Intrinsic Correlations Among Characteristics of Neutron-rich Matter Imposed by the Unbound Nature of Pure Neutron Matter
aa r X i v : . [ nu c l - t h ] M a r Intrinsic Correlations Among Characteristics of Neutron-rich MatterImposed by the Unbound Nature of Pure Neutron Matter
Bao-Jun Cai ∗ and Bao-An Li † Quantum Machine Learning Laboratory, Shadow Creator Inc., Shanghai 201208, China Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, TX 75429-3011, USA (Dated: March 3, 2021)
Background:
The equation of state (EOS) E ( ρ, δ ) of neutron-rich nucleonic matter at density ρ and isospinasymmetry δ can be approximated as the sum of the symmetric nuclear matter (SNM) EOS E ( ρ ) and apower series in δ with the coefficient E sym ( ρ ) of its first term the so-called nuclear symmetry energy. Greatefforts have been devoted to studying the characteristics of both the SNM EOS E ( ρ ) and the symmetryenergy E sym ( ρ ) using various experiments and theories over the last few decades. While much progresseshave been made in constraining parameters characterizing the E ( ρ ) and E sym ( ρ ) around the saturation density ρ of SNM, such as the incompressibility K of SNM EOS as well as the magnitude S and slope L ofnuclear symmetry, the parameters characterizing the high-density behaviors of both E ( ρ ) and E sym ( ρ ) , suchas the skewness J and kurtosis I of SNM EOS as well as the curvature K sym and skewness J sym of thesymmetry energy are still poorly known. Moreover, most attention has been put on constraining the characteris-tics of the SNM EOS and the symmetry energy separately as if the E ( ρ ) and E sym ( ρ ) are completely independent. Purpose:
Since the EOS of PNM is the sum of the SNM EOS and all symmetry energy coefficientsin expanding the E ( ρ, δ ) as a power series in δ , the unbound nature of PNM requires intrinsic correlationsbetween the SNM EOS parameters and those characterizing the symmetry energy independent of any nuclearmany-body theory. We investigate these intrinsic correlations and their applications in better constraining thepoorly known high-density behavior of nuclear symmetry energy. Method:
We first derive an expression for the saturation density ρ sat ( δ ) of neutron-rich matter up toorder δ in terms of the EOS parameters. Setting ρ sat ( δ ) =0 for PNM as required by its unbound nature leads toa sum rule for the EOS parameters. We then analyze this sum rule at different orders in δ to find approximateexpressions of the high-density symmetry energy parameters in terms of the relatively better determined SNMEOS parameters and the symmetry energy around ρ . Results:
Several novel correlations connecting the characteristics of SNM EOS with those of nuclearsymmetry energy are found. In particular, at the lowest-order of approximations, the bulk parts of the slope L , curvature K sym and skewness J sym of the symmetry energy are found to be L ≈ K / , K sym ≈ LJ / K and J sym ≈ I L / K , respectively. High-order corrections to these simple relations can be written in termsof the small ratios of high-order EOS parameters. The resulting intrinsic correlations among some of theEOS parameters reproduce very nicely their relations predicted by various microscopic nuclear many-bodytheories and phenomenological models constrained by available data of terrestrial experiments and astrophysicalobservations in the literature. Conclusion:
The unbound nature of PNM is fundamental and the required intrinsic correlations among the EOSparameters characterizing both the SNM EOS and symmetry energy are universal. These intrinsic correlationsprovide a novel and model-independent tool not only for consistency checks but also for investigating the poorlyknown high-density properties of neutron-rich matter by using those with smaller uncertainties.
PACS numbers: 21.65.-f, 21.30.Fe, 24.10.Jv
I. INTRODUCTION
The equation of state (EOS) of cold asymmetric nucleonicmatter (ANM) given in terms of the energy per nucleon E ( ρ, δ ) at density ρ and isospin asymmetry δ ≡ ( ρ n − ρ p ) /ρ betweenneutron and proton densities ρ n and ρ p is a fundamental quan-tity in nuclear physics and a basic input for various applicationsespecially in astrophysics [1]. The E ( ρ, δ ) is often expanded ∗ [email protected] † Bao-An.Li @ tamuc.edu around δ = to further define the EOS E ( ρ ) of symmetricnucleonic matter (SNM) and various orders of the so-callednuclear symmetry energy according to E ( ρ, δ ) = E ( ρ ) + ∞ X i = E sym , i ( ρ ) δ i . (1)Setting δ = , the above equation reduces to an approximaterelation among the EOS E PNM ( ρ ) ≡ E ( ρ, of pure neutronmatter (PNM), the SNM EOS E ( ρ ) and all of the symmetryenergy terms, i.e., E PNM ( ρ ) = E ( ρ ) + P ∞ i = E sym , i ( ρ ) . Nor-mally the expansion in δ ends at the quadratic term with i = in the so-called parabolic approximation, and one generallyrefers the E sym , ( ρ ) ≡ E sym ( ρ ) as the symmetry energy by ne-glecting all higher order terms. One then expands the SNMEOS E ( ρ ) and the symmetry energy E sym ( ρ ) around the satu-ration density ρ of SNM with their characteristic coefficients,e.g., the incompressibility K , skewness J and kurtosis I ofSNM as well as the magnitude S , slope L , curvature K sym andskewness J sym of the symmetry energy E sym ( ρ ) .Much efforts in both experiments and theories have beendevoted to investigating the characteristics of SNM duringthe last four decades and those of the symmetry energy overthe last two decades, respectively. Indeed, much progresseshave been made in constraining both the E ( ρ ) and nuclearsymmetry energy especially around and below ρ , see, e.g,Refs. [2–9] for reviews. On the other hand, much progresseshave also been made in recent years in understanding proper-ties of low-density PNM using the state-of-the-art microscopicnuclear many-body theories and advanced computational tech-niques, e.g., chiral effective field theories [10] and quantumMonte Carlo techniques [11]. All calculations indicate thatboth the energy and pressure in PNM approach zero smoothlyand monotonically as the density vanishes, reflecting the un-bound nature of PNM [12–16]. Moreover, it has been shownthat the unitary Fermi gas EOS E UG ( ρ ) = ξ E FG ( ρ ) in termsof the EOS of a free Fermi gas for neutrons E FG ( ρ ) and aBertsch parameter ξ ≈ . [17] provides a lower bound tothe E PNM ( ρ ) at low densities [18], demonstrating vividly thedeep quantum nature of the system in the so-called contactregion [19].Imposing constraints on theories or fitting model predictionswith experimental observables will naturally introduce corre-lations among the EOS parameters. Indeed, some interestingcorrelations have been found especially among the character-istics of either the SNM EOS E ( ρ ) or the symmetry energy E sym ( ρ ) . It is also interesting to note that the low-densityPNM EOS from the microscopic theories has been used as aboundary condition to calibrate some phenomenological mod-els [20–22] and to explore some correlations among the EOSparameters [23, 24]. In particular, the condition that the energyper neutron in PNM vanishes at zero density naturally leads to alinear correlation between K sym and S – L [25]. However, mostof the correlations among the EOS parameters found so far arevery model dependent especially those involving the high-order coefficients, see, e.g., Refs. [6, 18, 25–32]. Since thesecorrelations are known to have significant effects on propertiesof both nuclei and neutron stars, see, e.g., Refs. [23, 26, 35],a better understanding of the correlations among the EOS pa-rameters have significant ramifications in both nuclear physicsand astrophysics.In this work, we show that the unbound nature of PNMalone, especially its vanishing pressure P at zero density (i.e.,the saturation density of ANM approaches zero as δ → ),naturally leads to a sum rule linking intrinsically the EOS pa-rameters independent of any theory. Analyses of this sum ruleat different orders of the expansions in δ and ρ lead to novel cor-relations relating the characteristics of SNM EOS with thoseof nuclear symmetry energy. In particular, at the lowest-orderof approximations, we found that L ≈ K / , K sym ≈ LJ / K and J sym ≈ I L / K , respectively. Corrections to these sim- ple relations by considering high-order terms reproduce nicelythe empirical correlations among some of the EOS parametersreported previously in the literature.The rest of the paper is organized as follows. In sectionII A, we recall the basic definitions of ANM EOS parameters.Intrinsic correlations among the EOS parameters and theirgeneral implications are given in section II B. We then discusspotential applications of the intrinsic correlations. As the firstexample, we give in section III A a scheme for estimating theslope L of nuclear symmetry energy. Section III B is devoted toinvestigating the correlation between the curvature K sym of thesymmetry energy and the L , K and J . Relevant comparisonswith the empirical K sym – L relations in the literature will alsobe given in this section. In section III D and section III C, twodirect implications of the K sym – L relation on the correlationbetween L and the symmetry energy magnitude S at ρ aswell as the isospin-dependent part of the incompressibilitycoefficient K sat,2 will be studied. Section III E gives a possibleconstraint on the skewness J sym of the symmetry energy. Ashort summary is finally given in section IV. II. UNIVERSAL CONSTRAINTS ON THE EOSPARAMETERS OF NEUTRON-RICH MATTER BY THEUNBOUND NATURE OF PURE NEUTRON MATTERA. Characteristics of Isospin Asymmetric Nuclear Matter
For easy of our discussions in the following, we summarizehere the necessary notations and recall some terminologies weadopted. Using the δ and χ = ( ρ − ρ ) / ρ as two pertur-bative variables, the EOS of ANM can be expanded aroundSNM at ρ generally as E ( ρ, δ ) = w i j δ i χ j where the repeatedindices are summed over, with each term characterized by δ i χ j . In the following, we call “ i + j ” the order of thequantity considered [36]. In this sense, E ( ρ ) = w is theonly zeroth-order term from i = j = , and the first-orderterm is absent due to the vanishing pressure at ρ in SNM bydefinition of its saturation point. At second order, we have K = ρ E ′′ ( ρ ) = w as well as S ≡ E sym ( ρ ) = w .Very similarly, we have at third order L = ρ E ′ sym ( ρ ) = w and J = ρ E ′′′ ( ρ ) = w . While at fourth order, wehave I = ρ E ′′′′ ( ρ ) = w , K sym = ρ E ′′ sym ( ρ ) = w and S ≡ E sym,4 ( ρ ) = w . Finally, the skewness J sym = ρ E ′′′ sym ( ρ ) = w of the symmetry energy E sym ( ρ ) and theslope coefficient L sym,4 = ρ E ′ sym,4 ( ρ ) = w of the fourth-order symmetry energy E sym,4 ( ρ ) are both at order five. B. Intrinsic Correlations of EOS parameters and TheirGeneral Implications
