RAPID COMMUNICATIONS

PHYSICAL REVIEW C 90, 022801(R) (2014)

Quantifying correlations between isovector observables and the density dependence ofthe nuclear symmetry energy away from saturation density

F. J. Fattoyev,* W. G. Newton,† and Bao-An Li‡Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, Texas 75429, USA

(Received 4 May 2014; published 19 August 2014)

According to the Hugenholtz-Van Hove theorem, the nuclear symmetry energy S(ρ) and its slope L(ρ)at arbitrary densities can be decomposed in terms of the density and momentum dependence of the single-nucleon potentials in isospin-asymmetric nuclear matter. We quantify the correlations between several well-knownisovector observables and L(ρ) to locate the density range in which each isovector observable is most sensitiveto the density dependence of the S(ρ). We then study the correlation coefficients between those isovectorobservables and all the components of the L(ρ). The neutron skin thickness of 208Pb is found to be stronglycorrelated with the L(ρ) at a subsaturation density of ρ = 0.59ρ0 through the density dependence of thefirst-order symmetry potential. Neutron star radii are found to be strongly correlated with the L(ρ) over a widerange of suprasaturation densities mainly through both the density and momentum dependence of the first-ordersymmetry potential. Finally, we find that although the crust-core transition pressure has a complex correlationwith the L(ρ), it is strongly correlated with the momentum derivative of the first-order symmetry potential andthe density dependence of the second-order symmetry potential.

DOI: 10.1103/PhysRevC.90.022801 PACS number(s): 21.65.Cd, 21.65.Mn, 21.65.Ef, 26.60.Kp

Improving knowledge of the density dependence of thenuclear symmetry energy is an active endeavor due to itsmultifaceted impact in many areas of nuclear physics andastrophysics [1–5], as well as in some issues regarding possiblenew physics beyond the standard model [6–10]. Despiteintensive efforts aimed at constraining the density dependenceof the nuclear symmetry energy on both the experimentaland theoretical fronts [11–13], its knowledge still remainslargely uncertain even around nuclear saturation density ρ0

[14]. Traditionally, the nuclear symmetry energy, which is theenergy required to convert all protons in symmetric nuclearmatter (SNM) to neutrons, is expanded around ρ0 as S(ρ) =J + Lχ + 1

2Ksymχ2 + O(χ3) with χ ≡ (ρ − ρ0)/3ρ0, andthen individual parameters of this expansion—particularly Jand L—are probed using experimental observables that aresensitive to their variation. In this way correlations betweenthe density slope of the symmetry energy L(ρ0) and a multitudeof isovector observables have been established, with neutronskins of heavy nuclei and radii of neutron stars [15–17]exhibiting notably strong correlations with L.

In this Rapid Communication we begin examining thedensity dependence of these correlations, as well as theorigin of the correlations, by decomposing L in terms ofquantities that have a more direct physical meaning in finitenuclei and thus open up further potential experimental probes.By employing a least-squares covariance analysis with thecorrelation coefficient defined as

CAB = Cov(AB)

Var(A)Var(B), (1)

*farrooh.fattoyev@tamuc.edu†william.newton@tamuc.edu‡bao-an.li@tamuc.edu

we provide for the first time a proper statistical measureof correlations [18–20] between the density slope of thesymmetry energy as a function of baryon density L(ρ)and a selected number of isovector observables: (a) neutronskin thickness, (b) radii of neutron stars, and (c) crust-coretransition pressure. We will discuss the emergence of thesecorrelations by decomposing the L(ρ) in terms of the single-nucleon potentials in asymmetric nuclear matter as shown inRefs. [21–23]. While in general the correlation coefficient CAB

at a given density may not be able to assess systematic errorsreflecting limitations of the model, the strongest correlationcoefficient of almost +1 (or anticorrelation of almost −1) ata particular density should deliver a more universal model-independent message. For this reason, we scan the correlationcoefficients between the isovector observables and the L(ρ)over a wide range of baryon densities.

