Symmetry energy systematics and its high density behavior

We explore the systematics of the density dependence of nuclear matter symmetry energy in the ambit of microscopic calculations with various energy density functionals, and find that the symmetry energy from subsaturation density to supra-saturation density can be well determined by three characteristic parameters of the symmetry energy at saturation density $\rho_0 $, i.e., the magnitude $E_{\text{sym}}({\rho_0 })$, the density slope $L$ and the density curvature $K_{\text{sym}}$. This finding opens a new window to constrain the supra-saturation density behavior of the symmetry energy from its (sub-)saturation density behavior. In particular, we obtain $L=46.7 \pm 12.8$ MeV and $K_{\text{sym}}=-166.9 \pm 168.3$ MeV as well as $E_{\text{sym}}({2\rho _{0}}) \approx 40.2 \pm 12.8$ MeV and $L({2\rho _{0}}) \approx 8.9 \pm 108.7$ MeV based on the present knowledge of $E_{\text{sym}}({\rho_{0}}) = 32.5 \pm 0.5$ MeV, $E_{\text{sym}}({\rho_c}) = 26.65 \pm 0.2$ MeV and $L({\rho_c}) = 46.0 \pm 4.5$ MeV at $\rho_{\rm{c}}= 0.11$ fm$^{-3}$ extracted from nuclear mass and the neutron skin thickness of Sn isotopes. Our results indicate that the symmetry energy cannot be stiffer than a linear density dependence.In addition, we also discuss the quark matter symmetry energy since the deconfined quarks could be the right degree of freedom in dense matter at high baryon densities.


Introduction
The nuclear matter symmetry energy, which essentially characterizes the isospin dependent part of the equation of state (EOS) of asymmetric nuclear matter, is important for understanding many questions in nuclear physics and astrophysics, including the nuclear effective interactions in asymmetric nuclear matter, the structure and stability of exotic nuclei, the reaction dynamics induced by rare isotopes, the nature and evolution of neutron stars, and the mechanism of supernova explosion [1][2][3][4][5][6][7]. The symmetry energy also plays an important role in some interesting issues of new physics beyond the standard model [8][9][10][11][12]. During the last decade, a lot of experimental, observational and theoretical efforts have been devoted to constraining the density dependence of the symmetry energy [13][14][15][16][17][18]. While significant progress has been made in determining the density behavior of the symmetry energy around saturation density ρ 0 (∼ 0.16 fm −3 ), its supra-saturation density behavior is still poorly known and remains the most uncertain property of isospin asymmetric nuclear matter. Theoretically, many experimental and observational probes have been proposed to extract information on the supra-saturation density behavior of the symmetry energy [13]. In terrestrial laboratories, a e-mail: lwchen@sjtu.edu.cn European Physical Journal Web of Conferences heavy-ion collisions provide the only way to explore the supra-saturation density behavior of the symmetry energy under controlled conditions. To the best of our knowledge, the constraints on the supra-saturation density behavior of the symmetry energy obtained so far are all from the transport model analyses on the data of π − /π + ratio [19][20][21][22][23][24][25] and n/p elliptic flows [26,27] in heavy-ion collisions but unfortunately they are contradictive with each other, leaving a confusing situation for the community.
Conventionally, the nuclear matter EOS is defined as the binding energy per nucleon as a function of the density and a number of bulk characteristic parameters defined at saturation density ρ 0 are usually introduced to quantitatively characterize the energy of symmetric nuclear matter and the symmetry energy [28,29]. For example, the energy E 0 (ρ 0 ) and incompressibility K 0 of symmetric nuclear matter are the two lowest order bulk parameters for the EOS of symmetric nuclear matter while the symmetry energy magnitude E sym (ρ 0 ) and its slope parameter L are the two lowest order bulk parameters of the symmetry energy. While several lower order bulk characteristic parameters of asymmetric nuclear matter, such as E 0 (ρ 0 ), K 0 , E sym (ρ 0 ) and L have been relatively well constrained or in significant progress [13][14][15][16][17][18]30], yet the higher order bulk characteristic parameters are still poorly known. Actually, there has little experimental information on the third-order derivative parameter J 0 of symmetric nuclear matter at ρ 0 [31] and the symmetry energy curvature parameter K sym [29]. However, the higher order bulk characteristic parameters have been shown to be closely related to some important issues in nuclear physics and astrophysics, such as the determination of the isobaric incompressibility of asymmetric nuclear matter [28,32] and the core-crust transition density and pressure in neutron stars [33][34][35]. In particular, within the Skyrme energy density functional, it has been proposed [29] that the higher-order curvature parameter K sym may play an important role in the determination of the supra-saturation density behaviors of the symmetry energy.
