Bayesian inference of nuclear symmetry energy from measured and imagined neutron skin thickness in 116,118,120,122,124,130,132 Sn, 208 Pb, and 48 Ca
aa r X i v : . [ nu c l - t h ] S e p Bayesian inference of nuclear symmetry energy from measured and imagined neutronskin thickness in , , , , , , Sn,
Pb, and Ca Jun Xu ∗ ,
1, 2
Wen-Jie Xie † , and Bao-An Li ‡ Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Department of Physics, Yuncheng University, Yuncheng 044000, China Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, TX 75429, USA (Dated: September 30, 2020)The neutron skin thickness ∆ r np in heavy nuclei has been known as one of the most sensitiveterrestrial probes of the nuclear symmetry energy E sym ( ρ ) around of the saturation density ρ of nuclear matter. Existing neutron skin data mostly from hadronic observables suffer from largeuncertainties and their extraction from experiments are often strongly model dependent. Whilewaiting eagerly for the promised model-independent and high-precision neutron skin data for Pband Ca from the parity-violating electron scattering experiments (PREX-II and CREX at JLab aswell as MREX at MESA), within the Bayesian statistical framework using the Skyrme-Hartree-Fockmodel we infer the posterior probability distribution functions (PDFs) of the slope parameter L ofthe nuclear symmetry energy at ρ from imagined ∆ r np ( Pb) = 0 .
15, 0.20, and 0.30 fm with a 1 σ error bar of 0.02, 0.04, and 0.06 fm, respectively, as well as ∆ r np ( Ca) = 0 .
12, 0.15, and 0.25 fmwith a 1 σ error bar of 0.01 and 0.02 fm, respectively. The results are compared with the PDFs of L inferred using the same approach from the available ∆ r np data for , , , , , , Sn fromhadronic probes. They are also compared with results from a recent Bayesian analysis of the radiusand tidal deformability data of canonical neutron stars from GW170817 and NICER. The neutronskin data for Sn isotopes gives L = 45 . +26 . − . MeV surrounding its mean value or L = 53 . +18 . − . MeVsurrounding its maximum a posteriori value, respectively, with the latter smaller than but consistentwith the L = 66 +12 − MeV from the neutron star data within their 68% confidence intervals. Wefound that ∆ r np = 0 . − .
18 fm in
Pb with an error bar of about 0.02 fm leads to a PDF of L compatible with that from analyzing the Sn data. In order to provide additionally useful informationon L extracted from the ∆ r np of Sn isotopes, the experimental error bar of ∆ r np in Pb shouldbe at least smaller than 0.06 fm aimed by some current experiments. In addition, the ∆ r np ( Ca)needs to be larger than 0.15 fm but smaller than 0.25 fm to be compatible with the Sn and/orneutron star results. To further improve our current knowledge about L and distinguish its PDFsin the examples considered, even higher precisions leading to significantly less than ±
20 MeV errorbars for L at 68% confidence level are necessary. I. INTRODUCTION
Nuclear symmetry energy E sym ( ρ ) encodes the infor-mation about the energy necessary to make nuclear sys-tems, such as nuclei, neutron stars, and matter createdduring collisions of two nuclei or neutron stars, more neu-tron rich [1]. As such, reliable knowledge about the sym-metry energy has broad impacts on many critical issuesin both nuclear physics and astrophysics [2–4]. Thanksto the great efforts of many people in both communi-ties over the last two decades, much progress has beenmade in constraining both the magnitude E sym ( ρ ) andthe slope parameter L = 3 ρ ( dE sym /dρ ) ρ at the sat-uration density ρ of nuclear matter [5–12]. For exam-ple, fiducial values of E sym ( ρ ) = 31 . ± . L = 58 . ± . ∗ [email protected] † [email protected] ‡ [email protected] tions. In comparison, using a novel Bayesian approachto quantify the truncation errors in chiral effective fieldtheory (EFT) predictions for pure neutron matter and amany-body perturbation theory with consistent nucleon-nucleon and three-nucleon interactions up to fourth or-der in the EFT expansion, the E sym ( ρ ) and L werefound very recently to be E sym ( ρ ) = 31 . ± . L = 59 . ± . L = 66 +12 − MeVat 68% confidence level was found [16] while the E sym ( ρ )remains the same as the fiducial value. Clearly, these re-sults are all highly consistent while the error bars fromthe data analyses are significantly larger than the EFTpredictions. One of the possible reasons for the largererror bars of the extracted L values is that the extrac-tion of the symmetry energy from terrestrial experimentsoften involves large and sometimes unqualifiable theoret-ical uncertainties. Moreover, exiting experimental dataare mostly from hadronic probes that are known to sufferfrom large statistical and systematical errors. Thus, inthe continuous strive to better constrain the density de-pendence of nuclear symmetry energy, significant effortsare being made in the nuclear physics community to bet-ter quantify theoretical uncertainties and/or to find moreclean experimental probes, see, e.g., Refs. [17, 18].The neutron skin thickness ∆ r np = R n − R p is thedifference in root-mean-square neutron R n and proton R p radii. The ∆ r np values of heavy nuclei have beenknown as one of the most sensitive terrestrial probes ofthe nuclear symmetry energy E sym ( ρ ) at subsaturationdensities around ρ , see, e.g., Refs. [19–27]. For re-cent reviews, we refer the readers to Refs. [28, 29]. Ithas been shown using various nuclear many-body theo-ries that the ∆ r np is approximately proportional to thedensity slope within theoretical uncertainties, see, e.g.,Refs. [30, 31] for reviews. In fact, considerable ef-forts have been devoted continuously to measuring the∆ r np in Pb for decades [32]. For earlier reviews,see, e.g., Refs. [33, 34]. More recently, for example,∆ r np = 0 . +0 . − . fm and 0 . ± .
