From homogeneous matter to finite nuclei: Role of the effective mass
Abstract
We present a method for extracting an energy density functional (EDF) for nuclei starting from a given immutable equation of state (EoS) of homogeneous matter. The scheme takes advantage of a natural Ansatz for homogeneous nuclear matter and basic EDF theory tenets. We apply it to closed(sub)shell nuclei to show that a goodquality Skyrme force model can easily be reverseengineered from a goodquality EoS without refinement or extensive fitting. Then the bulk and static nuclear properties are found practically to be independent of the assumed value for the effective mass, which is a unique result in bridging EDF for finite and homogeneous systems. Our methodology can be applied to study the effect of the EoS parameters and of the effective mass on nuclear observables including excitations without inducing artificial correlations amongst the EoS parameters and the effective mass, thanks to the freedom to vary them independently.
Introduction
Atomic nuclei can be viewed as charged, selfbound droplets of nuclear matter. Unlike other quantum liquids such as atomic Helium, nuclear matter’s thermodynamic limit is hardly realized in nature but represents a fictitious system which is a very useful theoretical playground ST2 (); FW (). With certain qualifications such as the stabilizing and binding effects of charged leptons and gravity, “infinite” nuclear matter is found in compact stars so that many astronomical observations serve as nuclear physics observations as well Piekarewicz18 ().
Energy density functional (EDF) theory provides a unified framework for both finite, selfbound nuclei and the equation of state (EoS) of infinite nuclear matter. The founding theorems HK64 (); KS65 (), originally for an externally bound, nondegenerate, unpolarized electron gas in its ground state, were followed over the years by numerous generalizations to situations including degenerate, polarized, and finite systems, and to excited states, as well as by practical justifications and refinements in a hierarchy of approximations PS01 (); Bechstedt (). Attempts exist to formalize intrinsic EDFs for selfbound systems Engel06 (); Messud11 (); Messud12 (). Formal justifications aside, the KohnSham framework is widely used in nuclear physics BHR03 (); NMMY16 (), as exemplified by (though not restricted to) Skyrme models. There are various approaches based on the HartreeFock (HF) approximation, with or without explicit correlations beyond mean field or pairing.
Current major goals of nuclear EDF research include 1) establishing connections between EoS parameters and nuclear observables and 2) bridging phenomenological EDF approaches with ab initio approaches, which contain the use of precise phenomenological two and threenucleon potentials in Monte Carlo simulations FP81 (); APR98 (); GC09 () as well as the recent intense efforts devoted to nuclear potentials obtained from effective field theories EKLM08 (); DSS13 (); DHS15 (). After decades of work and hundreds of EDF models, the search for a universal functional is ongoing and stumbling blocks remain in reaching the above goals. Traditional EDF models demonstrate spurious correlations amongst parameters, in particular, involving the inmedium effective mass BDMNP17 (); DNMBP17 (). It is also commonly said that certain observables “prefer” a low or high effective mass. (Respective examples are radii or energies BHR03 ().) It turns out that most available models fail to reproduce reasonably constrained EoS properties and nuclear observables simultaneously DLSD12 (); SGSD12 (). The situation is certainly unsatisfactory: if an EoS is “realistic,” then by definition it should be able to reproduce nuclear properties.
This situation may give an impression that the true density dependence of nuclear EDF is likely so complex that in the construction of EDF guesswork prevails. Starting with the analysis of Ref. PPLH16 () we are trying to make a systematic construction of EDF. In that work a power expansion of a homogeneousmatter EDF in Fermi momentum was investigated. The Ansatz and related strategy are dubbed KIDS (Korea: IBSDaeguSKKU) after the locale or institute of the original developers GPHPO16 (); GOHP17 (). A statistical analysis showed that three terms suffice for isospinsymmetric nuclear matter (SNM) (the same is concluded in Ref. BFJPS17 () for nuclei) in a converging hierarchy, and that four terms suffice for pure neutron matter (PNM) in a broad regime of densities. An EDF set of parameters could be selected for applications.
In the present work, we address the question: Given such a realistic parameterization for homogeneous matter, can we apply it to nuclei with no refitting? This is the major motivation of the present work and we find that key to a positive answer is the treatment of the inmedium effective mass. In the following we present our method for extracting an EDF for nuclei from a given, immutable EoS, and its applications to nuclei.
Homogeneous matter
The basic idea of the KIDS model for nuclear EDF is the expansion of the EDF in terms of the Fermi momentum or the cubic root of density in homogeneous infinite nuclear matter PPLH16 ():
(1) 
where is the baryon density ( and are neutron and proton densities, respectively), and asymmetry is defined as . Kinetic energy per particle is written as
(2) 
where () is the proton (neutron) mass. In Ref. PPLH16 (), a set of SNM parameters was determined by using the established properties at saturation: the saturation density , the binding energy per particle at saturation , and the compression modulus . This information can uniquely fix three unknowns . (Because it gives a marginal contribution, we set PPLH16 (); GPHPO16 (); GOHP17 ().) The four PNM parameters were fitted PPLH16 () to a microscopic EoS of PNM APR98 (). Thus was generated the set of parameters called “KIDSad2” in Ref. PPLH16 ().
Focusing on high densities, in Ref. PPLH16 (), the efficiency of the scheme was demonstrated in the regime of neutron stars, while the convergence of the expansion was explored in Ref. GOHP17 (). The nuclear symmetry energy is calculated as shown in Fig. 1. The results show that the symmetry energy may not be soft or stiff but may have a nontrivial density dependence. It is interesting that a similar behavior is seen in Refs. LPR10 (); PKLMR17 () where the cusp originates from skyrmion–halfskyrmion phase transition.
Very relevant for nuclei, on the other hand, is the behavior at low densities. Having used the PNM EoS of Ref. APR98 () in the fitting, with pseudodata down to fm, we test the validity of extrapolation to the region of fm. The results are shown in Fig. 2 which are compared with the EoS’s obtained from ab initio methods (QMC AV4 GC09 () and EFT DSS13 ()) and from a resummation formula YGLO YGL16 () fitted to ab initio pseudodata in fm GC09 () and to the PNM EoS of Ref. APR98 () at fm and representative Skyrme models DLSD12 (). This shows that the KIDS model reproduces the lowdensity curvature best with respect to EFT.
Finite nuclei
We now take this EoS, namely the KIDSad2 parameterization of Ref. PPLH16 (), which has a wide range of applicability in density, to investigate nuclei without altering its parameter values. Additionally, for the present “proof of principle” we use a similarly obtained EoS but with MeV. For the application to nuclei we reverseengineer a Skyrmetype force for HF calculations. The HF equations can then be solved with a straightforward extension of a standard numerical code Reinhard91 (). Minimally, and to reproduce the EDF of Eq. (1), we adopt the form of generalized Skyrme force given as
(3) 
where and , is the relative coordinate, and is the spinexchange operator. The strength of the spinorbit coupling, which is absent in the EoS of Eq. (1), is introduced by the term. It should be noted that the above “force” is an auxiliary entity, with no direct relation to a true Hamiltonian, being used as a stepping stone to the equivalent independentparticle “external” potential and resulting KohnSham orbitals.
The Skyrme force of Eq. (3) resembles other generalized Skyrme models with multiple densitydependent couplings CBBDM04 (); ADK06 (); XPC15 (). However, our strategy for determining both the precise form and the strength of the parameters is completely different, as elaborated in Ref. PPLH16 () and below. By comparing the EDF of Eq. (1) and the EDF corresponding to the above Skyrme force BHR03 (); DLSD12 (), relations among parameters can be straightforwardly obtained as
(4) 
with
(5) 
This reveals that the above EDF for nuclei has two sources for the term (): one from the densitydependent term in Eq. (3) of (), and the other from the momentumdependent term in Eq. (3) of (). The latter parameters determine the isoscalar and isovector effective masses as CBHMS97 ()
(6)  
(7) 
where we have used, for simplicity, the average nucleon mass . Thus the values for the effective masses determine the parameters , while the total coefficients of the EoSs of SNM and PNM can remain unchanged. This is different from the traditional Skyrme force model that has a priori , making the whole term to be momentumdependent.
The unknowns to be determined for nuclei, besides , are then the momentum dependence proportions in and and, correspondingly, the precise values of and . Our procedure is to