Based on the Hugenholtz-Van Hove theorem [24], itwas shown that both the nuclear symmetry energy S(ρ)and its density slope L(ρ) can be decomposed in terms ofthe single-nucleon potentials without considering effects ofnucleon-nucleon correlations [23]. For convenience, we willrewrite those expressions here as S(ρ) = S1(ρ) + S2(ρ) andL(ρ) = L1(ρ) + L2(ρ) + L3(ρ) + L4(ρ) + L5(ρ), where

L1(ρ) = 2�2k2

F

6m∗0(ρ,kF)

≡ 2S1(ρ), (2)

L2(ρ) = − �2k3

F

6m∗20 (ρ,kF)

∂m∗0(ρ,k)

∂k

∣∣∣∣k=kF

, (3)

L3(ρ) = 3

2Usym,1(ρ,kF) ≡ 3S2(ρ), (4)

L4(ρ) =(

∂Usym,1(ρ,k)

∂kk

)∣∣∣∣k=kF

, (5)

L5(ρ) = 3Usym,2(ρ,kF). (6)

0556-2813/2014/90(2)/022801(6) 022801-1 ©2014 American Physical Society

RAPID COMMUNICATIONS

F. J. FATTOYEV, W. G. NEWTON, AND BAO-AN LI PHYSICAL REVIEW C 90, 022801(R) (2014)

1 2 3 4 5 60

20

40

60

80

100

0 1 2 3 4 5 6

0

40

80

120

160

200 SLy4 NRAPR APR

S(ρ

) (M

eV)

(b)

SLy4 NRAPR APR

E/N

(MeV

)

ρ/ρ0

Symmetric Nuclear Matter

(a)

FIG. 1. (Color online) Binding energy per nucleon in SNM (a)and the symmetry energy (b) as a function of baryon density ρ/ρ0 forSLy4 and NRAPR Skyrme EDFs.

The expressions above are valid at arbitrary baryon densities,where m∗

0(ρ,k) is the nucleon effective mass in SNM, whileUsym,1 and Usym,2 are the first- and the second-order nuclearsymmetry potentials defined as

Usym,i(ρ,k) ≡ 1

i

∂iUn(ρ,α,k)

∂αi

∣∣∣∣α=0

= (−1)i

i

∂iUp(ρ,α,k)

∂αi

∣∣∣∣α=0

. (7)

Here Un and Up are the single-neutron and single-protonpotentials, respectively, which generally depend on the baryondensity ρ, the isospin asymmetry α, and the amplitude ofthe nucleon momentum k. Physically, S1(ρ) and accordinglyL1(ρ) represent the kinetic energy part of the symmetryenergy that includes the isoscalar effective mass contribution,L2(ρ) describes the momentum dependence of the nucleon

0 1 2 3 4 5

-200

-150

-100

-50

0

50

100

150

200

1 2 3 4 5

-200

-150

-100

-50

0

50

100

150

200

(b) NRAPR

L i

ρ/ρ0

(a) SLy4

L3

Ltot

L1

L5 L4

L5

L3

Ltot

L4

L1

FIG. 2. (Color online) Density dependence of the L(ρ) and itsdecomposition for SLy4 and NRAPR Skyrme EDFs.

effective mass, S2(ρ) and hence L3(ρ) are due to the first-order symmetry potential contribution, L4(ρ) comes from themomentum dependence of the first-order symmetry potential,and L5(ρ) comes from the second-order symmetry potential.Since by definition, L(ρ) ≡ 3ρ ∂S(ρ)

∂ρ, it is then obvious that

there is also a required closure relation between the densityderivative of the first-order symmetry potential Usym,1 and themagnitude of Usym,2, in particular.

The decomposition method considered above is quitegeneral. In this exploratory study, we report our results fortwo Skyrme energy density functionals (EDFs) SLy4 [25] andNRAPR [2] that have been widely used in the literature, inboth nuclear physics and astrophysics. Since our aim is tostudy the isospin-dependent properties of asymmetric nuclearmatter, SLy4 and NRAPR are natural choices for the followingreasons. First, they predict an almost identical equation of state(EOS) of SNM [see Fig. 1(a)]. Therefore, the L1(ρ) term of thedensity slope is almost indistinguishable for these interactionsdue to the equivalence of the isoscalar effective mass, and thenuclear saturation density ρ0 (see Fig. 2). Notice, however,that since the nucleon effective mass in Skyrme EDFs doesnot depend on momentum, the L2(ρ) term becomes identicallyzero. Second, both Sly4 and NRAPR reproduce the symmetryenergy predicted by the APR EOS [26] up to ∼1.5ρ0 [seeFig. 1(b)]. Whereas a similar density dependence of L3(ρ)—hence S2(ρ) or Usym,1—is observed in these interactions, bothL4(ρ) and L5(ρ) terms have a completely different densitydependence (see Fig. 2). Consequently these models have atotally different density dependence of the symmetry energyat suprasaturation densities of ρ � 1.5ρ0.