07 fm were obtainedfrom proton [35] and pion [36] scatterings, respectively.Studies from the annihilation of antiprotons on the nu-clear surface gave ∆ r np = 0 . ± . . ) ± . . )fm [37, 38], while the isospin diffusion data in heavy-ioncollisions imply ∆ r np to be around 0 . ± .
04 fm [39, 40],and ∆ r np = 0 . ± . . ) +0 . − . (sys . ) fm was ob-tained from coherent pion photoproductions [41]. Ob-viously, both the mean and error bar of ∆ r np ( Pb)are not well determined. Consequently, in studying im-pacts of ∆ r np ( Pb) on neutron stars, sometimes a fidu-cial value of ∆ r np ( Pb) = 0 . ± .
04 fm was used[2, 42]. Among the available data for heavy nuclei, the∆ r np of Sn isotopes have been most extensively mea-sured using isovector spin-dipole resonances excited bythe charge-exchange reactions [43], antiproton annihila-tions [44], and proton elastic scatterings [45], etc. We willtherefore first use the measured ∆ r np values of Sn iso-topes to establish a reference PDF for L in our Bayesiananalyses, and compare the results with the informationfrom a traditional approach using forward-modeling with χ minimization. This reference serves as a quantita-tive measure of our current knowledge about inferringthe L value using available neutron skin data. We willthen measure possible improvements to this knowledgeby using anticipated high-precision neutron skin datafor ∆ r np ( Pb) and ∆ r np ( Ca) from parity-violatingelectron-nucleus scattering experiments.While most of the available neutron skin data fromhadronic probes suffer from large statistical and system-atic errors as well as model dependence, parity-violatingelectron scatterings were shown theoretically to providemodel-independent and high-precision measures of neu-tron skin thickness [46, 47]. However, these experi-ments are extremely difficult. While the pioneering Lead(
Pb) Radius EXperiment (PREX) at the JefersonLaboratory (JLab), i.e., PREX-I experiment, has demon-strated an excellent control of systematic errors, the re-sulting ∆ r np ( Pb) = 0 . +0 . − . fm still has a large errorbar [48]. The PREX-II experiment and the Calcium Ra-dius EXperiment (CREX) at Jlab are expected to dra- matically reduce the error bars to the level of ± .
06 fmfor
Pb and ± .
02 fm for Ca, respectively [28]. Evenbetter, the planned Mainz Radius EXperiment (MREX)at the Mainz Energy recovery Superconducting Acceler-ator (MESA) aims to determine the neutron radius in
Pb with a 0.5% (or 0.03 fm) precision; while for Cathe sensitivity is similar to the one expected from theCREX at JLab [28]. If realized, these experiments mayimprove dramatically our knowledge about the nuclearsymmetry energy and help constrain tightly nuclear the-ories.Wishing the experimentalists all the best luck in theworld and eagerly waiting for their new results from theparity-violating electron scattering experiments, hintedby existing results and the planned experiments, weimagine a few mean values and error bars for the neu-tron skin thickness in
Pb and Ca in our Bayesianinference of the symmetry energy slope parameter L .We compare the resulting PDFs of L with those fromBayesian analyses of neutron star observations and theneutron skin thickness in Sn isotopes. Following the spiritof a recent work conducting covariant analysis to obtainanalytic insights on the information content of new ob-servables [49], we also try to answer the two questionsposted by Reinhard and Nazarewicz [50]: (1) Consider-ing the current theoretical knowledge, what novel infor-mation does new measurements bring in? and (2)
Howcan new data reduce the uncertainties of current theo-retical models?
More specifically, we study (1)
How theuncertainties of the neutron skin measurements affect theextraction of the symmetry energy? and (2)
What addi-tional information about the symmetry energy can newmeasurements bring to us?
In order to address thesequestions, besides comparing with results from Bayesiananalyses of the very recent data from neutron star obser-vations and the old neutron skin data of Sn isotopes, wefreely dreamed that the experimentalists would some daymeasure the ∆ r np ( Pb) and ∆ r np ( Ca) at precisionseven better than they already planned at Jlab and/orMESA. We understand that these will be extremely chal-lenging, but we assume that they are not more difficultthan measuring nuclear matter effects on the strain am-plitude and frequency of gravitational waves from merg-ing neutron stars.The rest of the paper is arranged as follows. We shallfirst summarize in Section II the most relevant aspectsof the standard Skyrme-Hartree-Fock (SHF) model andinteractions we use, and then recall the main formalismsand prior information we use in the Bayesian analyses.In Section III we present and discuss our results. Thesummary and conclusions are given in Section IV.
