Constraining isovector nuclear interactions with giant resonances within a Bayesian approach
aa r X i v : . [ nu c l - t h ] J un Constraining isovector nuclear interactions with giant resonances within a Bayesianapproach
Jun Xu ∗ ,
1, 2
Jia Zhou,
2, 3
Zhen Zhang, 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 University of Chinese Academy of Sciences, Beijing 100049, China Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China Department of Physics, Yuncheng University, Yuncheng 044000, China Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, TX 75429, USA (Dated: June 11, 2020)We put a stringent constraint on the isovector nuclear interactions in the Skyrme-Hartree-Fockmodel from the centroid energy E − of the isovector giant dipole resonance in Pb as well as itselectric polarizability α D . Using the Bayesian analysis method, E − and α D are found to be mostlydetermined by the nuclear symmetry energy E sym at about ρ ⋆ = 0 .
05 fm − and the isovector nucleoneffective mass m ⋆v at the saturation density. At 90% confidence level, we obtain E sym ( ρ ⋆ ) = 16 . +1 . − . MeV and m ⋆v /m = 0 . +0 . − . . Understanding properties of nuclear interactions is oneof the main goals of nuclear physics. So far the uncer-tainties mainly exist in isovector channels of nuclear in-teractions, and they manifest themselves in the isospin-dependent part of the nuclear matter equation of state(EOS) and the single-nucleon potential. The isospin-dependent part of the EOS, i.e., the symmetry energy E sym , although still not well determined, is around 30MeV at the saturation density ρ , while its density de-pendence characterized by the slope parameter L =3 ρ ( dE sym /dρ ) ρ has recently been constrained within L = 58 . ± . m ⋆n − p ≡ ( m ⋆n − m ⋆p ) /m is even less constrained de-pending on the approaches used in the analysis (see, e.g.,Table 2 in Ref. [3]). Both the symmetry energy and theisospin splitting of the nucleon effective mass have sig-nificant ramifications in nuclear reactions, nuclear struc-tures, and nuclear astrophysics [3–7]. They are relatedto each other through the Hugenholtz-Van Hove theo-rem [1, 8], and most isospin tracers are sensitive to boththe E sym and the m ⋆n − p . This adds difficulties to ac-curately extracting the information of isovector nuclearinteractions, unless a proper analysis using multiple ob-servables is employed.Observables of finite nuclei serve as good probes of nu-clear interactions and nuclear matter properties at andbelow the saturation density. It has been found that use-ful information about isovector nuclear interactions can ∗ [email protected] be extracted from the isovector giant dipole resonance(IVGDR) [9–17], an oscillation mode in which neutronsand protons move collectively relative to each other. Thecentroid energy E − of the IVGDR is a good probe of the E sym around and below the saturation density [9, 14],while the product of the electric polarizability α D andthe E sym at the saturation density shows a good lineardependence on L [13, 15, 18]. Recently, it has been foundthat both the E − and the α D can be affected by m ⋆n − p as well [19, 20], once the isoscalar nucleon effective massis determined by the excitation energy of the isoscalargiant quadruple resonance (ISGQR) [19–26]. Since the E − and the α D are sensitive to both the E sym and the m ⋆n − p , we employ the Bayesian analysis as in Ref. [27]to extract E sym and m ⋆n − p as well as their correlations,with properties of giant resonances calculated from therandom-phase approximation (RPA) method based onthe Skyrme-Hartree-Fock (SHF) model [28].The standard SHF functional originating from the fol-lowing effective Skyrme interaction is used in the SHF-RPA calculation [28] 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 ) / t , t , t , t , x , x , x , x , and α can be solved inversely fromthe macroscopic quantities [29], i.e., the saturation den-sity ρ , the binding energy at the saturation density E ,the incompressibility K , the isoscalar and isovector nu-cleon effective mass m ⋆s and m ⋆v at the Fermi