed and compared to recent astrophysical and theoretical constraints. We argue that the developed model of color-superconducting quarkyonic matter provides a new, consistent tool for studying the scenario of smooth quark-hadron transition in NSs.

Figures (11)

• Figure 1: Schematic illustration of CSQY matter in momentum space. The orange dashed lines indicate the maximum momentum of quarks confined in hadrons $k_h$ and the Fermi momentum of unbound quarks $k_q$. The red, green, and blue spheres represent quarks of the corresponding color, with the spins indicated by the arrows. The flavor states are not indicated. Gray glass-like spheres depict three quarks confined to color-singlet nucleons of two possible spins. Dimming the red, green, and blue quarks by these gray spheres depicts confinement of colored degrees of freedom. The golden rings represent scalar diquarks formed by the Cooper pairing of deconfined red and green quarks, while deconfined blue quarks below the quark Fermi sphere remain unpaired. At small densities below the onset of QY matter (left panel), all quarks remain bound to hadrons, and quark Fermi momentum vanishes. At intermediate densities above the onset of QY matter (middle panel), unbound quarks form the Fermi sphere with finite $k_q$, while the quarks bound to nucleons exist in the momentum shell between $k_q$ and $k_h$. At high densities (right panel) the width of this shell vanishes.
• Figure 2: The single-particle distribution function of paired quarks $f_{fc{\bf k}}$ as a function of their momentum given in units of the Fermi momentum, i.e. $|{\bf k}|/k_{fc}$. The dotted, dashed, and solid curves correspond to the Fermi-Dirac distribution of unpaired quarks, the Nambu-Gorkov distribution of paired quarks with the high momentum states above the Fermi sphere, and the distribution of paired quarks in CSQY matter with a cut-off of the high momentum tail, respectively. The CSQY and Nambu-Gorkov distributions are calculated for constant $\Delta_{c{\bf k}}=0.15\epsilon_{fc}$. The green, light blue, and light orange shaded areas represent the occupation of the quark states in CSQY matter; the Fermi-Dirac exceeds these states below the Fermi sphere, and the Nambu-Gorkov exceeds above it.
• Figure 3: Scaled momentum-dependent pairing gap $\Delta_{\bf k}/\Delta$ used in this work (solid curve) compared to the Gaussian one (dashed curve) as functions of the scaled momentum $|{\bf k}|/\Lambda$.
• Figure 4: Pressure $p$ of electrically neutral CSQY matter at $\beta$-equilibrium as a function of the energy density $\varepsilon$ at the onset density $n_{B}^{\rm onset}=2n_0$ (upper panels) and $n_{B}^{\rm onset}=3n_0$ (lower panels) shown without zooming the region right after the onset of CSQY matter (left panels) and with that region zoomed (right panels). The calculations are performed for several values of the exponent $\kappa$ and pairing gap $\Delta$, which are indicated in the legend. The purely hadronic EoS is shown by the black dash-dotted curve in both panels. The filled circles indicate the maximum energy densities reached in the centers of the heaviest NSs modeled with the corresponding EoSs. The shaded areas represent the results of the chiral effective field theory Hebeler:2013nza, perturbative QCD Kurkela:2009gj and the constraints extracted from the binary millisecond pulsar PSR J1614-2230 Demorest:2010bx and PSR J0740+6620 Fonseca:2021wxt.
• Figure 5: The speed of sound $c_S^2$ (left column) and dimensionless interaction measure $\delta$ (right column) as a functions of energy density $\varepsilon$ obtained for the EoSs shown in Fig. \ref{['fig3']}. The filled circles indicate the maximum energy densities reached in the centers of the heaviest NSs modeled with the corresponding EoSs. The shaded areas represent the constraints from Ref. Altiparmak:2022bkeMarczenko:2022jhl and perturbative QCD Fraga:2013qra. The idea behind the shaded regions will be explained in the text.