The saturation density ρ sat ( δ ) for ANM with isospin asym-metry δ is defined as the point where the pressure vanishes,namely P ( ρ sat ( δ )) = , or equivalently ∂ E ( ρ, δ ) /∂ρ | ρ = ρ sat ( δ ) = .After a long but straightforward calculation by expanding allterms in Eq. (1) as functions of χ and the saturation density ρ sat as a function of δ (see the Appendix in Ref. [37]), one canobtain ρ sat ( δ ) /ρ ≈ + Ψ δ + Ψ δ + Ψ δ + O ( δ ) , (2)with Ψ = − LK , (3) Ψ = K sym LK − L sym,4 K − J L K , (4) Ψ = K sym LK − J L K − L sym,4 K J LK − K sym K + I L K − J sym L K + K sym,4 LK − L sym,6 K , (5)here K sym,4 is the curvature of the fourth-order symmetry en-ergy and L sym,6 is the slope of the sixth-order symmetry energy.The expressions for Ψ and Ψ were first given in Ref. [37].Required by the unbound nature of PNM as shown by allexisting nuclear many-body theories, the saturation density ofANM eventually decreases from ρ to zero as δ goes fromzero (SNM) to 1 (PNM). Quantitatively, to the order δ thisboundary condition requires + Ψ + Ψ + Ψ = . It canbe rewritten using the characteristic EOS coefficients as anapproximate sum rule = LK − K sym LK − J L K + L sym,4 K + J LK + K sym K − I L K + J sym L K − K sym,4 LK + L sym,6 K . (6)Clearly, it establishes an intrinsic correlation among the EOSparameters. Compared to the equation one would obtain fromsetting the energy per neutron E PNM (0) = at zero density,through couplings between isospin-independent and isospin-dependent coefficiens the above relation provides a more strin-gent constraint for the high-order EOS parameters without in-volving the binding energy E ( ρ ) of SNM and the magnitude S of the symmetry energy at ρ . In fact, the latter has been welldetermined to be around S = E sym ( ρ ) = . ± . MeV byextensive analyses of terrestrial experiments and astrophysicalobservations [8, 38] as well as ab initio nuclear-many theorypredictions [39].The analytic expressions for the saturation density at dif-ferent orders of δ characterized by Ψ , Ψ and Ψ , etc., arevery useful since they fundamentally encapsulate the intrinsiccorrelations among the characteristic EOS parameters. In thissense, intrinsic equations at different orders of δ could beobtained. Specifically, if one truncates the saturation density ρ PNMsat /ρ in PNM to order δ , an intrinsic equation + Ψ = is obtained. More accurately this should be an intrinsic in-equality . + Ψ . . However, it is known that at thelowest orders of δ in the so-called parabolic approximation,the ρ PNMsat is not necessarily zero in some of the many-body cal-culations, see examples given in Ref. [45]. Nevertheless, theEq. (2) provides a scheme to improve gradually the accuracyof calculating the saturation density of ANM. For example, an intrinsic equation + Ψ + Ψ = or + Ψ + Ψ + Ψ = could be obtained if the ρ PNMsat /ρ is truncated at order δ or δ ,respectively. As the order of truncation increases, the quantity + P j = Ψ j = + Ψ + Ψ + · · · becomes more and moreclose to zero, and the intrinsic correlations or the sum rule + P j = Ψ j = obtained becomes more and more accurate.Moreover, as the accuracy in estimating the ρ sat ( δ ) increase,more high-order EOS parameters get involved as one expects.We point out the following two basic usages of the intrinsicequations (at different orders of δ ):1. The intrinsic equations can be effectively used to estab-lish useful connections between parameters characterizingthe SNM EOS and those describing the symmetry energy.For example, one can immediately obtain from the intrinsicequation + Ψ = at order δ the relation L ≈ K / , andif the coefficient K is better constrained then the L coef-ficient could by subsequently inferred. Moreover, if bothof them are independently determined, this simple relationallows for a consistency check. Naturally, if higher ordercontributions are included, this simple relation is expectedto be modified. In section III A, we investigate this issuein more details. The most important physics outcome isthat the approximate relations, e.g., L ≈ K / , provide auseful guide for developing phenomenological models andmicroscopic theories.2. As mentioned above, as the truncation order of the expan-sion (2) increases, more and more characteristic coefficientswould be included, i.e., they will emerge in the intrinsicequations as the order increases. Then, the correlations canpotentially allow one to extract high-order (poorly known)coefficients from the lower-order (better known) ones. Anexample given in section III B on extracting the K sym fromits correlation with L will demonstrate this point in