As a starting point, using the currently accepted uncertaintyranges by the community we fix the isoscalar properties ofbulk nuclear matter such as the nuclear saturation density ρ0,the binding energies per nucleon in SNM at saturation B(ρ0)and at twice saturation density B(2ρ0), the incompressibilitycoefficient of nuclear matter K0, the nucleon isoscalar effectivemass m∗

s (ρ0), and the macroscopic gradient coefficient Gs [25]at a 2% level, while allowing the isovector effective mass

0.2 0.4 0.6 0.8 1.0 1.20.80

0.85

0.90

0.95

1.00

0.4 0.6 0.8 1.0 1.20.80

0.85

0.90

0.95

1.00208Pb

48Ca48Ca

208Pb

(b) NRAPR(a) SLy4

C(R

skin,L

)

ρ/ρ0

FIG. 3. (Color online) Correlation coefficients between the L(ρ)and the neutron skin thicknesses of 208Pb and 48Ca calculated usingSLy4 and NRAPR Skyrme EDFs.

022801-2

RAPID COMMUNICATIONS

QUANTIFYING CORRELATIONS BETWEEN ISOVECTOR . . . PHYSICAL REVIEW C 90, 022801(R) (2014)

m∗v(ρ0) and the symmetry-gradient coefficient Gv to have 20%

theoretical error bars [27]. We do not a priori assume any errorbars on the symmetry energy parameters, hence they have notbeen included in the χ2. Rather we include a conservativerange of theoretical data points for the neutron-matter energyat densities 0.04 < ρ/ρ0 < 0.12 fm−3 that were calculatedusing quantum Monte Carlo calculations with chiral effectivefield theory interactions [28]. It is worth mentioning that usingthe properties of doubly magic nuclei and ab initio calculationsof low-density neutron matter, several Skyrme parameters havebeen recently constrained [29].

In Fig. 3 we display the correlation coefficients between theL(ρ) as a function of the baryon density and the neutron skinthicknesses of 208Pb and 48Ca. An updated measurement of theskin thickness in 208Pb and a proposal for measuring the skinthickness in 48Ca have been recently approved at the ThomasJefferson National Accelerator Facility [30]. As evident fromthe figure, although the strong correlation between Rskin(208Pb)and the slope of the symmetry energy at saturation L(ρ0) ispresent, consistent with previous studies [15,17,31–33], thestrongest correlation coefficient of almost +1 appears only at amuch lower density of about ρ/ρ0 = 0.59 for both EDFs. Thismeans that a measurement of the neutron skin in 208Pb woulduniquely determine the slope of the symmetry energy at thisparticular subsaturation density. Indeed, a recent systematicstudy of correlations between Rskin(208Pb) and L(ρr ) at threedensities of ρr = 0.06, 0.11, 0.16 fm−3 also showed [34] thatthe strongest correlation coefficient appears at a subsaturationcross density of 0.11 fm−3 in agreement with our results shownin Fig. 3. This result should not come as a surprise, since onlyabout one-third of the nucleons in 208Pb occupy the saturationdensity area, which therefore explains why the neutron skinthickness should constrain the L(ρ) not at ρ0, but at acharacteristic density in finite nuclei, which is localized closeto a mean value of the density of nuclei [35]. One should notethat the neutron skin is formed as a result of the competitionbetween the surface tension and the pressure of neutrons inheavy nuclei, the latter being closely related to the L(ρ).Hence the greater is the L(ρ) the thicker is the skin [15–17].This simple picture is of course relevant for heavy nuclei,where the mean-field approximation is adequate. In the caseof light nuclei such as 48Ca the mean-field approximation maynot be sufficient, and one should expect beyond-mean-fieldeffects to crop up. Indeed, as seen in Fig. 3 although thecorrelation remains as strong at a subsaturation density, itis not close to +1, leaving a very rich unexplored physicsbehind. Moreover, the strongest correlation coefficients occurat different densities of ρ = 0.63ρ0 for SLy4 and at ρ = 0.39ρ0