momentumin normal nuclear matter, the symmetry energy and itsslope parameter at the saturation density E sym and L ,and the isoscalar and isovector density gradient coeffi-cient G S and G V . The spin-orbit coupling constant isfixed at W = 133 . . In the present study, wedetermine the value of m ⋆s from the excitation energy ofthe ISGQR in Pb, and calculate the posterior proba-bility distribution functions (PDFs) of the isovector in-teraction parameters, i.e., E sym , L , and m ⋆v , through theBayesian analysis, while keeping the values of the othermacroscopic quantities the same as the empirical onesfrom the MSL0 interaction [29], since the E − and α D have been shown to be most sensitive to E sym , L , and m ⋆v [9, 13–15, 18–20, 30].The operators for the IVGDR and ISGQR are chosenrespectively asˆ F = NA Z X i =1 r i Y (ˆ r i ) − ZA N X i =1 r i Y (ˆ r i ) , (2)and ˆ F = A X i =1 r i Y (ˆ r i ) , (3)where N , Z , and A are respectively the neutron, proton,and nucleon numbers in a nucleus, r i is the coordinateof the i th nucleon with respect to the center-of-mass ofthe nucleus, and Y (ˆ r i ) and Y (ˆ r i ) are the sphericalBessel functions. The choice of the magnetic quantumnumber M is related to the parity, with the later being − S ( E ) = X ν |h ν || ˆ F J || ˜0 i| δ ( E − E ν ) (4)of a nucleus resonance can be obtained, where the squareof the reduced matrix element |h ν || ˆ F J || ˜0 i| represents thetransition probability from the ground state | ˜0 i to theexcited state | ν i . The moments of the strength functioncan then be calculated from m k = Z ∞ dEE k S ( E ) . (5)The centroid energy E − of the IVGDR and the electricpolarizability α D can be obtained from the moments ofthe strength function through the relation E − = p m /m − , (6) α D = 8 πe m − . (7)The moments are not used in the ISGQR analysis, sincethe excitation energy E x is the peak energy of thestrength function to be compared with the correspondingexperimental result. The Bayesian analysis is used to obtain the PDFs ofmodel parameters from the experimental data. SuchPDFs can be formally calculated from the Bayes’ the-orem P ( M | D ) = P ( D | M ) P ( M ) R P ( D | M ) P ( M ) dM . (8)In the above, P ( M | D ) is the posterior probability for themodel M given the data set D , P ( D | M ) is the likelihoodfunction or the conditional probability for a given the-oretical model M to predict correctly the data D , and P ( M ) denotes the prior probability of the model M be-fore being confronted with the data. The denominatorof the right-hand side of the above equation is the nor-malization constant. For the prior PDFs, we choose themodel parameters p = E sym uniformly within 25 ∼ p = L uniformly within 0 ∼
120 MeV, and p = m ⋆v /m uniformly within 0 . ∼
1, with m being thebare nucleon mass. The theoretical results of d th = E − and d th = α D from the SHF-RPA method are used tocalculate the likelihood of these model parameters withrespect to the corresponding experimental data d exp and d exp according to P [ D ( d , ) | M ( p , , )]= 12 πσ σ exp (cid:20) − ( d th − d exp ) σ − ( d th − d exp ) σ (cid:21) , (9)where σ , denote the widths of the likelihood function.The posterior PDF of a single model parameter p i is givenby 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 , (10)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 . (11)The calculation of the posterior PDFs is based on theMarkov-Chain Monte Carlo (MCMC) approach using theMetropolis-Hastings algorithm [31, 32]. Since the MCMCprocess does not start from an equilibrium distribution,initial samples in the so-called burn-in period have to bethrown away.The mean values of the experimentally measured exci-tation energy E x = 10 . Pbcan be reproduced by using m ⋆s /m = 0 .
83 approximatelyindependent of other macroscopic quantities, whose val-ues remain unchanged as those from the MSL0 interac-tion [29] for the IVGDR analysis. The small experimen-tal error bars of E x are neglected in the present study.For the given m ⋆s /m , the experimental results of the cen-troid energy E − = 13 .