detail.Moreover, a few interesting points related to Eq. (2) andEq. (6) are worth emphrasing:1. Since the saturation density ρ sat ( δ ) /ρ is obtained orderby order, one naturally expects the higher order term con-tributes less. If the ρ sat ( δ ) /ρ is truncated at order δ ,then one has Ψ = − . Actually what we have obtainedis & Ψ & − . Next if the Ψ contribution is takeninto consideration, we have ρ PNMsat /ρ = + Ψ + Ψ = ,i.e., Ψ = − − Ψ . Simultaneously setting | Ψ | . and | Ψ | . | Ψ | gives the constraint on Ψ as − / . Ψ . .When the Ψ term is included, a similar analysis on thevalue of Ψ could be made.2. Although equation (6) implies a complicated intrinsic cor-relation among all the parameters involved, certain termscould directly be found to be less important. For exam-ple, the characteristic coefficients related to the fourth-ordersymmetry energy E sym,4 ( ρ ) , namely L sym,4 and K sym,4 areexpected to be small, since the value of S ≡ E sym,4 ( ρ ) isknown to be smaller than about 2 MeV [46–49]. Particu-larly, by adopting a density dependence similar to the sym-metry energy E sym ( ρ ) [50] as E sym,4 ( ρ ) = S ( ρ/ρ ) γ with γ an effective density-dependence parameter, one obtains L sym,4 = γ S and K sym,4 ≈ γ ( γ − S . A conservativeestimate with . γ . indicates that MeV . L sym,4 . MeV and − . MeV . K sym,4 . MeV. Consequently,one can safely neglect the term “ − K sym,4 L / K ” in Eq. (6)since its magnitude is smaller than 1.5% with the currentlyknown most probable values of L [8, 38] and K [9]. Forsimilar reasons, the last term in Eq. (6) involving the slope L sym,6 of the six-order symmetry energy E sym,6 ( ρ ) can alsobe neglected. However, the term involving the L sym,4 mightbe important and it is kept in the following discussions.3. By truncating Eq. (2) at different orders of δ , differentintrinsic correlation equations are obtained. But they arenot all independent. As the truncation order increases, theintrinsic correlation becomes more general since more andmore characteristic coefficients are taken into consideration.Of course, the resulting dependences/correlations amongthe EOS parameters look gradually more complicated. III. APPLICATIONS OF THE INTRINSICCORRELATIONS AMONG EOS PARAMETERS
In this section, we present a few applications of the sumrule in Eq. (6) at different orders of δ . To justify some ofour numerical approximations and ease of our discussions, itis useful to note here the currently known ranges of the lower-order parameters and the rough magnitudes of the high-orderparameters. In particular, we shall use the empirical valuesof K ≈ ± MeV [6, 9, 40–42], J ≈ − MeV [36]and I ≈ − ± MeV [43] for the SNM EOS. For thesymmetry energy, it is known that L ≈ ± MeV [38], K sym ≈ − MeV [44] and J sym ≈ MeV [29]. While thecommunity has not reached a consensus on the exact rangesof some of the high-order parameters, the reference valuesenable us to make rough estimates and judge if some of theratios/terms in our analyses can be neglected.
A. Estimating the Slope L of Nuclear Symmetry Energy As the first application of the intrinsic equations, we derivean expression for the slope L of the symmetry energy in termsof better known quantities corrected by some ratios of otherparameters that are small. The main purpose of this analysisis to show how the unbound nature of PNM naturally gives agood estimate for L .As discussed in section II B, the intrinsic equation at order δ gives the lowest-order approximation L ≈ K / . Thus,by introducing the dimensionless quantity x = L / K (conse-quently x (0) ≈ corresponding to L ≈ K / could be found) and the following combinations ψ = − L sym,4 K − L sym,6 K , (7) ψ = − K sym K + L sym,4 J K − K sym,4 K + K sym K , (8) ψ = − K sym K + J sym J , (9) ψ = − J K I , (10)we obtain the equation for x by rewriting Eq. (6) as ψ − ψ x − J K ψ x + I K ψ x = . (11)Rewriting the Eq. (6) in the form (11) is mainly for the conve-nience of discussing its relevance for estimating L . If on theother hand the interested quantity is K sym , then Eq. (6) couldbe cast into the form AK sym + BK sym + C = with A , B and C being some relevant coefficients independent of K sym . Wenote that the J appearing in ψ comes from the Ψ , while the J sym and I appearing in ψ and ψ come from the Ψ .Different approximations of Eq. (11) and analysis could bedeveloped. In the following, we discuss approximate solutionsof Eq. (11) at δ , δ and δ order, separately. At the δ order,besides the simplest estimation L ≈ K / by taking ψ ≈ ψ ≈ in Eq. (11) and simultaneously neglecting the x and x terms, one can also find that L / K > , i.e., the sign of theincompressibility K is the same as that of the slope L . Thelatter can be seen from the relation ρ sat ( δ ) /ρ ≈ +Ψ δ + O ( δ ) and Ψ = − L / K . As the isospin asymmetry δ increases fromzero to a small finite value, the saturation density has to bereduced. Thus, Ψ has to be negative, requiring L / K > .Systematically, we can generalize the relation L ≈ K / as L ≈ K × + “higher order corrections” ! , (12)when the higher order terms Ψ δ , Ψ δ , etc., are taken intoaccount in the saturation density ρ sat /ρ . For example, at δ order, by considering the Ψ term, but simultaneously neglect-ing the Ψ term, the coefficients related to the fourth-ordersymmetry energy as well as the small terms ( K sym / K ) and J K sym / K , one can obtain J K x + − K sym K ! x − = . (13)Its solution leads to L = − K sym K ! K J s + J K − K sym K ! − − , (14) ≈ K × − K sym K ! − − J K − K sym K ! − , (15)where the second approximation is obtained by noticing that | + (2 J / K )(1 − K sym / K ) − | ≪ using the empirical valuesof the involved parameters given earlier. In addition, the pos-itiveness of the discriminant of Eq. (13) limits the skewness J to J & − K (1 − K sym / K ) / . Since K sym / K ≈ − / and J / K ≈ − / are both small and at the same level, theabove expression for L can be further approximated as L ≈ K × + K sym K − J K ! . (16)Similarly, at δ order, again by neglecting the coefficientsrelated to the fourth-order and the sixth-order symmetry ener-gies as well as the other small terms, the solution of Eq. (11)leads to L ≈ K × " + K sym K − J + J sym K + I K + K sym K − J K + I K ! . (17)While the kurtosis I and the incompressibility K have similarorders of magnitude, the second line of (17) contributes onlyabout 2% compared to the leading contribution “1”. Usingthe empirical values of the EOS parameters given earlier, the“high order contributions” in (17) is estimated to be about − %. Consequently, the L is about . MeV, comparedto the simplest estimation of about 80 MeV from L ≈ K / .Moreover, the coefficient / together with the small ratiosenable us to further approximate the expression (17) to L ≈ K × + K sym K − J K − J sym K + I K ! . (18)As noticed before, the J sym and I appear only in the Ψ .Neglecting them, the above expression naturally reduces toEq. (16) valid at the δ order as one expects.Our analyses above indicate clearly that as the truncationorder of ρ PNMsat increases, the higher order terms become even-tually irrelevant although they appear in a complicated manner.To understand the results intuitively, it is useful to look at theanalogy with the period T of small angle oscillations of asimple pendulum T ( θ ) ≈ π s l g × + θ + θ ! , (19)where θ is the maximum angle and l is the length of the pen-dulum. For θ = , we obtain from the above expressionthe period T (1) ≈ . × T (0) . While the exact result is T (1) ≈ . × T (0) , indicating that although the apparentperturbation element θ is not much smaller than 1, the per-turbative expansion is still very effective (useful) near θ ≈ due to the small in-front coefficients / and / in theexpansion of T ( θ ) . B. Intrinsic Correlation between K sym and L Besides estimating the slope L , the intrinsic equations couldalso be used to investigate the correlations among certain EOS parameters. As K sym first appears at δ order in the Ψ term,through the intrinsic equation + Ψ + Ψ = , we obtain thefollowing relation for K sym , K sym ≈ K − K L + J K LK + L sym,4 L ! . (20)One can first check the magnitude of K sym using this equation.By taking K ≈ MeV , J ≈ − MeV , L ≈ MeV and L sym,4 ≈ MeV, one obtains K sym ≈ − MeV. In addition, ifone assumes | Ψ / Ψ | . , then one finds − MeV . K sym . MeV. These results are consistent with the constraint on K sym obtained recently from Bayesian analyses of neutron starobservations [44, 51].It is necessary to point out that the relations (14) or (15) and(20) are not independent, and in fact they are effectively twodifferent presentations of the same intrinsic equation. How-ever, they are usefully different in the sense that they giveseparately the expressions for L and K sym in terms of theirmain parts plus small corrections determined by the ratios ofother EOS parameters. The analysis above indicates that the K sym is closely correlated with K , L and J while the L sym,4 / L has negligible effects as we discussed earlier. Moreover, tak-ing the lowest-order approximation for L , i.e., L ≈ K / , theexpression (20) can be further reduced to K sym ≈ K J K LK + L sym,4 L ! ≈ LJ / K . (21)One then immediately finds qualitatively that K sym correlatespositively with L and J but negatively with K .In the following, we investigate more quantitatively thecorrelations of K sym with other EOS parameters using themore accurate relation (20). For this purpose, we performa Monte Carlo sampling of the EOS parameters in their cur-rently known ranges. Shown in Fig. 1 are the correlationsbetween K sym – L sym,4 , K sym – L , K sym – K and K sym – J , respec-tively. In this study, the parameters are randomly sampled si-multaneously and uniformly in the range of MeV . L sym,4 . MeV , MeV . L . MeV , MeV . K . MeVand − MeV . J . − MeV. From the results shownwe clearly observe that the K sym has weak positive correlationswith both L sym,4 and J but a strong positive (negative) correla-tion with L ( K ). When more experimental constraints on theEOS parameters become available, the accuracy and robust-ness of the correlations are expected to be improved. However,the correlation patterns revealed here should stay the same.The strengths of correlations between the K sym and theother EOS parameters can be