for NRAPR. Thus, simultaneous measurement of Rskin in both208Pb and 48Ca are very important, because they not onlyhelp map out the density dependence of the symmetry energyin a broader subsaturation density region, but also shouldprovide complementary information on the structure of theneutron-rich calcium isotope 48Ca [30].

Next, in Fig. 4 we display correlation coefficients betweenRskin and the individual decomposed terms of L(ρ) at varioussubsaturation densities. Except for L3(ρ) or Usym,1, all otherterms show almost no sign of a correlation. In fact, one shouldnot expect any correlation with the L1(ρ) term as it is purely

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.2 0.4 0.6 0.8 1.0 1.2-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.4 0.6 0.8 1.0 1.2-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

L1

L4

L5

L3L3

L4

L5

L1

L4

L1

L5

L3L3

L5

L4

L1

(b) NRAPR(a) SLy4

208Pb208Pb

C(R

skin,L

i)

48Ca48Ca

ρ/ρ0

FIG. 4. (Color online) Correlation coefficients between variousterms in the L(ρ) and the neutron skin thicknesses of 208Pb and 48Cacalculated using SLy4 and NRAPR Skyrme EDFs.

isoscalar in nature. The little correlation with L4(ρ) and L5(ρ)indicates that the size of the neutron skin is not particularlysensitive to the second-order symmetry potential and themomentum dependence of the first-order symmetry potential.Since at low densities the contribution to the total L(ρ) comesmostly from the L3(ρ) component (see Fig. 2), we observe asimilar rise of correlation as a function of density (compareFigs. 3 and 4). However, at higher densities contributions fromother components of L(ρ) become significant. Because of theirlittle correlation with the neutron skin, an effective decrease ofthe correlation measure with the total L(ρ) at higher densities isdeveloped resulting in a peak of correlation at the subsaturationdensity.

Since the pressure of neutron-rich matter also supportsneutron stars against gravitational collapse albeit at differentdensities than found at the center of finite nuclei, one shouldalso expect the strong correlation to emerge between the L(ρ)and the neutron star radii. This correlation is complicatedby the fact that the radius of a neutron star samples thesymmetry energy over a range from half saturation densityup to several times saturation density; one should expectpotentially significant variations in the correlation coefficientas a function of density from EOS to EOS. We demonstrate thisby displaying the correlation coefficients between the neutron

022801-3

RAPID COMMUNICATIONS

F. J. FATTOYEV, W. G. NEWTON, AND BAO-AN LI PHYSICAL REVIEW C 90, 022801(R) (2014)

0.5 1.0 1.5 2.0 2.5 3.00.90

0.92

0.94

0.96

0.98

1.00

1.0 1.5 2.0 2.5 3.00.90

0.92

0.94

0.96

0.98

1.00

R1.4=11.86 km

R1.8=11.05 km

R1.0=12.22 km

R1.8=11.30 kmR1.4=11.75 km

R1.0=11.91 km

C(R

NS,L

)

(b) NRAPR(a) SLy4

ρ/ρ0

FIG. 5. (Color online) Correlation coefficients between L(ρ) andthe 1.0M�, 1.4M�, and 1.8M� neutron star radii as a function of thebaryon density calculated using SLy4 and NRAPR Skyrme EDFs.

star radii and the L(ρ) as a function of density for both SLy4and NRAPR models in Fig. 5. In the case of SLy4, the radiusof a 1.0M� neutron star shows a strong correlation with thedensity slope at saturation. As the mass of the neutron starincreases, the strongest correlation shifts to the L(ρ) at higherdensities, e.g., at 1.5ρ0 for a 1.4M� neutron star, and at 2.5ρ0

for a 1.8M� neutron star. Moreover, the correlation coefficientremains almost flat for higher densities in a 1.8M� neutronstar. Higher mass stars sample the internal pressure at a higheraverage density. SLy4 has monotonically increasing L(ρ), andthus the internal pressure at higher densities will continue tobe dominated by the symmetry energy contribution.