46 MeV of the IVGDR from pho-toneutron scatterings [33], and the electric polarizability α D = 19 . ± . from polarized proton inelastic scat-terings [34] and with the quasi-deuteron excitation con-tribution subtracted [15], are used in the Bayesian analy-sis. Larger σ and σ values are used to evaluate the like-lihood function in the early stage of the MCMC process,in order to accelerate the convergence procedure, and itgradually decreases to the 1 σ error from the experimen-tal measurement, after which the results are analyzed.An artificial 1 σ error of 0 . E − value of the IVGDR is used in the analysis afterconvergence. step number < m * v / m > (a) (b) step number < L > ( M e V ) m *v /m L ( M e V ) L (MeV) P D F m *v /m (c) (d) (e) PDF
FIG. 1: (Color online) Upper: Mean values of m ⋆v /m (a) and L (b) as a function of the step number at a fixed E sym = 30MeV for 10 parallel runs; lower: The PDFs of m ⋆v /m (c) and L (d) as well as their correlations (e) at a fixed E sym = 30MeV. By fixing E sym = 30 MeV, we first study the posteriorPDFs of m ⋆v /m and L , and their mean values as a func-tion of the step number are plotted in the upper panelsof Fig. 1, for 10 parallel runs. It is seen that the con-vergence is generally reached after a few thousand steps.The PDFs are thus from analyzing the results after about2000 steps, and until 10000 steps there are totally about800 accepted data samples for each run. The posteriorPDFs of m ⋆v /m and L as well as their correlations areplotted in the lower panels of Fig. 1. It is seen that thePDF of m ⋆v /m peaks around 0.8, while that of L peaksaround 30 MeV. The anticorrelation between m ⋆v /m and L for a fixed E sym is observed. A narrower anticorrela-tion is expected to be observed by using a smaller artifi-cial 1 σ error for E − , but with the slope of the anticorre-lation unchanged. Although the L values from the aboveBayesian analysis at a fixed E sym = 30 MeV are smallcompared with the average ones extracted from variousapproaches in Refs. [1, 2], they are consistent with theIVGDR result in Ref. [9] (see also the ”GDR” band inFig. 1 of Ref. [35]).The uncertainties of E sym are expected to affect theextracted PDFs of m ⋆v /m and L as well as their corre-lations. The upper panels of Fig. 2 compare the corre-lations between m ⋆v /m and L at E sym = 28, 30, and 32 L ( M e V ) m *v /m E = 32 MeVE = 30 MeV E = 28 MeV PDF (a) (b) (c)(d) (e) (f) E sy m ( * ) ( M e V ) FIG. 2: (Color online) PDFs in the m ⋆v /m − L plane (upperpanels) and in the m ⋆v /m − E sym ( ρ ⋆ ) plane (lower panels) at E sym = 28, 30, and 32 MeV, with ρ ⋆ = 0 .
05 fm − . MeV, respectively. For a larger E sym , the PDF moves tothe upper side of the figure with a larger L value, whilethe PDF of m ⋆v /m as well as the anticorrelation between m ⋆v /m and L remain almost unchanged. Similarly, it isalso possible to study the correlation between m ⋆v /m and E sym at a fixed L , and the results are shown in the upperpanels of Fig. 3. The positive correlation between m ⋆v /m and E sym for a given L is observed. For a larger L , thePDF moves to the upper side of the figure with a larger E sym value, while the shape of PDF remains almost thesame. E sy m ( M e V ) L = 30 MeV
PDF(a) (b) (c)
L = 20 MeV L = 40 MeV (e)(d) (f) E sy m ( * ) ( M e V ) m *v /m FIG. 3: (Color online) PDFs in the m ∗ v − E sym plane (upperpanels) and in the m ⋆v /m − E sym ( ρ ⋆ ) plane (lower panels) at L = 20, 30, and 40 MeV, with ρ ⋆ = 0 .
05 fm − . Inspired by the regular behaviors observed above,we have further studied the correlation between L and E sym ( ρ ⋆ ) for different ρ ⋆ at a fixed m ⋆v /m in Fig. 4. For ρ ⋆ = 0 .
16 fm − , a nearly linear and positive correlationbetween L and E sym = E sym ( ρ ⋆ ) is observed. Similar lin-ear relations were extracted in Ref. [15] from the neutron-skin thickness and α D for various nuclei. This positivelylinear correlation between L and E sym is consistent withthe behaviors observed in the upper panels of Figs. 2 and3, showing that properties of the IVGDR are sensitiveto the E sym ( ρ ⋆ ) at ρ ⋆ other than the saturation density.The latter can be calculated from the SHF functional,with given values of L , m ⋆v /m , E sym , and other defaultquantities from the MSL0 interaction. From positive toslightly negative correlations between L and E sym ( ρ ⋆ )are observed with the decreasing value of ρ ⋆ . It is inter-esting to see that for ρ ⋆ = 0 .