II. THEORETICAL FRAMEWORK
Within the Bayesian statistical framework we inferfrom the neutron skin thickness data the posterior PDFsof isovector nuclear interactions used in the standardSHF model. These isovector interactions determine thedensity dependence of nuclear symmetry energy, whilethe isoscalar parameters are fixed at their currentlyknown most probable values. Consequently, the poste-rior PDF of the symmetry energy slope parameter L canbe obtained. For completeness and ease of discussions, wesummarize in the following the most important aspectsof the SHF model and the Bayesian approach as well asthe specific inputs used in this work. We skip most ofthe details that can be found easily in the literature. A. Skyrme-Hartree-Fock model
We start from the following standard effective Skyrmeinteraction between nucleon 1 and nucleon 2 [51] v ( ~r , ~r ) = t (1 + x P σ ) δ ( ~r )+ 12 t (1 + x P σ )[ ~k ′ δ ( ~r ) + δ ( ~r ) ~k ]+ t (1 + x P σ ) ~k ′ · δ ( ~r ) ~k + 16 t (1 + x P σ ) ρ α ( ~R ) δ ( ~r )+ iW ( ~σ + ~σ )[ ~k ′ × δ ( ~r ) ~k ] . (1)In the above, ~r = ~r − ~r and ~R = ( ~r + ~r ) / ~r and ~r , ~k = ( ∇ −∇ ) / i is the relative momentum operator and ~k ′ is its complexconjugate acting on the left, and P σ = (1 + ~σ · ~σ ) / ~σ being the Paulimatrics. The parameters t , t , t , t , x , x , x , x , and α determine macroscopic quantities describing the sat-uration properties of symmetric nuclear matter, densitydependence of nuclear symmetry energy, and structuresof finite nuclei. Inversely, they can be expressed analyt-ically in terms of several macroscopic quantities, facili-tating the Bayesian inference of the latter directly fromthe neutron skin data. In this work, we use the MSL0interaction [52]. Specifically, the macroscopic quantitiesused are: the saturation density ρ , the binding energy E at the saturation density, the incompressibility K ,the isoscalar and isovector nucleon effective mass m ⋆s and m ⋆v at the Fermi momentum in normal nuclear matter,the symmetry energy E sym ≡ E sym ( ρ ) and its slope pa-rameter L at the saturation density, and the isoscalarand isovector density gradient coefficient G S and G V .The spin-orbit coupling constant is fixed at W = 133 . . In the present study, we calculate the posteriorPDFs of the isovector interaction parameters, i.e., E sym , L , and m ⋆v , by varying them randomly with equal prob-ability within their respective prior ranges, while fixingthe other macroscopic quantities at their empirical valuesas in the original MSL0 interaction [52].The potential energy density can be calculated fromthe above effective interaction [Eq. (1)] based on theHartree-Fock method, and the single-particle Hamito-nian can then be obtained using the variational princi-ple. Here we assume that the nucleus is spherical and only time-even contributions are considered. Solving theSchr¨odinger equation leads to the wave functions of eachnucleon, and the density distributions for neutrons andprotons can be calculated accordingly. The neutron skinthickness can then be obtained from the difference of theroot-mean-square radii between neutrons and protons.For details of this standard procedure, we refer the readerto Ref. [53]. In the present work, we use Reinhard’s SHFcode described in Ref. [54]. B. Bayesian analysis
Compared to the traditional approach of forward-modeling together with a χ minimization to fit the ex-perimental data and empirical properties of nuclear mat-ter, the advantages of Bayesian analysis in the uncer-tainty quantification and evaluating correlations of modelparameters have been well documented in the literature,see, e.g. Ref. [55] for a very recent overview of theBayesian approach and its applications in studying nu-clear structures. We adopt it here to infer the posteriorPDFs of the isovector interaction parameters and the cor-responding nuclear symmetry energy from the neutronskin data. The Bayes’ theorem describes how new exper-imental data may improve a hypothesis reflecting priorknowledge via P ( M | D ) = P ( D | M ) P ( M ) R P ( D | M ) P ( M ) dM . (2)In the above, P ( M | D ) is the posterior PDF for the model M given the data set D , P ( D | M ) is the likelihood func-tion or the conditional probability for a given theoreticalmodel M to predict correctly the data D , and P ( M )denotes the prior PDF of the model M before being con-fronted with the data. The denominator of the right-hand side of the above equation is the normalization con-stant.For the prior PDFs, we choose the model parameters p = E sym uniformly within 25 ∼
35 MeV, p = L uni-formly within 0 ∼
120 MeV, and p = m ⋆v /m uniformlywithin 0 . ∼
1, with m being the bare nucleon mass.Our choice of the large prior range and the uniform PDFfor the symmetry energy slope parameter L is intention-ally ignorant with respect to our current knowledge frommany earlier analyses of both terrestrial and astrophysi-cal data as well as the state-of-the-art EFT predictions aswe outlined in the introduction. Without belittling theinvaluable prior knowledge from the hard work of manypeople over two decades, this choice helps us reveal howthe neutron skin data alone may narrow down the uni-form prior PDF of L in the artificially enlarged range of0 ∼
