Constraints on the Symmetry Energy from PREX-II in the Multimessenger Era
CConstraints on the Symmetry Energy from PREX-II in the Multimessenger Era
Tong-Gang Yue , Lie-Wen Chen , ∗ Zhen Zhang , and Ying Zhou School of Physics and Astronomy, Shanghai Key Laboratory for Particle Physics and Cosmology,and Key Laboratory for Particle Astrophysics and Cosmology (MOE),Shanghai Jiao Tong University, Shanghai 200240, China Sino-French Institute of Nuclear Engineering and Technology,Sun Yat-sen University, Zhuhai 519082, China and Quantum Machine Learning Laboratory, Shadow Creator Inc., Shanghai 201208, China (Dated: February 11, 2021)The neutron skin thickness ∆ r np of heavy nuclei is essentially determined by the symmetry energydensity slope L ( ρ ) at ρ c = 0 . / . ρ ( ρ is nuclear saturation density), roughly corresponding tothe average density of finite nuclei. The PREX collaboration recently reported a model-independentextraction of ∆ r = 0 . ± . fm for the ∆ r np of Pb, which suggests a rather stiff symmetryenergy E sym ( ρ ) with L ( ρ c ) ≥ MeV. We demonstrate that the E sym ( ρ ) cannot be too stiff and L ( ρ c ) ≤ MeV is necessary to be compatible with (1) the ground-state properties and giantmonopole resonances of finite nuclei, (2) the constraints on the equation of state of symmetricnuclear matter at suprasaturation densities from flow data in heavy-ion collisions, (3) the largestneutron star (NS) mass reported so far for PSR J0740+6620, (4) the NS tidal deformability extractedfrom gravitational wave signal GW170817 and (5) the mass-radius of PSR J0030+045 measuredsimultaneously by NICER. This allow us to obtain ≤ L ( ρ c ) ≤ MeV and . ≤ ∆ r ≤ . fm, and further E sym ( ρ ) = 34 . ± . MeV, L ( ρ ) = 85 . ± . MeV, and E sym (2 ρ ) =63 . ± . MeV. A number of critical implications on nuclear physics and astrophysics are discussed.
Introduction. — The Lead Radius Experiment (PREX)collaboration recently reported a model-independent ex-traction of ∆ r = 0 . ± . fm [1] for the neutronskin thickness (the difference between the rms radii ofthe neutron and proton distributions, ∆ r np ≡ r n − r p )of Pb by combining the original PREX result [2] withthe new PREX-II measurement [1, 3]. This updated re-sult (hereafter referred to as simply “PREX-II") reachesa precision close to , much more precise than the orig-inal ∆ r = 0 . +0 . − . fm [2]. In PREX, the neutrondensity distribution in Pb is determined by measuringthe parity-violating electroweak asymmetry in the elas-tic scattering of polarized electrons off
Pb and thusis free from the strong interaction uncertainties. Sincethe proton is charged and its distributions are well de-termined, the . ± . fm may represent the clean-est and most accurate ∆ r so far although more pre-cise measurement has been planned at MESA [4]. Thecoherent elastic neutrino-nucleus scattering [5] providesanother clean way to extract the ∆ r np , but the currentuncertainty is too large [6, 7]. The . ± . fm meansa rather thick ∆ r , significantly larger than those ex-tracted from other approaches that suffer from the un-certainties of strong interaction (see, e.g., Ref. [8] for arecent review).Besides its fundamental importance for nuclear struc-ture, the ∆ r np has been identified as an ideal probe onthe symmetry energy E sym ( ρ ) — a key but poorly-knownquantity that encodes the isospin dependence of nuclearmatter equation of state (EOS) and plays a critical rolein many issues in nuclear physics and