NRAPR exhibits a different evolution of the correlationcoefficient with neutron star mass, with a much less pro-nounced increase in its peak as we move to higher masses.The correlation for 1.0M� and 1.4M� stars peak at densities ofaround ∼2.5ρ0 while the correlation for a 1.8M� star peaks atthe slightly higher density of ∼3.5ρ0. L(ρ) at suprasaturationdensities for NRAPR is nonmonotonic, peaking at around1.5ρ0 before falling to zero at 3.5ρ0—the density range withinwhich the radius shows its peak correlation with L. Beyond thepeak L(ρ), the contributions to the internal pressure from thehigher-order symmetry coefficients and from the symmetricpart of the EOS will become steadily more important, thusquenching the correlation with L. The two behaviors ofL(ρ) exhibited by the two Skyrme models here broadlybracket the types of behaviors seen in all Skyrme models,and the behaviors of the correlation coefficient betweenradius and L(ρ) with increasing neutron star mass can beexpected to similarly bracket the range of possible behaviorsin such models. Nevertheless, for both models the strongestcorrelation coefficient for a 1.4M� star emerges with L(ρ) atsuprasaturation densities, between 1.5ρ0 and 2ρ0. The needfor the symmetry energy in this range for the determination ofneutron star radii was also empirically observed in Ref. [4].Thus measurements of the neutron skin in finite nuclei and theradius of neutron stars probe quite different density regimes,

0.5 1.0 1.5 2.0 2.5 3.0-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.0 1.5 2.0 2.5 3.0-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

L3

L4

L1

L5

L3

L4

L5

L1

(b) NRAPR(a) SLy4

C(R

1.4,L

i)

ρ/ρ0

FIG. 6. (Color online) Correlation coefficients between variousterms in the L(ρ) as a function of the baryon density and canonical1.4M� neutron star radii are calculated using SLy4 and NRAPRSkyrme EDFs.

unsurprisingly. The latter of these regimes may be tested withcollisions of very neutron-rich heavy ions at different beamenergies [36].

We show the correlation coefficients between the radius ofa 1.4M� star and the components of L(ρ) in Fig. 6. Similar tothe neutron skin case, no correlation is found with L1(ρ), andrelatively mild correlations or anticorrelations are found withL4(ρ) and L5(ρ). Also notice that for NRAPR the correlationwith L4(ρ) is now much stronger thus pushing the strongcorrelation measure with the total L(ρ) to the much higherdensities. It is, however, again the L3(ρ) term that appears tohave the strongest correlation at higher densities. Recall thatL3(ρ) = 3

2Usym,1. This indicates that refinements in extractingUsym,1 from the measurement of the nucleon optical modelpotentials, and from heavy-ion collision observables, wouldimprove our predictions of neutron skins and neutron star radii.

Finally, we discuss our results for the crust-core transitionpressure; the most important correlate with the thickness, mass,and moment of inertia of the neutron star’s crust [37,38].Several methods have been used to determine the crust-coretransition properties [16,39–41]. Unlike the neutron skinthickness and the radii of neutron stars, the crust-core transitionpressure cannot be determined entirely by the pressure ofpure-neutron-rich matter itself. Indeed, in the simplest caseof the thermodynamical approach the following mechanicalstability condition, ( ∂P

∂ρ)μ > 0, must be satisfied in order for

the system to be stable against small density fluctuations,where μ is the chemical potential. Since the density derivativeof the pressure is quite complicated, a complex correlationbetween the transition pressure and the L(ρ) [37,39,41,42]must therefore emerge. We determine the transition pressurefrom a compressible liquid drop model of the crust outlinedin Ref. [43]. By plotting the correlation coefficients betweenvarious terms in the L(ρ) and the crust-core transition pressurein Fig. 7 we observe that the values of L4(ρ) and L5(ρ)—thatis, the momentum derivative of Usym,1 and the magnitude

022801-4

RAPID COMMUNICATIONS

QUANTIFYING CORRELATIONS BETWEEN ISOVECTOR . . . PHYSICAL REVIEW C 90, 022801(R) (2014)