120 MeV.For a given set of the MSL0 interaction parameters,the theoretical neutron skin thickness d th = ∆ r (1) np , d th =∆ r (2) np , ... for different nuclei from the SHF calculationsare used to calculate the likelihood of these model pa-rameters with respect to the corresponding experimentaldata d exp , d exp , ... according to P [ D ( d , ,... ) | M ( p , , )]= Π i √ πσ i exp (cid:20) − (∆ d i ) σ i (cid:21) , (3)where ∆ d i and σ i denote respectively the deviation oftheoretical results from the experimental data and thewidth in the likelihood function from an independent ex-perimental data sample i . In principle, the likelihoodfunction depends on uncertainties of both the experi-mental data and model predictions. For the neutron-skin thickness of Pb or Ca, we use the imaginedexperimental error bar (which is varied and could beconsidered as due to both experimental and model un-certainties) as the width σ i as often done in the liter-ature, and ∆ d i = | d thi − d expi | being the deviation ofthe theoretical result from the mean value of the imag-ined experimental data. For the neutron skin thicknessesof , , , , Sn [45] and , Sn [56], they aretreated as two independent experimental data sampleswith each series extracted from a correlated method. Thedeviation and the width for each series are calculated ac-cording to ∆ d i = vuut j ∈ i X j ( d thj − d expj ) , (4) σ i = vuut j ∈ i X j σ j , (5)where d thj , d expj , and σ j are the theoretical result, themean value of the experimental data, and the experi-mental 1 σ error bar of the neutron skin thickness for aSn isotope j , respectively.The posterior PDF of a single model parameter p i isgiven by P ( p i | D ) = R P ( D | M ) P ( M )Π j = i dp j R P ( D | M ) P ( M )Π j dp j , (6)while the correlated PDF of two model parameters p i and p j is given by P [( p i , p j ) | D ] = R P ( D | M ) P ( M )Π k = i,j dp k R P ( D | M ) P ( M )Π k dp k . (7)For the one-dimensional PDF, the range of the model pa-rameter at the 68% confidence level is obtained accordingto [57] Z p iU p iL P ( p i | D ) dp i = 0 . , (8)where p iL ( p iU ) is the lower (upper) limit of the cor-responding narrowest interval of the parameter p i sur-rounding its mean value h p i i = Z p i P ( p i | D ) dp i (9) or its maximum a posteriori (MAP) value. The calcula-tion of the posterior PDFs is based on the Markov-ChainMonte Carlo approach using the Metropolis-Hastings al-gorithm [58, 59]. The calculation generally takes about10 − steps, and the analysis is carried out after thefirst 10 steps when the convergence is mostly reached. III. RESULTS AND DISCUSSIONSA. L from measured neutron skin thickness in , , , , , , Sn To establish a reference for comparisons, we firstperform Bayesian analyses with the real experimentaldata of neutron skin thicknesses ∆ r np in Sn isotopes.It is one of the most complete ∆ r np data sets alongthe longest isotope chain available. The ∆ r np data of , , , , Sn and , Sn as two independent ex-perimental data samples from Refs. [45, 56] are listed inTable I. The mean values and the experimental 1 σ errorstogether with the theoretical results are used in calculat-ing the likelihood function according to Eqs. (3), (4), and(5).After integrating one of the isovector model parame-ters L , m ⋆v /m , or E sym according to Eq. (7), the resultingcorrelated PDFs of the other two parameters are shownin the upper panels of Fig. 1. It is seen that the L pa-rameter is strongly correlated with the isovector effectivemass m ⋆v /m , with the latter weakly correlated with the E sym within their prior ranges considered, due to the de-compositions of the L and E sym parameters [60] accord-ing to the Hugenholtz-Van Hove theorem [61], see, theextensive review in Ref. [12]. The anti-correlated PDFin the L − E sym plane is similar to the L − E sym correla-tion observed in Fig. 6 of Ref. [52], where the traditional χ fit was performed using the same MSL0 interactionwithin SHF to the empirical properties of nuclear matterand some properties of finite nuclei as well as the same setof the neutron skin thickness data of Sn isotopes. Thisconsistency is what one expects. However, the Bayesiananalysis can go beyond what the traditional analysis canprovide. The posterior PDFs of each model parameterafter integrating all the others according to Eq. (6) areshown in the lower panels of Fig. 1, where the ranges of L and E sym from fiducial values [13, 14], from the EFTanalysis [15], and from the neutron star analysis [16] arealso plotted for comparisons. It is seen that with theneutron skin thickness data of Sn isotopes, the uniformprior distribution of L within (0 , m ⋆v /m and E sym are not improved by much com-pared to their prior PDFs. More quantitatively, the L is determined to be within (23 . , .