05 fm − values of E sym ( ρ ⋆ )become approximately uncorrelated with L , showing that E − and α D are most sensitive to the symmetry energyat that density. This is consistent with the conclusionfrom Ref. [30] that α D of Pb is strongly correlatedwith the symmetry energy at about ρ /
3. The observedcutoffs in the PDFs are due to the choice of the priordistribution, i.e., E sym within 25 ∼
35 MeV. The corre-lations are shifted at different fixed m ⋆v /m values, whilethe strong correlation between properties of the IVGDRin Pb and E sym at ρ ⋆ = 0 .
05 fm − remains robust. * =0.16fm -3 * =0.05fm -3* =0.09fm -3 * =0.03fm -3 E sy m ( * ) ( M e V ) m *v /m=0.75 * =0.16fm -3* =0.05fm -3* =0.09fm -3* =0.03fm -3 L (MeV) m *v /m=0.80(a) (b) (c) * =0.16fm -3* =0.05fm -3 * =0.09fm -3* =0.03fm -3 PDFm *v /m=0.85 FIG. 4: (Color online) PDFs in the L − E sym ( ρ ⋆ ) plane at m ⋆v /m = 0 .
75 (a), 0.80 (b), and 0.85 (c) for different valuesof ρ ⋆ . The above finding shows that the sensitivity of IVGDRproperties to the E sym at ρ ⋆ = 0 .
05 fm − instead of L or E sym as well as m ⋆v /m is a robust feature based onthe Bayesian analysis. To further confirm this finding,we have calculated E sym ( ρ ⋆ ) from L , m ⋆v /m , E sym , andother default quantities from the MSL0 interaction, andreplotted the upper panels of Fig. 2 and Fig. 3. For dif-ferent values of L and E sym , the resulting correlations inthe m ⋆v /m − E sym ( ρ ⋆ ) plane are displayed in the lowerpanels of the corresponding figures. It is seen that thesePDFs are almost the same and thus approximately inde-pendent of L and E sym .The final resulting PDFs of m ⋆v /m and E sym ( ρ ⋆ ) aswell as their correlations in the present study are shownin Fig. 5. It is seen that m ⋆v /m are positively corre-lated with E sym ( ρ ⋆ ). We obtain m ⋆v /m = 0 . +0 . − . and E sym ( ρ ⋆ ) = 16 . +0 . − . MeV at 68% confidence level, and m ⋆v /m = 0 . +0 . − . and E sym ( ρ ⋆ ) = 16 . +1 . − . MeV at 90% confidence level. The 90% confidence interval of m ⋆v /m together with m ⋆s /m = 0 .
83 leads to the neutron-protoneffective mass splitting m ⋆n − p ≈ . +0 . − . δ in normalnuclear matter with the isospin asymmetry δ . PDF
FIG. 5: (Color online) The PDFs of m ⋆v /m (a) and E sym at ρ ⋆ = 0 .
05 fm − (b) as well as their correlations (c) roughlyindependent of L and E sym . In conclusion, we have studied the IVGDR in
Pbfrom the random-phase approximation method basedon the Skyrme-Hartree-Fock model, and employed theBayesian analysis to extract the posterior PDFs of isovec-tor parameters from the centroid energy of the IVGDRand the electric polarizability. Inspired by the similarshape of the PDFs for a given symmetry energy at thesaturation density E sym or the slope parameter L of thesymmetry energy as well as the linear correlation between L and E sym , we found that properties of IVGDR aremostly determined by the positive correlation betweenthe symmetry energy at ρ ⋆ ≈ .