astrophysics [9–14]. Indeed, it has been established [15–20] that the ∆ r np exhibits a strong positive linear correlation with the symmetry energy density slope L ( ρ ) at nuclear sat-uration density ρ ≈ . fm − , i.e., L ≡ L ( ρ ) . Aneven stronger correlation is found between the ∆ r np ofheavy nuclei and the L ( ρ ) at a subsaturation cross den-sity ρ c = 0 . / . ρ ≈ . fm − [21], roughly cor-responding to the average density of finite nuclei, i.e., L c ≡ L ( ρ c ) . Furthermore, the L ( ρ ) around ρ stronglyinfluences the mass-radius (M-R) relation and tidal de-formability of neutron stars (NSs), and thus provides aunique bridge between atomic nuclei and NSs [22–25].The large value of ∆ r = 0 . ± . fm suggests avery stiff E sym ( ρ ) (a large L ( ρ ) ) around ρ , which gen-erally leads to a very large NS radius and tidal deforma-bility. However, an upper limit of Λ . ≤ [26] for thedimensionless tidal deformability of . M (cid:12) NS has beenobtained from the gravitational wave signal GW170817,which requires a relatively softer E sym ( ρ ) . In addi-tion, the heaviest NS with mass . +0 . − . M (cid:12) for PSRJ0740+6620 [27] also strongly limits the E sym ( ρ ) [25],especially under the constraints on the EOS of symmet-ric nuclear matter (SNM) at suprasaturation densitiesfrom flow data in heavy-ion collisions [28], which is rela-tively soft and strongly restricts the NS maximum mass M max [24, 25, 29]. Furthermore, two independent si-multaneous M-R determination from NICER [30, 31] forPSR J0030+0451 with mass around . M (cid:12) has been ob-tained, further constraining the E sym ( ρ ) . Given the richmultimessenger data, it is extremely important to de-velop a unified framework that can simultaneously de-scribe the finite nuclei and NSs which involve a verylarge density range. Actually, serious tension between ∆ r = 0 . ± . fm and the limits from GW170817and NICER has been observed in a covariant density a r X i v : . [ nu c l - t h ] F e b functional study [32].In this work, within a single unified framework of theextended Skyrme-Hartree-Fock (eSHF) model [33, 34]which includes momentum dependence of effective many-body forces, we find the L c cannot be larger than MeVunder the constraints from GW170817, NICER, the NSmass . +0 . − . M (cid:12) , flow data in heavy-ion collisions, andthe data of ground-state properties and giant monopoleresonances (GMR) of finite nuclei. Our findings producean upper limit of ∆ r ≤ . fm, and this togetherwith the ∆ r = 0 . ± . fm lead to stringent con-straints of . ≤ ∆ r ≤ . fm and correspondingly ≤ L c ≤ MeV, which have a number of critical im-plications in nuclear physics and astrophysics.
Model and method. — The EOS of nuclear matter atdensity ρ = ρ n + ρ p and isospin asymmetry δ = ( ρ n − ρ p ) /ρ with ρ n ( ρ p ) denoting the neutron(proton) density,defined by the binding energy per nucleon, can be ex-pressed as E ( ρ, δ ) = E ( ρ ) + E sym ( ρ ) δ + O ( δ ) , (1)where E ( ρ ) = E ( ρ, δ = 0) is SNM EOS and E sym ( ρ ) = ∂ E ( ρ,δ ) ∂δ (cid:12)(cid:12)(cid:12) δ =0 is the symmetry energy. At ρ , the E ( ρ ) can be expanded in χ = ( ρ − ρ ) / ρ as E ( ρ ) = E ( ρ ) + K χ + J χ + O ( χ ) , in terms of incom-pressibility K and skewness J . The E sym ( ρ ) can beexpanded at a reference density ρ r in terms of the slopeparameter L ( ρ r ) and the curvature parameter K sym ( ρ r ) as E sym ( ρ ) = E sym ( ρ r )+ L ( ρ r ) χ r + K sym ( ρ r ) χ r + O ( χ r ) ,with χ r = ( ρ − ρ r ) / (3 ρ r ) . Setting ρ r = ρ leads to theconventional L ≡ L ( ρ ) and K sym ≡ K sym ( ρ ) .Within the eSHF model [33, 34] which includes Skyrme interaction parameters α , t ∼ t , x ∼ x andthe spin-orbit coupling constant W , we have E ( ρ ) = 3 (cid:126) m k F + 38 t ρ + 380 [3 t + t (4 x + 5)] ρk F + 116 t ρ α +1 + 380 [3 t + t (4 x + 5)] ρ k F , (2)and E sym ( ρ ) = (cid:126) m k F − t (2 x + 1) ρ − t (2 x + 1) ρ α +1 −