0) MeV around themean value 45 . r np data of Sn iso-topes. Due to the asymmetric PDF of L , the mean valueis smaller than the MAP value, with the latter consistent L (MeV) m * v / m (a)
26 28 30 32 340.50.60.70.80.91.0 (b)E (MeV) m * v / m (c)E (MeV) L ( M e V ) Snm *v /m (d) EFT neutron star (e)
SnL (MeV)
EFT (f)
SnE (MeV)
FIG. 1: (Color online) Upper: Correlated posterior PDFs from neutron skin thicknesses in Sn isotopes in the L − m ⋆v /m plane(a), the E sym − m ⋆v /m plane (b), and the E sym − L plane (c); Lower: Prior (dotted lines) and posterior (solid lines) probabilitydistributions of m ⋆v /m (d), L (e), and E sym (f), with the band of fiducial values [13, 14], the 68% confidence band from theEFT analysis [15], and the PDF from the neutron star analysis [16] (dashed line) plotted for comparisons. with the fiducial value L = 58 . ± . L = 59 . ± . L = 66 +12 − MeV from the neutron star analysis [16].The anti-correlation between the E sym and L inFig. 1(c) deserves some discussions. As noticed before[25, 52, 60, 62], this correlation is opposite to the posi-tive correlation from studying nuclear giant resonances,heavy-ion collisions, and the electric dipole polarizabil-ity [62–64]. The overlapping area of these opposite cor-relations played a critical role in finding the common con-straints on the E sym − L plane [25, 60, 62, 65]. However,its origin needs further understanding. For this purpose,shown in the upper panels of Fig. 2 are the correlatedPDFs between the E sym and L ( ρ ⋆ ) = 3 ρ ⋆ ( dE sym /dρ ) ρ ⋆ at different subsaturation densities ρ ⋆ using the sameBayesian analysis method. Here the L ( ρ ⋆ ) calculated at ρ ⋆ from the same density dependence of E sym ( ρ ) dependson the SHF parameters in the same way as the E sym and L at ρ . They are thus all correlated. It is interestingto see that at the density ρ ⋆ smaller (larger) than 0.10fm − the L ( ρ ⋆ ) and E sym are positively correlated (anti-correlated), while at ρ ⋆ = 0 .
10 fm − the PDF of L ( ρ ⋆ )is approximately independent of E sym . Moreover, it isseen that at the 68% confidence level, L ( ρ ⋆ = 0 .
10 fm − )is tightly constrained to 43 . +5 . − . MeV with an symmet-ric distribution, while the PDFs of L ( ρ ⋆ ) are generallybroader especially at higher densities. This shows that the neutron skin thicknesses in Sn isotopes determinesmost tightly the value of L ( ρ ⋆ ) around ρ ⋆ = 0 .
10 fm − ,which is approximated the average density of a nucleus.We note that this finding is robust for different nuclei,since the neutron skin thickness in Pb and Ca is alsofound to be mostly determined by L ( ρ ⋆ = 0 .
10 fm − ) aswell, and this is consistent with that observed in Ref. [25]within the traditional approach.So, why are the E sym and L at ρ anti-correlated?As shown in the upper panels of Fig. 2, there is a cleartendency that their correlation changes from positive tonegative as the density increases towards ρ . One canunderstand these numerical results by analytically inves-tigating how the E sym and L at ρ are correlated when aconstraint is applied to the function E sym ( ρ ) at a subsat-uration density ρ ∗ . In the Appendix A, using a generalform of the symmetry energy E sym ( ρ ) = E sym · (cid:16) ρρ (cid:17) γ describing those predicted by SHF very well, we haveshown analytically that the E sym and L at ρ are pos-itively correlated if the observable used constrains themagnitude of E sym ( ρ ∗ ), while a negative correlation ap-pears if the observable constrains the L ( ρ ∗ ) at ρ ∗ . In thesituation here, the neutron skin thickness constrained the L ( ρ ∗ ) but not E sym ( ρ ∗ ) around ρ ⋆ = 0 .
10 fm − . Conse-quently, the neutron skin constraint leads to a negativecorrelation between the E sym and L at ρ . We havealso noticed that the strength of the anti-correlation be-
26 28 30 32 342030405060 E (MeV) L ( * = . f m - ) ( M e V ) (a)
26 28 30 32 342030405060 E (MeV) L ( * = . f m - ) ( M e V ) (b)
26 28 30 32 342030405060 E (MeV) L ( * = . f m - ) ( M e V ) (c) SnL ( *=0.08 fm -3 ) (MeV) (d) 0 20 40 60 80 100 1200.000.050.10 SnL ( *=0.10 fm -3 ) (MeV) (e) 0 20 40 60 80 100 1200.000.050.10 SnL ( *=0.12 fm -3 ) (MeV) (f) FIG. 2: (Color online) Upper: Correlated posterior PDFs from neutron skin thicknesses in Sn isotopes in the E sym − L ( ρ ⋆ )plane for ρ ⋆ = 0 .
08, 0.10, and 0.12 fm − ; Lower: Prior (dotted lines) and posterior (solid lines) probability distributions of L ( ρ ⋆ ). tween L ( ρ ⋆ ) ρ ⋆ = 0 .