0.0 0.2 0.4 0.6 0.8 1.0-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.2 0.4 0.6 0.8 1.0-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

L5

L1

Ltot

L4

(b) NRAPR

C(P

t,Li)

ρ/ρ0

(a) SLy4

L3

L4

L1

L3

Ltot

L5

FIG. 7. (Color online) Correlation coefficients between variousterms in the L(ρ) and the crust-core transition pressure are calculatedusing SLy4 and NRAPR models.

of Usym,2—are most important in the determination of thetransition pressure. The complex correlation between the crust-core transition pressure and the L(ρ) appears to originate fromthe complicated behavior of the correlation with L3(ρ) coupledwith the balance between the strong correlation between thecrust-core transition pressure and L4(ρ), and the similarlystrong anticorrelation with L5(ρ), which together complicatethe emergence of correlation with the total density slope. Thusextracting the density and momentum dependence of Usym,1

and the value of Usym,2 are both very crucial in determiningthe crust-core transition pressure, which plays a critical rolein understanding many phenomena related to the neutron starcrust [44].

In summary, we have quantitatively mapped the correlationsbetween L(ρ) and the neutron skin thickness, radii of neutronstars, and the crust-core transition pressure over a wide range

of densities for two widely used Skyrme models SLy4 andNRAPR which have similar symmetric nuclear matter EOSsbut diverging behaviors of L(ρ) at suprasaturation densities.We have also calculated the correlation coefficients betweenthe symmetry potential component of L at saturation densityand the same set of isovector observables. We found that forthe neutron skin thickness of 208Pb the strongest correlationappears at a subsaturation density of ρ = 0.59ρ0, and that theorigin of this correlation is found to be tied to the magnitude ofthe first-order symmetry potential Usym,1. A similarly strongcorrelation exists with the radius of neutron stars and L(ρ)over a wide range of suprasaturation densities. The behaviorof the correlation is seen to depend on the behavior of L(ρ) atsuprasaturation densities. If L(ρ) continues to monotonicallyincrease, then the peak correlation of radius with L occurs athigher densities for higher mass stars. If L(ρ) increases to amaximum and starts decreasing above a certain suprasaturationdensity, then the peak correlation of radius with L occurswithin a similar density range independent of mass. The radiusis also found to correlate most strongly with the magnitude ofthe first-order symmetry potential Usym,1.

The crust-core transition pressure, on the other hand, isfound to be strongly correlated not with the value of Usym,1

but with its momentum derivative term and the magnitudeof Usym,2. Future improvements in extracting the density andmomentum dependence of first- and second-order symmetrypotentials from optical model analysis of nucleon-nucleusscattering and at facilities for rare isotope beams will thereforeprovide strong constraints on the density dependence of thesymmetry energy.

We thank Prof. Lie-Wen Chen and Prof. Jorge Piekarewiczfor fruitful discussions. This work was supported in part bythe National Aeronautics and Space Administration underGrant No. NNX11AC41G issued through the Science MissionDirectorate, and the National Science Foundation under GrantNo. PHY-1068022.

[1] Topical Issue on Nuclear Symmetry Energy, edited by B.-A. Li,A. Ramos, G. Verde, and I. Vidana, Eur. Phys. J. A 50, 2 (2014).

[2] A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J. Ellis, Phys.Rep. 411, 325 (2005).

[3] V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys. Rep.410, 335 (2005).

[4] J. M. Lattimer and M. Prakash, Phys. Rep. 442, 109 (2007).[5] B.-A. Li, L.-W. Chen, and C. M. Ko, Phys. Rep. 464, 113 (2008).[6] C. J. Horowitz, S. J. Pollock, P. A. Souder, and R. Michaels,

Phys. Rev. C 63, 025501 (2001).[7] T. Sil, M. Centelles, X. Vinas, and J. Piekarewicz, Phys. Rev. C

71, 045502 (2005).[8] P. G. Krastev and B.-A. Li, Phys. Rev. C 76, 055804 (2007).[9] D.-H. Wen, B.-A. Li, and L.-W. Chen, Phys. Rev. Lett. 103,

211102 (2009).[10] W. Lin, B.-A. Li, L.-W. Chen, D.-H. Wen, and J. Xu, J. Phys. G

41, 075203 (2014).[11] J. M. Lattimer and Y. Lim, Astrophys. J. 771, 51 (2013).