So far (very likely also in future), essentially all the obtained constraints on E sym (ρ) are based on some energy density functionals or phenomenological parameterizations of E sym (ρ). Therefore, it would be very interesting to see whether there exist some universal laws (systematics) for the density dependence of the symmetry energy within these functionals or parameterizations and whether one can get some useful information on the high density symmetry energy from the relatively well-known knowledge of the symmetry energy around saturation density. For all the energy density functionals or phenomenological parameterizations, the E sym (ρ) increases from ρ = 0 up to a certain density around ρ 0 and then either continuously increases or decreases depending on the parameters of the energy density functionals or phenomenological parameterizations. While the parameters E sym (ρ 0 ), L and K sym accurately characterize the symmetry energy density behaviors around ρ 0 , their relation to the density behaviors at sub-and supra-saturation densities in various energy density functionals or phenomenological parameterizations of E sym (ρ) are still unclear.
In the present talk, we report the preliminary results of the study on the systematics of the density dependence of nuclear matter symmetry energy in the ambit of microscopic calculations with various energy density functionals. We systematically analyze the relation between the parameters E sym (ρ 0 ), L and K sym defined at saturation density ρ 0 and the symmetry energy density behaviors at sub-and supra-saturation densities in various energy density functionals. In addition, since the dense matter at high baryon densities could be quark matter, we also discuss briefly the quark matter symmetry energy.
The strong linear correlation between E sym (ρ) and E * sym (ρ) as well as between L(ρ) and L * (ρ) for the 60 interactions at different densities shown in Fig. 2 and Fig. 3 leads to the following relations The values of the coefficients A(ρ), B(ρ), A L (ρ) and B L (ρ) generally depend the density. In particular, one can see from Fig. 2 and Fig. 3 that A(ρ) (A L (ρ)) is generally nonzero and B(ρ) (B L (ρ)) usually deviates from unit, reflecting the higher-order effects beyond the expansion in Eq. (4) and Eq. (5). We note A(ρ) ≈ 0 (A L (ρ) ≈ 0) and B(ρ) ≈ 1 (B L (ρ) ≈ 1) for ρ ≈ ρ 0 as expected. Eq. (6) and Eq. (7) establish the systematics of E sym (ρ) and L(ρ) in terms of three characteristic parameters E sym (ρ 0 ), L and K sym . It should be noted that in principe Eq. (7) can also be deduced from Eq. (6) according to the definition. We would like to point out the Eq. (6) for the systematics of E sym (ρ) can be safely applied in the density region from ρ 0 /5 to 3ρ 0 where the Pearson linear correlation coefficient r is always larger than 0.96. Similarly,     the Eq. (7) for the systematics of L(ρ) can be safely applied in the density region from ρ 0 /2 to 2.5ρ 0 where the Pearson linear correlation coefficient r is always larger than 0.96.

Supra-saturation density behaviors of the symmetry energy
The systematics of E sym (ρ) and L(ρ) in Eq. (6) and Eq. (7) imply that the three characteristic parameters E sym (ρ 0 ), L and K sym (and thus E sym (ρ) and L(ρ)) can be determined once three values of either E sym (ρ) or L(ρ) are known. This means that one can extract information on the high density behaviors of the symmetry energy from the relatively well constrained (sub-)saturation density behaviors of the symmetry energy.
In recent years, several accurate constraints on the symmetry energy at subsaturation density have been obtained through analyzing nuclear structure properties of heavy nuclei. Indeed, a quite accurate constraint on the symmetry energy at the subsaturation cross density ρ c = 0.11 fm −3 , i.e., E sym (ρ c ) = 26.65 ± 0.20 MeV, has been recently obtained from analyzing the binding energy difference of heavy isotope pairs [37]. At the same time, an accurate constraint on the density slope at ρ c , i.e, L(ρ c ) = 46.0 ± 4.5 MeV has been obtained from analyzing the neutron skin data of Sn isotopes [37]. At density ρ c = 0.11 fm −3 , the systematics of E sym (ρ) and L(ρ) in Eq. (6) and Eq. (7) lead to the following expressions The values of coefficients a(ρ c ) and b(ρ c ) (a L (ρ c ) and b L (ρ c )) can be obtained from linear fitting to the correlation between E * sym (ρ c ) and E sym (ρ c ) (L * (ρ c ) and L(ρ c )). Shown in Fig. 4 are E sym (ρ c ) vs E * sym (ρ c ) and L(ρ c ) vs L * (ρ c ) with the 60 interactions, and one can observe a very strong linear correlation (the Pearson linear correlation coefficient r is about 0.999 for both cases) between E * sym (ρ c ) and E sym (ρ c ) as well as between L * (ρ c ) and L(ρ c ), and these linear correlations lead to a(ρ c ) = −0.111 ± 0. Besides E sym (ρ c ) and L(ρ c ), one needs another constraint condition to determine E sym (ρ 0 ), L and K sym and thus the supra-saturation density behaviors of the symmetry energy. In the present work, we further use the constraint of E sym (ρ 0 ) = 32.5 ± 0. MeV is in very good agreement with other constraints extracted from terrestrial experiments, astrophysical observations, and theoretical calculations with controlled uncertainties [13][14][15][16][17][18]. The obtained K sym = −166.9 ± 168.3 MeV also agrees well with the result K sym = −100 ± 165 MeV [29] obtained from a correlation analysis within SHF approach.