12 fm − and E sym is much weakercompared with that at smaller ρ ⋆ . This indicates the dif-ficulty of constraining the symmetry energy at the sat-uration density using the neutron skin data. Basically,the latter determines the slope of E sym at 0.10 fm − ,while the information about the E sym at higher densitiesis from extrapolating the underlying energy density func-tional. Thus, while the neutron skin data may be modelindependent and very precise, the extraction of E sym or L at ρ from the neutron skin data is also model depen-dent. The correlation between the neutron-skin thicknessof nuclei and the radii of neutron stars is even weakerand very model dependent as demonstrated numericallyalready in Ref. [29]. Here we used the SHF functionalin our Bayesian analysis, and it would be interesting tostudy in the future with other models.It is interesting to note that in a recent Bayesian anal-ysis [64] using the centroid energy E − of the isovectorgiant dipole resonance in Pb as well as its electric po-larizability α D , it was found that these data determinethe nuclear symmetry energy E sym at about ρ ⋆ = 0 . − and the isovector nucleon effective mass m ⋆v at ρ .At 90% confidence level, E sym ( ρ ⋆ ) = 16 . +1 . − . MeV and m ⋆v /m = 0 . +0 . − . were obtained around their meanvalues. Compared to what we have learned from theBayesian analysis of neutron skin thicknesses of Sn iso-topes, the results are complimentary for mapping out thedensity dependence of nuclear symmetry energy while their difference is completely understandable. Specifi-cally, the neutron skin thickness is mostly dominated bythe neutron pressure related to L [19, 22], while the giantresonances are affected by both the restoring force fromthe EOS and the nucleon effective mass [64, 66, 67]. r np =0.15 0.02 0.04 fm 0.06 Pb (c)(b) L (MeV) r np =0.20 0.02 0.04 fm 0.06 Pb (a) r np =0.30 0.02 0.04 fm 0.06 Pb FIG. 3: (Color online) Prior (dotted lines) and posterior prob-ability distributions of L from the imagined neutron skinthickness 0.15 (a), 0.20 (b), and 0.30 (c) fm in Pb withdifferent error bars. B. L from imagined neutron skin thickness in Pb As discussed in the introduction, we want to know ifand how new measurements can improve our knowledge
TABLE I: The slope parameter L of the symmetry energyat 68% confidence level from real and imagined neutron skinthickness data of various nuclei used in this study, with themean values calculated according to Eq. (9), the MAP valuesfrom fitting the peaks of the PDFs using a Gaussian function,and the confidence intervals obtained by using Eq. (8).Nucleus ∆ r np (fm) L (mean) (MeV) L (MAP) (MeV) Sn 0 . ± . Sn 0 . ± . Sn 0 . ± . Sn 0 . ± .
016 45 . +26 . − . . +18 . − . Sn 0 . ± . Sn 0 . ± . Sn 0 . ± . Pb 0 . ± .
02 35 . +19 . − . . +19 . − . Pb 0 . ± .
04 42 . +18 . − . . +24 . − . Pb 0 . ± .
06 48 . +16 . − . . +28 . − . Pb 0 . ± .
02 65 . +26 . − . . +16 . − . Pb 0 . ± .
04 64 . +36 . − . . +25 . − . Pb 0 . ± .
06 63 . +41 . − . . +29 . − . Pb 0 . ± .
02 112 . +7 . − . . +0 . − . Pb 0 . ± .
04 102 . +17 . − . . +0 . − . Pb 0 . ± .
06 91 . +29 . − . . +0 . − . Ca 0 . ± .
01 14 . +3 . − . . +18 . − . Ca 0 . ± .
02 23 . +6 . − . . +29 . − . Ca 0 . ± .
01 30 . +9 . − . . +24 . − . Ca 0 . ± .
02 37 . +10 . − . . +27 . − . Ca 0 . ± .
01 114 . +5 . − . . +0 . − . Ca 0 . ± .
02 106 . +14 . − . . +0 . − . Pb and 0 . ± . . +23 . − . . +19 . − . Pb and 0 . ± . . +25 . − . . +18 . − . Pb and 0 . ± . . +25 . − . . +18 . − . Pb and 0 . ± . . +24 . − . . +16 . − . Pb and 0 . ± . . +26 . − . . +18 . − . Pb and 0 . ± . . +26 . − . . +18 . − . Pb 0 . ± . Ca 0 . ± .
01 24 . +13 . − . . +12 . − . Pb 0 . ± . Ca 0 . ± .
01 32 . +19 . − . . +11 . − . Pb 0 . ± . Ca 0 . ± .
01 37 . +22 . − . . +14 . − . Pb 0 . ± . Ca 0 . ± .
01 47 . +26 . − . . +13 . − . about the symmetry energy, especially its slope parame-ter L at ρ with respect to what we learned from analyz-ing the Sn isotopes and neutron star data. Since the neu-tron skin thickness in Pb is still not well determined,to illustrate how the uncertainties of ∆ r np in Pb mayaffect the extraction of L , we display in Fig. 3 its poste-rior PDFs by using the imagined neutron skin thicknessdata of ∆ r np = 0 .