124 [3 t x − t (4 + 5 x )] ρk F −
124 [3 t x − t (4 + 5 x )] ρ k F , (3)where m is the nucleon rest mass and k F = (3 π / ρ ) / is the Fermi momentum. The last term in Eqs. (2)and (3) is from the momentum dependence of three-body forces which is not considered in the standard SHFmodel (see, e.g., Ref. [35]). The eSHF provides a success-ful framework to describe simultaneously nuclear mat-ter, finite nuclei, and NSs [34]. The Skyrme param-eters α , t ∼ t , x ∼ x can be expressed explicitly in terms of the following macroscopic quantities (pseudo-parameters) [34]: ρ , E ( ρ ) , K , J , E sym ( ρ r ) , L ( ρ r ) , K sym ( ρ r ) , the isoscalar effective mass m ∗ s, , the isovec-tor effective mass m ∗ v, , the gradient coefficient G S , andthe symmetry-gradient coefficient G V , the cross gradientcoefficient G SV , and the Landau parameter G (cid:48) of SNMin the spin-isospin channel. In this work, instead of di-rectly using the Skyrme parameters, we use the macroscopic model parameters in the eSHF calculationsfor nuclear matter, finite nuclei and NSs [34].For NSs, we consider the conventional NS model, i.e.,the NS contains core, inner crust and outer crust withthe core including only neutrons, protons, electrons andpossible muons ( npeµ ). For the details, one is referredto Refs. [24, 25, 34]. We would like to emphasize that inthe following NS calculations, the core-crust transitiondensity ρ t is determined self-consistently by a dynamicalapproach [36] and the causality condition is guaranteed. Result and discussion. — For the macroscopic modelparameters in eSHF, we fix E sym ( ρ c ) = 26 . MeV sinceit has been obtained with high precision by analyzingthe binding energy difference of heavy isotope pairs [21].Furthermore, the L c essentially determines the ∆ r np ofheavy nuclei [21], while the higher-order parameters J and K sym only weakly affect the properties of finite nucleibut are critical for NS properties [24, 25]. To explorethe ∆ r np and NSs, therefore, our strategy is to searchfor the parameter space of L c , J and K sym under theconstraints on the SNM EOS from flow data as well asthe limits from GW170817 and NS observations, whilewith the other parameters ( ρ , E ( ρ ) , K , m ∗ s, , m ∗ v, , G S , G V , G SV , G (cid:48) , and W ) being obtained by fitting thenuclear data on the binding energies, charge rms radii,GMR energies, and spin-orbit energy level splittings (seeRefs. [24, 25, 34] for details) to guarantee that the eSHFcan successfully describe nuclear properties (the relativedeviations of charge radii and total binding energies formedium and heavy nuclei from data are less than . ).From the obtained L c , J and K sym , one can extractinformation on EOS, ∆ r np , and NSs.A larger L c generally leads to a larger ∆ r and cor-respondingly a larger Λ . . For fixed L c and J , reducingthe K sym can effectively reduce the Λ . but also reducesthe NS maximum mass M max [24, 25]. Furthermore, in-creasing J can enhance significantly the M max but the J cannot be too large [24, 25, 29] due to relatively softSNM EOS