[12] J. M. Lattimer, Ann. Rev. Nucl. Part. Sci. 62, 485 (2012).[13] B.-A. Li and X. Han, Phys. Lett. B 727, 276 (2013).[14] F. J. Fattoyev and J. Piekarewicz, Phys. Rev. Lett. 111, 162501

(2013).[15] B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000).[16] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86, 5647

(2001).[17] R. J. Furnstahl, Nucl. Phys. A 706, 85 (2002).[18] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C 81, 051303

(2010).[19] F. J. Fattoyev and J. Piekarewicz, Phys. Rev. C 84, 064302

(2011).[20] F. J. Fattoyev and J. Piekarewicz, Phys. Rev. C 86, 015802

(2012).[21] C. Xu, B.-A. Li, and L.-W. Chen, Phys. Rev. C 82, 054607

(2010).[22] C. Xu, B.-A. Li, L.-W. Chen, and C. M. Ko, Nucl. Phys. A 865,

1 (2011).

022801-5

RAPID COMMUNICATIONS

F. J. FATTOYEV, W. G. NEWTON, AND BAO-AN LI PHYSICAL REVIEW C 90, 022801(R) (2014)

[23] R. Chen, B.-J. Cai, L.-W. Chen, B.-A. Li, X.-H. Li, and C. Xu,Phys. Rev. C 85, 024305 (2012).

[24] N. Hugenholtz and L. van Hove, Physica 24, 363 (1958).[25] E. Chabanat, J. Meyer, P. Bonche, R. Schaeffer, and P. Haensel,

Nucl. Phys. A 627, 710 (1997).[26] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, Phys. Rev.

C 58, 1804 (1998).[27] L.-W. Chen, C. M. Ko, B.-A. Li, and J. Xu, Phys. Rev. C 82,

024321 (2010).[28] A. Gezerlis, I. Tews, E. Epelbaum, S. Gandolfi, K. Hebeler

et al., Phys. Rev. Lett. 111, 032501 (2013).[29] B. A. Brown and A. Schwenk, Phys. Rev. C 89, 011307(R)

(2014).[30] C. Horowitz, K. Kumar, and R. Michaels, Eur. Phys. J. A 50, 48

(2014).[31] M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys.

Rev. Lett. 102, 122502 (2009).[32] M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys.

Rev. C 82, 054314 (2010).[33] I. Vidana, C. Providencia, A. Polls, and A. Rios, Phys. Rev. C

80, 045806 (2009).

[34] Z. Zhang and L.-W. Chen, Phys. Lett. B 726, 234(2013).

[35] E. Khan, J. Margueron, and I. Vidana, Phys. Rev. Lett. 109,092501 (2012).

[36] C. Horowitz, E. Brown, Y. Kim, W. Lynch, R. Michaels et al.,J. Phys. G 41, 093001 (2014).

[37] F. J. Fattoyev and J. Piekarewicz, Phys. Rev. C 82, 025810(2010).

[38] J. Piekarewicz, F. J. Fattoyev, and C. J. Horowitz,arXiv:1404.2660.

[39] J. Xu, L.-W. Chen, B.-A. Li, and H.-R. Ma, Astrophys. J. 697,1549 (2009).

[40] S. S. Avancini, S. Chiacchiera, D. P. Menezes, and C. Providen-cia, Phys. Rev. C 82, 055807 (2010).

[41] C. Ducoin, J. Margueron, C. Providencia, and I. Vidana, Phys.Rev. C 83, 045810 (2011).

[42] B.-A. Li and C. Ko, Nucl. Phys. A 618, 498 (1997).[43] W. G. Newton, M. Gearheart, and B.-A. Li, Astrophys. J. Suppl.

204, 9 (2013).[44] W. G. Newton, J. Hooker, M. Gearheart, K. Murphy, D.-H. Wen,

F. J. Fattoyev, and B.-A. Li, Eur. Phys. J. A 50, 41 (2014).

022801-6