quantified using the quantity Θ ( φ i ) = δφ i ∂ K sym /∂φ i where φ i = L sym,4 , L , K , J . Morespecifically, we find that Θ ( L ) ≈ MeV, Θ ( L sym,4 ) ≈ MeV , Θ ( K ) ≈ − MeV, and Θ ( J ) ≈ MeV, respec-tively. These numbers clearly quantify the strengths of thecorrelations shown in Fig. 1. It is interesting to note here thatthe results of our model-independent analyses are very con-sistent with the findings of Ref. [25] (see Fig. 7 there). In thelatter, the correlations were obtained from some basic physicalconstraints imposed on the Taylor’s expansion of the bind-ing energy in ANM around ρ [25]. One of the constraints sym,4 (MeV)−600−400−2000200 K s y m ( M e V ) (a) 30 40 50 60 70 80 90L (MeV)−600−400−2000200 (b) 200 220 240 260 280K (MeV)−600−400−2000200 (c) −500 −400 −300 −200 −100J (MeV)−600−400−2000200 (d) Fig. 1: (Color Online). Correlations between K sym and L sym,4 , L , K and J according to Eq. (20). Ref. [25] used is that the EOS of PNM at zero density is zero.The present analysis thus shares with Ref. [25] the requirementthat the PNM is unbound. However, as we pointed out earlier,the vanishing pressure of PNM puts a more stringent con-straints on the high-order EOS parameters than its vanishingbinding energy at zero density.The near-linear correlations between K sym and the otherEOS parameters can also be described more quantitatively. Forexample, the correlation between K sym and L can be writtenas K sym ≈ a L + b . By minimizing the algebraic error, thecoefficients a and b can be obtained as a ≈ h K sym L i − h K sym ih L ih L i − h L i , (22)and b = h K sym i − a h L i , here the average h· · · i for one in-dependent simulation with total m points is defined simply as h k i = m − P mi = k ( i ) . By independently sampling n times oneobtains the standard uncertainty of a or b as σ f = vut n n X i = f ( i ) , − n n X i = f ( i ) , (23)where f ( i ) ↔ a ( i )1 , b ( i )1 . Using m = , n = in our simu-lations, we find a = n − P ni = a ( i )1 ≈ . ± . and similarly b = n − P ni = b ( i )1 ≈ − ± MeV. Taking L ≈ MeV in K sym ≈ a L + b gives K sym ≈ − MeV. For the correlationbetween K sym and L sym,4 , namely K sym ≈ a L sym,4 + b , one has a ≈ . ± . and b ≈ − ± MeV, similarly for K sym and K , namely K sym ≈ a K + b , one has a ≈ − . ± . and b ≈ ± MeV. Finally for the correlation between K sym and J , i.e., K sym ≈ a J + b , the results are found tobe a ≈ . ± . and b ≈ − ± MeV. It is necessaryto point out that while the central values of a i and b i (with i = ∼ ) will approach those determined by the centralvalues of the characteristic coefficients, e.g., L and K , forthe simulation as n → ∞ according to the law of large num-bers [52], the magnitude of the uncertainty is affected by thechoice of m . A smaller m leads to a larger uncertainty as oneexpects.When high order contributions are considered, we can simi-larly study the correlation between the K sym and the other EOS
30 40 50 60 70 80 90L (MeV)−600−400−2000200 K s y m ( M e V ) (a) order Ψ
200 220 240 260 280K (MeV)−600−400−2000200 (b) order Ψ Fig. 2: (Color Online). Correlations between K sym and L (left) andthat between K sym and K (right) to order Ψ . coefficients. For instance, at the δ order, the relevant equa-tion for calculations is obtained by neglecting the coefficients L sym,6 and K sym,4 in Eq. (6). Fig. 2 shows results of a MonteCarlos sampling of the correlations at δ order using MeV . J sym . MeV and − MeV . I . MeV as wellas those ranges given earlier for the low-order EOS parame-ters. It is seen that the positive (negative) correlation between K sym and L ( K ) is unchanged when the relevant higher ordercontributions are included. This means that the qualitativefeatures of the intrinsic correlations obtained earlier from theEq. (6) at the δ order are kept and stable, while the fittingcoefficients are affected quantitatively. More specifically, nowfor the correlation K sym ≈ a ′ L + b ′ the n -average of a ′ and b ′ are found to be a ′ ≈ . ± . and b ′ ≈ − ± MeV,respectively. Similarly for the correlation K sym ≈ a ′ K + b ′ ,we have a ′ ≈ − . ± . and b ′ ≈ ± MeV, respectively.It is also interesting to compare our K sym – L correlation withtypical results obtained from other approaches in the litera-ture [18, 29, 30]. More specifically, Ref. [29] gave a corre-lation K sym ≈ ( − . ± . S − L ) + ± MeV at 68%confidence level from analyzing over 520 predictions of theSkyrme–Hartree–Fock energy density functionals and the rel-ativistic mean field theories. Their result using S ≈ ± MeVis shown with the black lines in Fig. 3. The authors ofRef. [18] did a similar analysis as Ref. [29] but applied addi-tional constraints. Their result K sym ≈ . L − ± MeVis shown as the sky blue band in Fig. 3. In Ref. [30], a K sym – L correlation was derived by using the Fermi liquid Fig. 3: (Color Online). Correlation between K sym and L from intrinsicanalysis as well as the prediction on it from Refs. [18, 29, 30]. theory with its parameters calibrated by the chiral effectivefield theory at sub-saturation densities. Their correlation K sym ≈ . L − ± MeV is shown as the cyan band.While our result is shown