constrained by the flow data. Using the limitof Λ . ≤ from GW170817, M max ≥ . M (cid:12) fromPSR J0740+6620, and the flow data constraint on SNMEOS, one thus expects there should exist an upper limitfor L c (also for J and K sym ). Figs. 1 (a), (b), and (c)show the M max vs K sym at various J with L c = 57 , and MeV, respectively. The shadowed regions repre-sent the allowed parameter space of J and K sym , whichall satisfy the limits of Λ . ≤ , M max ≥ . M (cid:12) ,and the flow data constraint. We note that the allowed - 2 0 0 - 1 0 0 01 . 82 . 02 . 22 . 4 J = - 3 5 3 M e VJ = - 3 4 7 M e VJ = - 3 4 6 M e V2 . 0 5 M (cid:1) L = 5 8 0 J ( M e V ) - 3 0 0 - 3 5 0 - 4 0 0 Mmax/M (cid:1) ( a ) L c = 5 7 M e V - 2 0 0 - 1 0 0 0
U p p e r l i m i t o f J ( f l o w d a t a ) L = 5 8 0 K s y m ( M e V ) ( b ) L c = 6 5 M e V - 2 0 0 - 1 0 0 0 L = 5 8 0 ( c ) L c = 7 3 M e V FIG. 1. NS maximum mass M max vs K sym within the eSHFmodel in a series of extended Skyrme interactions with J and K sym fixed at various values for L c = 57 MeV (a), MeV (b)and MeV (c), respectively. The shadowed regions indicatethe allowed parameter space. See the text for details. parameter space agrees with ∆ r ≥ . fm. As ex-pected, one sees from Fig. 1 that the allowed parame-ter space becomes smaller and smaller with increasing L c (see also Ref. [25]), and it is essentially reduced toa point at L c = 73 MeV with K sym = − MeV and J = − MeV as shown in Fig. 1 (c) (the correspondingparameter set is denoted as “Lc73”). Therefore, our re-sults indicate the L c has an upper limit of L c ≤ MeV.We note that the eSHF with Lc73 predicts ∆ r =0 . fm, which is consistent with the ∆ r = 0 . ± . fm from PREX-II. On the other hand, a smaller L c will lead to a smaller ∆ r and thus may violate theconstraint ∆ r = 0 . ± . fm. Therefore, the lowerlimit of ∆ r = 0 . fm from PREX-II can set a lowerlimit of L c . To obtain a quantitative relation between L c and ∆ r , we construct a series parameter sets with L c from to MeV in a step of MeV. For the parametersets of L c = 30 ∼ MeV, the J and K sym are obtainedby requiring them to reach the largest NS mass underthe constraints of Λ . ≤ and M max ≥ . M (cid:12) aswell as the flow data constraints on SNM EOS. For the parameter sets of L c = 75 ∼ MeV, the constraint of Λ . ≤ is turned off since it cannot be satisfied for L c ≥ MeV under the constraints of M max ≥ . M (cid:12) and the flow data as discussed earlier.Using the constructed parameter sets, we plot the L c vs ∆ r in Fig. 2 (a). As expected, the L c displaysa very strong positive linear correlation with ∆ r , i.e., L c = ( − . ± .
02) + (353 . ± . r , (4)or ∆ r = (0 . ± . . ± . L c , (5)where the units of L c and ∆ r are MeV and fm, re-spectively. Using Eq. (4), one obtains a lower limit of L c = 54 . MeV with ∆ r = 0 . fm. It is interest-ing to note that using L c = 73 MeV in Eq. (5) leadsto an upper limit of ∆ r = 0 . fm, nicely consis- ( a ) L c 7 3 M e V £ L( r ) (MeV) r = r r = r c D r ( f m ) D r ( f m ) P R E X - I I ( b ) r = 2 r r = r Esym( r ) (MeV) FIG. 2. The correlation of ∆ r with L c and L (a) as well as E sym ( ρ ) and E sym (2 ρ ) (b) in the eSHF model. The limit of . ≤ ∆ r ≤ .
27 fm from PREX-II [1] and L c ≤ MeVobtained in this work is indicated by the orange band. tent with the eSHF prediction with Lc73. We thus con-clude .
22 fm ≤ ∆ r ≤ .
27 fm and correspondingly
55 MeV ≤ L c ≤
73 MeV .To see the implications of the constraint .
22 fm ≤ ∆ r ≤ .