15, 0.20, and 0.30 fm with different er-ror bars. As one expects, a larger neutron skin thicknessgenerally leads to a larger value of L . With ∆ r np = 0 . L would peak outside the prior range of (0 , L . The width of the PDF actually depends on therelative error bar of the experimental data, i.e., a smallerwidth in PDF is obtained with a larger mean value ofthe experimental data for the same absolute 1 σ error baras one expects. The L values at 68% confidence levelaround the mean values and the MAP values from thereal and imagined neutron skin thickness data of variousnuclei used in this study are compared in Table I.What further information on L can the measurement of∆ r np in Pb bring to us, in additional to our knowledgefrom analyzing the Sn isotopes? To answer this ques-tion, we compare in Fig. 4 the PDFs of L from using theimagined ∆ r np = 0 .
15 (0.20) fm data of
Pb with dif-ferent error bars, the measured ∆ r np data of Sn isotopes,and the combined data, respectively. Since ∆ r np = 0 . Pb leads to smaller (larger) L values com-pared to that extracted from the ∆ r np data of Sn iso-topes, the PDF of L from the combined data is shiftedand peaks at a smaller (larger) value. We found that a∆ r np = 0 . − .
18 fm of
Pb with an error bar of about0.02 fm leads to a PDF of L compatible with that fromanalyzing the Sn data. On the other hand, it is shownthat larger error bars of ∆ r np ( Pb) weaken the effectsof incorporating the
Pb data into the Bayesian analy-sis with the combined data. Quantitatively, an error baras large as 0.06 fm for ∆ r np ( Pb) leads to negligibleimprovements of the posterior PDF of L extracted fromthe ∆ r np of Sn isotopes. C. L from imagined neutron skin thickness in Ca An ab initio calculation in Ref. [68] has predicted thatthe neutron skin thickness in Ca is about 0 . − .
15 fm,while it is predicted to be about 0.25 fm from a nonlocaldispersive optical-model analysis [69]. Accordingly, herewe consider three cases of ∆ r np = 0 .
12, 0 .
15, and 0.25fm with an 1 σ error bar of 0.01 and 0.02 fm, respectively.The resulting PDFs of L are displayed in Fig. 5 with theprior range of (0 , r np = 0 .
12 or 0.25fm, the posterior PDF of L would peak out of the priorrange of (0 , L from earlier analyses [13, 14]. r np =0.15 0.02 fm in Pb r np in Sn isotopes r np in Pb&Sn isotopes (a) r np =0.20 0.02 fm in Pb r np in Sn isotopes r np in Pb&Sn isotopes (d) r np =0.15 0.04 fm in Pb r np in Sn isotopes r np in Pb&Sn isotopes (c) r np =0.20 0.04 fm in Pb r np in Sn isotopes r np in Pb&Sn isotopes (f)
L (MeV) r np =0.15 0.06 fm in Pb r np in Sn isotopes r np in Pb&Sn isotopes (e) r np =0.20 0.06 fm in Pb r np in Sn isotopes r np in Pb&Sn isotopes
FIG. 4: (Color online) Posterior probability distributions of L from imagined neutron skin thicknesses with different meanvalues and error bars in Pb (dashed lines), from real neu-tron skin thickness data of Sn isotopes (dot-dashed lines), aswell as from their combinations (solid lines). Ca r np =0.12 0.01 fm 0.02 fm Ca(b)
L (MeV) r np =0.15 0.01 fm 0.02 fm Ca(a) (c) r np =0.25 0.01 fm 0.02 fm FIG. 5: (Color online) Prior (dotted lines) and posterior prob-ability distributions of L from imagined neutron skin thick-nesses 0.12 (a), 0.15 (b), and 0.25 (c) fm in Ca with differenterror bars.
To compare the posterior PDFs of L from analyzingthe ∆ r np in Ca with those in the case of
Pb, oneneeds to compare results with approximately the samerelative values of both the neutron skin thicknesses andtheir error bars with respect to the radii of the two nuclei.Comparing the fractions ∆ r np /R of Ca and
Pb, theradius R of Ca is about 4.3 fm, while that of
Pbis about 7.0 fm. For the same ∆ r np = 0 .
15 fm, it isabout 3 .
5% of the radius for Ca but only 2% for
Pb. r np =0.15 0.02 fm in Pb r np =0.12 0.01 fm in Ca r np in Pb& Ca (d)(c) (b) r np =0.15 0.02 fm in Pb r np =0.15 0.01 fm in Ca r np in Pb& Ca (a) L (MeV) r np =0.20 0.02 fm in Pb r np =0.12 0.01 fm in Ca r np in Pb& Ca r np =0.20 0.02 fm in Pb r np =0.15 0.01 fm in Ca r np in Pb& Ca FIG. 6: (Color online) Posterior probability distributions of L from imagined neutron skin thicknesses in Pb (dashedlines), imagined neutron skin thicknesses in Ca (dot-dashedlines), as well as from their different combinations (solid lines).
For the same reason, with the same imagined 0 .