with the purple band. It is seenthat our correlation is highly consistent with the results fromthe three other studies. They all overlap largely around theupper boundary from the analysis of Ref. [29]. More quan-titatively, taking L ≈ MeV in K sym ≈ a ′ L + b ′ gives K sym ≈ − MeV, while the same value of L leads to a meanvalue of K sym ≈ − MeV , − MeV and − MeV usingthe constraints from Refs. [18, 29, 30], respectively. We alsonotice that a very recent study based on the so-called KIDSenergy functional [32] gave a K sym – L correlation consistentwith the results discussed above. More quantitatively, theyconcluded that the curvature K sym is probably negative and notlower than − MeV [32]. Finally, it is also interesting to notethat the analysis combining the tidal deformability of neutronstars via a χ -based covariance approach also showed that thecoefficients K sym and L are strongly correlated [33, 34]. C. Implications of the Intrinsic K sym – L Correlation on theIncompressibility of Neutron-rich Matter along its SaturationLine
The incompressibility of neutron-rich matter K sat ( δ ) ≡ ρ sat ∂ E ( ρ, δ ) ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ sat ≈ K + K sat,2 δ + O ( δ ) (24)is an important quantity directly related to the ongoing stud-ies of various collective modes and stability of neutron-richnuclei [9]. The strength of its isospin-dependent part can bewritten as [37] K sat,2 = K sym − L − J L / K . (25) By using Eq. (20) for K sym , the latter can be rewritten as K sat,2 ≈ + L sym,4 L − K L ! K − + J K ! L . (26)Numerically, we find K sat,2 ≈ − MeV using the empiricalvalues of EOS parameters given earlier. This value is consis-tent with the latest study on K sat,2 within the KIDS frameworkwhere it was found that the K sat,2 should roughly lie between − MeV and − MeV [32]. Similar to what are shownin Fig. 1, using the above expression for K sat,2 one can alsoanalyze its correlations with K , L , L sym,4 and J , separately.Moreover, noticing that | L sym,4 / L | ≪ and L ≈ K / , the K sat,2 can be further approximated as K sat,2 ≈ − K − J / . (27)This relation clearly demonstrates that the uncertainty of K sat,2 mainly comes from the poorly known J although its contri-bution has a shrinking factor of − / . D. Implications of the Intrinsic K sym – L Correlation on the L – S Correlation of the Symmetry Energy
The near-linear correlation between K sym and L also hasan important implication on the correlation between L and S of the symmetry energy at ρ . Noticing that at an arbitrarydensity ρ d L ( ρ )d E sym ( ρ ) = d L ( ρ )d ρ , d E sym ( ρ )d ρ = + K sym ( ρ ) L ( ρ ) (28)according to the basic definitions of the characteristic coeffi-cients of symmetry energy E sym ( ρ ) . By taking ρ = ρ , therelation (28) gives d L / d S = + K sym / L . For instance, the L and K sym in the free Fermi gas model are L = S and K sym = − S = − L , respectively, automatically fulfilling therelation (28). Using the intrinsic correlation K sym ≈ aL + b found earlier and by integrating R d S = R d L (3 + K sym / L ) − = R d L ( a + + b / L ) − , we obtain the following relation between S and LS ( L ) = a + L − ba + a + L + b ] ! + const. , (29)where the constant is determined via some reference point ( S H , L H ) , e.g., S H ≈ MeV and L H ≈ MeV at ρ accordingto the surveys of available data [8, 38].Shown in Fig. 4 with the pink circles are the Monte Carlosamplings of the L – S correlation within the empirical EOSparameter ranges and adopting a ′ ≈ . and b ′ ≈ − MeVfrom the δ order analysis discussed earlier. Due to the near-linear correlation between K sym and L , the correlation between L and S is also found to be near-linear, although it is not soobvious that the relation (29) is linear. Since the near-linearitybetween K sym and L is intrinsic (only the fitting coefficientsvary if the intrinsic equation is truncated at different orders),
24 28 32 36 40S (MeV)405060708090 L ( M e V ) S(L)∆L ≈ (a + 3 + b/L H )∆Ssampling Fig. 4: (Color Online). Correlation between L and S . the near-linearity between L and S is also expected to be in-trinsic. Specifically, one can easily show that ∆ L ≈ Φ∆ S h − (cid:16) b / L H (cid:17) ∆ S i , Φ = a + + b / L H , (30)where ∆ L = L − L H , ∆ S = S − S H , and Φ is simply the valueof + K sym / L taken at L H . Numerically, we obtain the relation L ≈ . ∆ S + . ∆ S + L H by using a = a ′ ≈ . and b = b ′ ≈ − MeV. The quadratic correction . ∆ S is small,thus ∆ L ≈ . ∆ S and consequently L ≈ . S − . MeV,which is very close to the free Fermi gas model predictionon the L – S relation L = S . The resulting linearized ∆ L – ∆ S correlation (30) (by neglecting the ∆ S correction) andthe full L – S correlation (29) are shown with the red and bluelines, respectively, in Fig. 4. Around the currently known mostprobable value of S = E sym ( ρ ) = . ± . MeV [8, 38], boththe L – S and the Monte Carlo samplings are approximatelylinear consistently. Moreover, by putting the L in terms of S back into the relation K sym ≈ aL + b , we obtain K sym ≈ . S − MeV, which has a large deviation from its freeFermi gas model counterpart, see also Ref. [30].