27 fm on the symmetry energy, we also showin Fig. 2 the ∆ r vs L , E sym ( ρ ) and E sym (2 ρ ) , whichdisplays strong linear correlations. To understand thesecorrelations, it is instructive to write down L ( ρ r ) ≈ Lρ r /ρ + K sym ρ r /ρ ( ρ r − ρ ) / (3 ρ ) , (6)by using E sym ( ρ ) ≈ E sym ( ρ )+ Lχ + K sym χ , which is avery good approximation to E sym ( ρ ) for density less thanabout ρ [37, 38]. Taking ρ c = 0 . / . ρ ≈ / ρ ,one can obtain the following relations L ≈ L c / K sym / , (7) E sym ( ρ ) ≈ E sym ( ρ c ) + L c / K sym / , (8) E sym (2 ρ ) ≈ E sym ( ρ c ) + 2 L c / K sym / , (9)which indicate the L , E sym ( ρ ) and E sym (2 ρ ) are alllinearly correlated with L c (and thus ∆ r ) for fixed E sym ( ρ c ) and small disturbance from K sym . From thestrong linear correlations shown in Fig. 2, one obtains L = 85 . ± . MeV, E sym ( ρ ) = 34 . ± . MeV,and E sym (2 ρ ) = 63 . ± . MeV. These results sug-gest a rather stiff symmetry energy around ρ , in con-trast to the constraints E sym ( ρ ) = 31 . ± . MeVand L = 58 . ± MeV [39, 40], or E sym ( ρ ) =31 . ± . MeV and L = 58 . ± . MeV [41], and E sym (2 ρ ) = 47 +23 − MeV [42], obtained by averaging es-sentially all the existing constraints. In addition, abinitio coupled-cluster calculations [43] predict a rathersoft symmetry energy of . ≤ L ≤ . MeV and . ≤ E sym ( ρ ) = 30 . MeV, which are significantlysmaller than our present constraints. It is interesting tomention that the present constraint L = 85 . ± . MeVis surprisingly in good agreement with the earlier con-straint L = 88 ± MeV [17] obtained from transportmodel analyses [44, 45] on the isospin diffusion data [46]in heavy-ion collisions. D r ( f m ) ( a ) r D U (fm -3 ) M DU (M (cid:1) ) -0.960 -0 .9 8 1 r t & DU Threshold -0.998 r t (fm -3 ) ( b ) (cid:1) (cid:1) (cid:1) RM (km)
P R E X - I I
L c 7 3 M e V £ ( c ) NICER (1.44 M (cid:1) ) (cid:1) (cid:1) D r ( f m ) NICER (1.34 M (cid:1) ) FIG. 3. Same as Fig. 2 but for the correlation with ρ t , ρ DU and M DU (a), the radius R M of NS with mass M = 1 . M (cid:12) , . M (cid:12) and . M (cid:12) (b), and the R M with M = 1 . M (cid:12) and . M (cid:12) (c). The NICER constraints [30, 31] are also includedin panel (c) for comparison. Figure 3 shows the correlation of ∆ r with thecrust-core transition density ρ t , the threshold density ρ DU and threshold NS mass M DU above which the di-rect Urca (DU) process ( n → p + e − + ¯ ν e , p + e − → n + ν e ) [47] becomes possible, the radius R M of NSwith mass M = 1 . M (cid:12) , . M (cid:12) and . M (cid:12) as well as M = 1 . M (cid:12) and . M (cid:12) . One sees the ∆ r ex-hibits a strong linear (anti-)correlation with all these NSproperties, which together with the constraint .
22 fm ≤ ∆ r ≤ .
27 fm allow us to obtain following information: ρ t = 0 . ± . fm − , ρ DU = 0 . ± . fm − , M DU = (1 . ± . M (cid:12) , R . = 14 . ± . km, R . = 13 . ± . km, R . = 13 . ± . km, R . = 13 . ± . km, and R . = 13 . ± . km.Our results suggest a relatively small ρ t , implying the NScrust will have a small thickness, fractional mass, andmoment of inertia [36]. The M DU = (1 . ± . M (cid:12) and ρ DU = 0 . ± . fm − indicate that the DUprocess will clearly occur in NSs with mass larger than . M (cid:12) (central density larger than . fm − ). Fur-thermore, if ∆ r were larger than .