05 fm − and the isovec-tor nucleon effective mass m ⋆v , but not directly by L and E sym . Moreover, m ⋆v /m = 0 . +0 . − . at the saturationdensity and E sym ( ρ ⋆ ) = 16 . +1 . − . MeV are obtained at90% confidence level from the present study.JX acknowledges the National Natural Science Foun-dation of China under Grant No. 11922514. ZZ ac-knowledges the National Natural Science Foundation ofChina under Grant No. 11905302. WJX acknowledgesthe National Natural Science Foundation of China un-der Grant No. 11505150. BAL acknowledges the U.S.Department of Energy, Office of Science, under AwardNumber de-sc0013702, the CUSTIPEN (China-U.S. The-ory Institute for Physics with Exotic Nuclei) under theUS Department of Energy Grant No. de-sc0009971. [1] B. A. Li and X. Han, Phys. Lett. B , 276 (2013).[2] M. Oertel, M. Hempel, T. Kl¨ahn, and S. Typel, Rev.Mod. Phys. , 015007 (2017).[3] B. A. Li, B. J. Cai, L. W. Chen, and J. Xu, Prog. Part.Nucl. Phys. , 29 (2018).[4] V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys.Rep , 335 (2005).[5] A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J.Ellis, Phys. Rep. , 325 (2005).[6] J. M. Lattimer and M. Prakash, Phys. Rep. , 109(2007).[7] B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. ,113 (2008).[8] C. Xu, B. A. Li, and L. W. Chen, Phys. Rev. C ,054607 (2010).[9] L. Trippa, G. Col`o, and E. Vigezzi, Phys. Rev. C ,061304(R) (2008).[10] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C ,051303(R) (2010).[11] J. Piekarewicz, B. K. Agrawal, G. Col`o, W. Nazarewicz,N. Paar, P.-G. Reinhard, X. Roca-Maza, and D. Vrete-nar, Phys. Rev. C , 041302(R) (2012).[12] D. Vretenar, Y. F. Niu, N. Paar, and J. Meng, Phys. Rev.C , 044317 (2012).[13] X. Roca-Maza, M. Brenna, G. Col`o, M. Centelles, X.Vi˜nas, B. K. Agrawal, N. Paar, D. Vretenar, and J.Piekarewicz, Phys. Rev. C , 024316 (2013).[14] G. Col`o, U. Garg, and H. Sagawa, Eur. Phys. J. A ,26 (2014).[15] X. Roca-Maza, X. Vi˜nas, M. Centelles, B. K. Agrawal,G. Col`o, N. Paar, J. Piekarewicz, and D. Vretenar, Phys.Rev. C , 064304 (2015).[16] Z. Zhang and L. W. Chen, Phys. Rev. C , 031301(R)(2015).[17] H. Zheng, S. Burrello, M. Colonna, and V. Baran, Phys.Rev. C , 014313 (2016). [18] E. Gebrerufael, A. Calci, and R. Roth, Phys. Rev. C ,031301(R) (2016).[19] Z. Zhang and L. W. Chen, Phys. Rev. C , 034335(2016).[20] H. Y. Kong, J. Xu, L. W. Chen, B. A. Li, and Y. G. Ma,Phys. Rev. C , 034324 (2017).[21] A. Bohr and B. R. Mottelson, Nuclear Stucture , Vols. Iand II (W. A. Benjamin Inc., Reading, MA, 1975).[22] O. Bohigas, A. M. Lane, and J. Martorell, Phys. Rep. , 267 (1979).[23] J.-P. Blaizot, Phys. Rep. , 171 (1980).[24] P. Kl¨upfel, P.-G. Reinhard, T. J. B¨urvenich, and J. A.Maruhn, Phys. Rev. C , 034310 (2009).[25] X. Roca-Maza, M. Brenna, B. K. Agrawal, P. F. Bor-tignon, G. Col`o, L. G. Cao, N. Paar, and D. Vretenar,Phys. Rev. C , 034301 (2013).[26] G. Bonasera, M. R. Anders, and S. Shlomo, Phys. Rev.C , 054316 (2018).[27] W. J. Xie and B. A. Li, Astro. Phys. J. , 174 (2019).[28] G. Col`o, L. Cao, N. Van Gia, and L. Capelli, Com. Phys.Com. , 142 (2013).[29] L. W. Chen, B. A. Li, C. M. Ko, and J. Xu, Phys. Rev.C , 024321 (2010).[30] Z. Zhang and L. W. Chen, Phys. Rev. C , 064317(2014).[31] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, andA. H. Teller, J. Chem. Phys. , 1087 (1953).[32] W. K. Hastings, Biometrika , 97 (1970).[33] S. S. Dietrich and B. L. Berman, At. Data Nucl. DataTables , 199 (1988).[34] A. Tamii, I. Poltoratska, P. vonNeumann-Cosel, et al. ,Phys. Rev. Lett. , 062502 (2011).[35] J. M. Lattimer and A. W. Steiner, Eur. Phys. J. A50