Based on E sym (ρ 0 ) = 32.5±0.5 MeV, L = 46.7±12.8 MeV and K sym = −166.9±168.3 MeV, one then can obtain E sym (ρ) and L(ρ) according to the systematics in Eq. (6) and Eq. (7), and the results are shown in Fig. 5. For comparison, we also include the results from the MDI interaction [39] with x = 1, 0, −1 and −2 and the phenomenological parameterizations of E sym,pot (ρ) ∼ (ρ/ρ 0 ) γ [26,40] for the potential energy part of the symmetry energy with γ = 0.5, 0.75, 1.0, 1.25 and 1.5. We would like to point out that the MDI interaction and the parameterizations of E sym,pot (ρ) ∼ (ρ/ρ 0 ) γ have been extensively applied in transport model simulations of heavy ion collisions. One can see from Fig. 5 that the present analysis based on the symmetry energy systematics with E sym (ρ 0 ) = 32.5 ± 0.5 MeV, L = 46.7 ± 12.8 MeV and K sym = −166.9 ± 168.3 MeV favors a softer symmetry energy and suggests that the symmetry energy cannot be stiffer than a linear density dependence.
In particular, at the supra-saturation density of 2ρ 0 , we find E sym (2ρ 0 ) = 40.2 ± 12.8 MeV and L(2ρ 0 ) = 8.9 ± 108.7 MeV. We note that these values are in nice agreement with the variational many-body theory calculation with WFF1 interaction [41] which can give a good description on the recent observation of heavy neutron stars with radius of 9.1 +1.3 −1.5 km [42].
At extremely high baryon density, the main degree of freedom could be the deconfined quark matter rather than the confined baryon matter, and there the quark matter symmetry energy should be involved for the properties of isospin asymmetric quark matter (isospin symmetry is still satisfied in quark matter). The isospin asymmetric quark matter could be produced in ultra-relativistic heavy ion collisions induced by neutron-rich nuclei and it could also exist in compact stars such as neutron stars or quark stars. Although significant progress has been made in understanding the density dependence of the nuclear matter symmetry energy, there has little information on the density dependence of the quark matter symmetry energy.
Theoretically, it is difficult to calculate the quark matter symmetry energy since the ab initio Lattice QCD simulations does not work at finite baryon density while perturbative QCD only works at extremely high baryon density.
Similarly as in the case of nuclear matter, the EOS of quark matter consisting of u, d, and s quarks, defined by its binding energy per baryon number, can be expanded in isospin asymmetry δ q as where E 0 (n B , n s ) = E(n B , δ q = 0, n s ) is the binding energy per baryon number in three-flavor u-d-s quark matter with equal fraction of u and d quarks; the quark matter symmetry energy E sym (n B , n s ) is expressed as E sym (n B , n s ) = 1 2! ∂ 2 E(n B , δ, n s ) ∂δ 2 q δq=0 .
The isospin asymmetry of quark matter is defined as which equals to −n 3 /n B with the isospin density n 3 = n u − n d and n B = (n u + n d )/3 for two-flavor u-d quark matter. We note that the above definition of δ q for quark matter has been extensively used in the literature [43][44][45][46][47], and one has δ q = 1 (−1) for quark matter converted by pure neutron (proton) matter according to the nucleon constituent quark structure, consistent with the conventional definition for nuclear matter, namely, ρn−ρp ρn+ρp = −n 3 /n B . In Eq. (10), the absence of odd-order terms in δ q is due to the exchange symmetry between u and d quarks in quark matter when one neglects the Coulomb interaction among quarks. The higher-order coefficients in δ q are shown to be very small in various model calculations [47].
It has been demonstrated recently [47] that the isovector properties of quark matter may play an important role in understanding the properties of strange quark matter and quark stars. If the recently discovered heavy pulsars PSR J1614-2230 [48] and PSR J0348+0432 [49] with mass around 2M ⊙ were quark stars, they can put important constraint on the isovector properties of quark matter, especially the quark matter symmetry energy. Within the confined-isospin-density-dependent-mass (CIDDM) model [47], in particular, it has been shown that the two-flavor u-d quark matter symmetry energy should be at least about twice that of a free quark gas or normal quark matter within conventional NJL model in order to describe the PSR J1614-2230 and PSR J0348+0432 as quark stars.