25 fm , one obtains M DU (cid:46) . M (cid:12) , and this means the DU process will occurin essentially all the observed NSs. The DU process willenhance the emission of neutrinos and make it a moreimportant process in the cooling of a NS [47]. This ob-servation is particularly interesting given the fact that afast neutrino-cooling process has been suggested by thedetected x-ray spectrum of the NS in the low-mass x-ray binary MXB 1659-29 [48]. Nevertheless, it shouldbe mentioned that the NS cooling can be significantlyinfluenced by nucleon pairing [49, 50].As for the NS radii, very strong limits with a precisionof have been obtained. In particular, our present re-sults R . = 13 . ± . km and R . = 13 . ± . kmare in agreement with the NICER constraints [30, 31] buthave much better precision. It is interesting to point outthat the NICER constraints have not been imposed inconstructing the parameter sets of L c = 30 ∼ MeV,implying in eSHF, they are compatible with Λ . ≤ L c 7 3 M e V £ C aC o u p l e d C l u s t e r D rnp (fm) C a D r ( f m ) P R E X - I I ( a ) 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 . 0 50 . 1 00 . 1 50 . 2 00 . 2 5( b )
A l l f i t t i n g s :0 . 9 9 9 I R u
Z r
X e
C s
FIG. 4. Same as Fig. 2 but for the correlation with ∆ r np of Ca (a) and Zr, Ru, I, Cs,
Xe (b). The ∆ r np of Ca from ab initio coupled-cluster calculations [43] is alsoincluded in panel (a) for comparison. and M max ≥ . M (cid:12) as well as the flow data constraintson SNM EOS. It should be noted that the NS radius de-pends on the poorly-known inner crust EOS [36, 51] but Λ . does not [24], and thus it is relatively safer to use Λ . as a constraint.Shown in Fig. 4 is the ∆ r vs the ∆ r np of Ca, Zr, Ru, I, Cs,
Xe, and very strong positive linearcorrelations are seen between the ∆ r np of these nuclei.Using .
22 fm ≤ ∆ r ≤ .
27 fm together with the linearcorrelations, we obtain ∆ r (Ca) = 0 . ± . fm for Ca, ∆ r (Zr) = 0 . ± . fm for Zr, ∆ r (Ru) =0 . ± . fm for Ru, ∆ r (I) = 0 . ± . fmfor I, ∆ r (Cs) = 0 . ± . fm for Cs, and ∆ r (Xe) = 0 . ± . fm for Xe.Particularly interesting is the ∆ r (Ca) as it has beenpredicted to be . ≤ ∆ r (Ca) ≤ . fm from ab initio coupled-cluster calculations [43], which is sig-nificantly smaller than our present result ∆ r (Ca) =0 . ± . fm. At this point, we must mention thatthe Calcium Radius EXperiment (CREX) [1] is expectedto finish the data analysis on ∆ r (Ca) soon with a pre-cision of . (or ± . fm). Therefore, CREX can pro-vide a unique bridge between ab initio approaches anddensity functional theory (DFT). This is particularly im-portant as the DFT (e.g., eSHF) is still the only realisticframework to investigate the physics of heavy nuclei andNSs.The ∆ r (Zr) and ∆ r (Ru) are also very interestingsince a recent study [52] has demonstrated that the iso-baric Zr+ Zr and Ru+ Ru collisions at relativisticenergies can be used to extract the ∆ r np of Zr and Ru with a weak model-dependence. The ∆ r (Zr) and ∆ r (Ru) are also crucial for the chiral magnetic effectsearch in isobaric collisions [53]. Our present results of ∆ r (Zr) and ∆ r (Ru) are particularly timely, becausethe data on these isobaric collisions at RHIC have beentaken in 2018 and have been subject to a blinded analy-sis to assess the chiral magnetic effect. In addition, ourresults of ∆ r (I) and ∆ r (Cs) are critical for the in-formation extraction of new physics [7] via coherent elas-tic neutrino-nucleus scattering in the COHERENT ex-periment [5], while the ∆ r (Xe) is important for darkmatter direct detection in liquid Xe detector [54]. Conclusion. — We have demonstrated the symmetryenergy slope parameter L c cannot be larger than MeV,and this leads to an upper limit of ∆ r ≤ . fm. Thislimit together with the recent model-independent mea-surement on ∆ r from PREX-II leads to a rather largebut very precise constraint of . ≤ ∆ r ≤ . fm,which suggests a rather stiff symmetry energy around ρ and has critical implications on many issues in nuclearphysics and astrophysics. In particular, our present con-straints on the symmetry energy and the neutron skin of Ca reveal serious tension with the predictions from abinitio coupled-cluster theory, and the soon coming datafrom CREX thus become extremely important.
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