Symmetry energy in the KIDS framework and extension to quarkionic matter

Recent and ongoing laboratory experiments on nuclei and astronomical observations of neutron stars and gravitational waves offer valuable information on the symmetry energy and its dependence on the baryonic density ρ, S(ρ). Analytical models abound for the behavior of S(ρ) in the nucleonic regime. At high densities, however, extrapolations from the nucleonic regime become unreliable and the functional form of S(ρ) should be properly adjusted. Conveniently, the pseudoconformal symmetry emergent in dense, topologically altered nuclear matter suggests a simple expression for the energy per baryon in terms of ρ. Here, I consider a rudimentary interpolation between the Korea-IBS-Daegu-SKKU (KIDS) nucleonic equation of state (EoS) and a pseudo-conformal one at zero temperature. I assume for simplicity that the conformal limit is reached abruptly, but under continuous energy and pressure. Application to neutron stars suggestes that a crossover to quarkionic matter can lead to more compact stars or even heavier stars, depending on the precise form of the nucleonic EoS and on the order of the transition to quarkionic matter. These results are in line with previous studies of hybrid EoSs and represent only a “baby step” to further explorations with hybrid-KIDS EoSs.

The properties of very neutron rich nuclear systems are largely determined by the density dependence of the nuclear symmetry energy, S(ρ).Recent and ongoing experiments aiming to measure the neutron skin thickness [1,2,3] and astronomical observations of neutron stars and gravitational waves [4,5] offer valuable information on the symmetry energy at sub-saturation and supra-saturation densities, respectively.Heavy ion collisions permit laboratory studies of the symmetry energy at different density regimes below and above saturation [6].The availability of highly asymmetric beams at new rare-isotope facilities such as RAON [7] generate even more opportunities to constrain the symmetry energy.
The symmetry energy at zero temperature is conventionally characterized by its value J and the coefficients of its Taylor expansion at normal (saturation) density ρ 0 , namely its slope L, curvature K sym , skewness Q sym , etc., where x = (ρ − ρ 0 )/3ρ 0 .Current knowledge suggests, roughly, J = 30 − 33 MeV and L = 40 − 70 MeV [8,9].Higher-order parameters have been much less constrained.The Korea-IBS-Daegu-SKKU (KIDS) theoretical framework for the nuclear equation of state (EoS) and energy density functional (EDF) [10,11,12], based on a Fermi-momentum power expansion, resembles in practice an extended Skyrme functional, but offers the possibility to explore the symmetry-energy parameters independently of each other and independently of assumptions about the in-medium effective mass.The formalism and existing applications are reviewed in Ref. [13].Within this versatile and physically motivated framework, any set of EoS parameters can be transposed into a corresponding EDF and readily tested in microscopic calculations of nuclear properties.Studies within KIDS of symmetry-energy parameters based on both astronomical observations and bulk nuclear properties [14,15] suggest, conservatively, −K sym = 0 − 200 MeV, but also that K sym cannot be constrained by nuclear data alone [16].Also shown was that traditional EoS/EDF models (more explicitly, Skyrme models) are overly restrictive, imposing extremely tight artificial correlations between EoS parameters, for instance between 3J − L and K sym [14,16].Extended functional forms may be required for resolving, among other puzzles, the PREX/CREX discrepancy [16,17].
An interesting finding of the KIDS analyses so far is the apparent necessity for the EoS to stiffen at high density in order to support a heavy neutron star [14,15].Specifically, the symmetry energy S(ρ) is predicted to have an inflection point below or at roughly 2ρ 0 , and signifies a possible phase transition or the activation of new degrees of freedom.An early example is the Akmal-Pandharipande-Ravenhal (APR) EoS [18] where pion condensation gives rise to an inflection point at ρ ≈ 2ρ 0 .Currently, it is conjectured that hadronic matter might cross over to quarkionic matter at high densities and support very heavy neutron stars [19,20] or very compact (small) third-family neutron stars [21,22].The crossover region could be inferred from measurements by next-generation gravitational waves detectors [23].In Ref. [24], the mechanism of parity doubling is proposed to be activated at about two times saturation density.A common prediction of hybrid EoSs is that the speed of sound is not a monotonic function of the density, but peaks at some density well above the saturation regime, reaching roughly v 2 /c 2 = 0.7 − 0.8, and then drops and stabilizes at the conformal limit, v 2 /c 2 = 1/3.In that case, the upper limit for the central density of a neutron star could increase considerably.