E. Estimating the K sym – J sym Correlation
The J sym first emerges in Ψ , one can thus investigate itsintrinsic correlations with the other EOS parameters startingfrom the order δ . Since it is a high-order parameter, its valueis poorly known and some of its correlations especially withthe low-order parameters are expected to be weak. Solving theintrinsic equation (6) by neglecting L sym,4 , K sym,4 and L sym,6 leads to the following estimation for J sym J sym ≈ K L − LK ! + K K sym L − J ! + J LK − K sym K + I L K . (31)Taking the lowest-order approximations for L ≈ K / and K sym ≈ LJ / K obtained earlier, the above equation can be further reduced to J sym ≈ I L / K . Numerically, we have J sym ≈ MeV by putting the empirical values of K , J , I , L and K sym given earlier into Eq. (31). This value is consistentwith the constrain J sym ≈ . ± . MeV from analyzing thesystematics of over 520 energy density functionals in Ref. [29].It is also consistent with the J sym = ± MeV at 68%confidence level from a recent Bayesian analysis in Ref. [53](see Tab. 4 there).Fig. 5 shows the J sym – K sym correlation from our MonteCarlo samplings in the range of − MeV . K sym . MeVwhile the other EOS parameters are taken randomly in theirempirical ranges given earlier. It is seen that there is a strongcorrelation between J sym and K sym . In fact, the J sym dependson the K sym quadratically in Eq. (31). Obviously, the uncer-tainty of J sym is very large and its strong dependence on thestill poorly constrained K sym partially explains why constrain-ing the J sym is difficult. Moreover, the last term in (31), namely I L / K ≈ I / also contributes significantly to the uncer-tainty of J sym since we have little knowledge on I . −400 −300 −200 −100 0K sym (MeV)−4000−3000−2000−1000010002000 J s y m ( M e V ) Fig. 5: (Color Online). Correlation between J sym and K sym accordingto the relation (31). One can obtain the fitting parameters p and q appearing inthe linear fit J sym ≈ pK sym + q by the same method adoptedin the analysis of the correlation between K sym and L . Tak-ing n = independent runs of the simulation with eachsimulation m = points, we obtain p ≈ . ± . and q ≈ ± MeV. The large slope p means that the J sym is very sensitive to the change in K sym . For example, J sym is979 MeV (193 MeV) if K sym is taken as − MeV ( − MeV).Thus, although the change in K sym is only 40 MeV, the cor-responding change in J sym is about − MeV. This showsagain obtaining the constraint on J sym is very difficult unlessthe K sym is very well constrained. It is interesting to note thatthe strong positive correlation between J sym and K sym was alsoreported in Ref. [25] (see Fig. 7 there). IV. SUMMARY
In summary, the unbound nature of PNM requires a sum rulelinking intrinsically the ANM EOS parameters independent ofany theory. By analyzing this sum rule at different orders in δ ,we found several novel correlations relating the characteristicsof SNM EOS with those of nuclear symmetry energy. In par-ticular, at the lowest-order of approximations, the bulk partsof the slope L , curvature K sym and skewness J sym of the sym-metry energy are found to be L ≈ K / , K sym ≈ LJ / K and J sym ≈ I L / K , respectively. High-order corrections to thesesimple relations can be written in terms of the small ratiosof high-order EOS parameters. The resulting intrinsic corre-lations among the magnitude S , slope L , curvature K sym andskewness J sym of the nuclear symmetry energy reproduce verynicely their empirical correlations from various microscopicnuclear many-body theories and phenomenological models inthe literature.The unbound nature of PNM is fundamental and the requiredintrinsic correlations among the characteristics of ANM aregeneral. Since the EOS of PNM is the sum of two sectors: theSNM EOS and different orders of nuclear symmetry energy from expanding the ANM EOS E ( ρ, δ ) in even powers of δ , thevanishing pressure of PNM at zero density naturally relates thecharacteristics of the two sectors. While much progress hasbeen made by the nuclear physics community in probing sep-arately characteristics of the two parts of the ANM EOS, verylittle is known about the correlations between the character-istics of SNM and those characterizing the symmetry energy.The intrinsic correlations among the characteristics of ANMEOS provide a novel and model-independent tool not only forconsistency checks but also for investigating the poorly knownhigh-density properties of neutron-rich matter by using thosewith smaller uncertainties. Acknowledgement
This work is supported in part by the U.S. Department of En-ergy, Office of Science, under Award Number DE-SC0013702,the CUSTIPEN (China-U.S. Theory Institute for Physics withExotic Nuclei) under the US Department of Energy Grant No.DE-SC0009971. [1] P. Ring and P. Schuck,
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