02 fmabsolute error bar the relative error for the neutron skinthickness is actually larger for Ca than for
Pb.What further information on L can the new neutronskin thickness measurement of Ca bring to us? To an-swer this question, we have done Bayesian analyses byusing both the imagined experimental data for
Pb and Ca. The resulting posterior PDFs of L from differentcombinations of ∆ r np in Pb and Ca are shown inFig. 6. Due to the different constraints on L from ∆ r np in Pb and Ca, it is seen that the posterior PDFs of L indicated by the solid lines are in-between those fromtwo separate analyses, with the dashed lines from onlythe ∆ r np in Pb and dot-dashed lines from only the∆ r np in Ca, respectively. The corresponding L valuesat 68% confidence level around the mean values and theMAP values are listed in Table I. Again, the final PDFsalso depend on the 1 σ error bar of ∆ r np . Using a larger1 σ error bar for the ∆ r np in Pb or Ca, the corre-sponding PDF of L becomes broader and less important,and the posterior PDF of L from the combined ∆ r np datais closer to the one with a smaller error bar. Our resultsindicate that it is better to analyze the Pb and Cadata separately, then compare the L values extracted, in-stead of combining the data and extracting a common L .This is because the two nuclei have very different chargeradii. Coulomb and other dynamical effects in the twonuclei may be very different unlike the neutron skins inisotope chains having the same charge. IV. SUMMARY AND CONCLUSIONS
In summary, within the Bayesian statistical frameworkusing both real and imagined neutron skin thickness datain heavy and medium nuclei, we have investigated howthe available and expected data may help improve ourknowledge about the density dependence of nuclear sym-metry energy, especially its slope parameter L at thesaturation density of nuclear matter. Using the avail-able data for Sn isotopes, we have not only extractedthe posterior PDF of L parameter as a useful referencefor future studies with new data of high precisions fromparity-violating electron scattering experiments, but alsofound the density region in which the neutron skin datais most sensitive to the variation of symmetry energy.We also demonstrated numerically and explained ana-lytically why the magnitude and the slope parameter ofsymmetry energy at ρ are anti-correlated when the ex-perimental constraint on the neutron skin thickness isapplied. Moreover, we compared the L values extractedfrom the Bayesian analyses of the neutron skin data inSn isotopes and observations of neutron stars. They arelargely compatible within their 68% confidence intervals.Furthermore, we found that a neutron skin of the size∆ r np = 0 . − .
18 fm in
Pb with an error bar ofabout 0.02 fm leads to a PDF of L compatible withthat from analyzing the Sn neutron skin data, while the∆ r np ( Pb) = 0 .
30 fm regardless of its error bar leadsto a posterior PDF of L largely incompatible with theresults from analyzing neither the neutron star data northe neutron skin data of Sn isotopes. In order to providenew information on L compared to our current knowledgeabout it, the experimental error bar of ∆ r np in Pbshould be at least smaller than 0.06 fm aimed by somecurrent experiments. On the other hand, the ∆ r np ( Ca)needs to be larger than 0.15 fm but smaller than 0.25 fmfor the extracted PDF of L to be compatible with theSn and/or neutron star results. To further improve ourcurrent knowledge about L and distinguish its posteriorPDFs in the examples considered in this work, betterprecisions of measurements leading to significantly lessthan ±
20 MeV error bars for L at 68% confidence levelare necessary. Acknowledgments
JX acknowledges the National Natural Science Foun-dation of China under Grant No. 11922514. WJX ac-knowledges the National Natural Science Foundation ofChina under Grant No. 11505150. BAL acknowledgesthe U.S. Department of Energy, Office of Science, underAward Number DE-SC0013702, the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei)under the US Department of Energy Grant No. DE-SC0009971.
Appendix A: Intuitive discussions on the correlationbetween L and E sym Here we discuss intuitively the correlation between thesymmetry energy E sym at the saturation density and the slope parameter L of the symmetry energy at the sat-uration density. We will show that their positive cor-relation means that the observable is dominated by thesymmetry energy at a subsaturation density, while theirnegative correlation means that the observable is domi-nated by the slope parameter of the symmetry energy ata subsaturation density.We illustrate the idea with a popularly used symmetryenergy of the following form E sym ( ρ ) = E sym · (cid:18) ρρ (cid:19) γ . (A1)Thus, the slope parameter L of the symmetry energy canbe expressed as L = 3 ρ (cid:20) dE sym ( ρ ) dρ (cid:21) ρ = 3 E sym γ. (A2)For a fixed symmetry energy at a subsaturation density ρ ⋆ E sym ( ρ ⋆ ) = E sym (cid:18) ρ ⋆ ρ (cid:19) γ , (A3)the expression of L in terms of E sym is L = 3 E sym ( ρ ⋆ ) " E sym E sym ( ρ ⋆ ) ln[ E sym /E sym ( ρ ⋆ )]ln( ρ /ρ ⋆ ) . (A4)It is obviously seen that L increases with increasing E sym (see Ref. [64] as an example). The slope parameter at ρ ⋆ can be expressed as L ( ρ ⋆ ) = 3 ρ ⋆ (cid:20) dE sym ( ρ ) dρ (cid:21) ρ ⋆ = L (cid:18) ρ ⋆ ρ (cid:19) γ , (5)where L ( ρ ⋆ ) is seen to be smaller than L . For a fixed L ( ρ ⋆ ), the expression of E sym in terms of L is E sym = L ( ρ ⋆ )3 ln( ρ ⋆ /ρ )[ L ( ρ ⋆ ) /L ] ln[ L ( ρ ⋆ ) /L ] . (6)The function x ln( x ) is negative for x < x for x > .
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