In this work, I perform a rudimentary interpolation between the KIDS EoS and a pseudoconformal (PC) EoS resulting from a skyrmion-half skyrmion topological change [24,25].For the KIDS EoS I choose two representative parameter sets, the KIDS-P4 set [12], which is based on a fit to the APR EoS, and one of the EoSs that were found in Ref. [14] to be consistent with gross nuclear properties and basic neutron star observations.I denote the latter "KIDS-65" referring to its corresponding value for the slope parameter, L = 65 MeV.At low densities, where mechanical instability is reached, I replace KIDS with an EoS for clusterized matter which is based on the D1M* functional [26].(Note that in previous studies with KIDS, a different prescription was used for dilute matter, which may lead to some differences in predictions, in particular for neutron star radii.) The energy density in the PC state, including the nucleon mass, has the form where δ is the isospin asymmetry.Expression ( 2) is obtained by demanding dP/dρ = (−1/3)dH/dρ so that v 2 /c 2 = 1/3.The following values are suggested in [24] for the coefficients B and D (based on Eq. ( 53) therein and the associated saturation density n 0 = 0.154 fm −3 ): Between the two extremes of asymmetry, δ = 0 and 1, one can resort to the quadratic approximation.More directly, the resulting EoS for β-stable matter in the same density region can be assumed to follow the same relation, with the coefficients B and D left to be determined.
For the interpolation between the KIDS EoS and the PC phase a number of prescriptions can be considered.One is to employ the parameter set already obtained in Ref. [24] and interpolate at some high density, e.g., at about two times saturation density as suggested in that work.I find that the hybrid KIDS-PC EoSs thus obtained are too soft to sustain a massive neutron star.(This is not a general result: Different nucleonic EoSs crossing to the PC state at that density could support heavy neutron stars [24,25].)Instead, at present I will simply assume that, beyond some onset density ρ x , the total energy density, including the nucleon rest mass, is given by ( 5), which for the total pressure gives P = (B/3)ρ 4/3 − D. I further assume that at ρ x both P and H are continuous, not only for simplicity, but also because the so-called Cheshire cat principle [27] suggests a crossover rather than a phase transition.One can then determine B and D from the values of P and H at the onset density according to the KIDS model, P x = P (ρ x ), H x = H(ρ x ).Then one gets I examine here two choices of ρ x : • ρ x = ρ 0.7 , the density at which v 2 /c 2 in the KIDS model reaches the value x = 0.7.
• ρ x = ρ 1 , the density at which v 2 /c 2 in the KIDS model reaches the limit x = 1.The resulting values for the onset density ρ x and for B, D are given in Table (1 1. EoS parameters used in this work.Numbers tabulated under "Symmetry Energy" correspond to (J, L, K sym , Q sym ) in units of MeV.Numbers tabulated under "Extension(x)" correspond to the onset density ρ x in units of fm −3 , where x = v 2 /c 2 the assumed onset condition for the PC and the associated values of (B, D) in units of (MeV fm 4 , MeV/fm 3 ), respectively.The density, energy per particle, and compression modulus of symmetric nuclear matter at saturation are set to the values 0.16 fm −3 , −16 MeV, and 240 MeV, respectively.
x = 0.7 is reached at 2.5 − 3 times the saturation density, as suggested also in [24], while the limit x = 1 is reached at approximately seven times the saturation density.
For the EoSs thus generated, I solve the Tolman-Oppenheimer-Volkoff (TOV) equations and obtain the mass-radius relations for a neutron star [28].Results for the pressure plotted against the energy density and for the neutron star mass-radius relation are shown in Fig. 1.The crossover lowers the pressure at high densities.A crossover at higher densities (x = 1) supports more massive stars than a crossover at low densities.Heavier stars are predicted more compact than in the purely nucleonic case, while the radius of canonical stars is not affected.
In this work, I used only two KIDS models for the nucleonic EoS and made some very simplistic choices for the interpolation to quarkionic matter: I assumed that P (ρ) and H(ρ) are continuous and that the conformal limit is reached abruptly.Nevertheless, some basic conclusions can be drawn from the present study: Conformality imposes a specific and convenient expression for the EoS.It is uncertain at what density the (pseudo-)conformal state might be reached and what the relevant intermediate phase might be, but otherwise its presence allows the neutron (hybrid) star EoS to be extended to higher densities than a nucleonic phase would permit.A crossover to quarkionic matter can lead to more compact stars or even heavier stars, depending on the precise form of the nucleonic EoS and on the order of the transition to quarkionic matter.These first results are in line with previous studies of hybrid EoSs and represent only a "baby step" to further explorations with hybrid-KIDS equations of state.Red: KIDS-P4 [12]; blue: KIDS-65 [14] (see text).Solid lines: nucleonic KIDS EoSs without crossover to the PC state.Dashed (dotted) lines: crossover at densities where v 2 /c 2 reaches x = 1 (0.7).Thin red line in (a): PC state described by the parameters of Ref. [24].