determine the momentumdependent terms by fitting to the energy and charge radius of \nuclide[40]Ca, with initialized to null,

determine from the energies and radii of the nuclei \nuclide[48]Ca and \nuclide[208]Pb,

iterate, i.e., examine \nuclide[40]Ca with the new value of and again determine anew and so on.
It turns out that iteration is largely unnecessary, because the bulk properties of the spinsaturated nucleus \nuclide[40]Ca are insensitive to .
We first determine the momentum dependence part (steps 1, 3 above). A simplistic procedure we explored before GOHP17 (); GPHPO16 () is to set , and encode the momentum dependence in a single parameter , corresponding to the portion of assigned to the momentum dependence part. The value of is then determined from the properties of \nuclide[40]Ca, which gives small values of . For KIDSad2 we obtained , corresponding to (, ), and .
A preferable way is to retain the freedom in . We now have four parameters to be explored. We consider specific values for () which will be varied for examining the results. This leaves us with only two unknowns in each case. In practice, the parameter space is further constrained by imposing 1) that the gradient coefficient be lower than (a handy and loose enough rule of thumb HPDBD13 ()) and 2) that polarized neutron matter remains stable at high densities KW94b (). From the acceptable combinations we choose the one that gives the best results for \nuclide[40]Ca. Finally, for each parameter set [i.e., essentially, for each pair of () and best fit to \nuclide[40]Ca] we determine the best value of by fitting to the energies and radii of \nuclide[48]Ca and \nuclide[208]Pb.
In the following applications we retain for comparison the KIDSad2 EoS with , henceforth labeled KIDS0:

KIDS0: , , , , , , , , , , MeV.
Parameters are also derived, as already described, for the following cases:

K240: Same EoS as KIDS0 for SNM and PNM, i.e., same , but allowing for nonvanishing () and for (with ) or of (with ).

K220: Same EoS as KIDS0 except that the SNM compression modulus is MeV; thus , , , , with different values of (for ).
In the above, all values are given in units . Resulting Skyrmetype parameters are collected in Table 1.
Model  \nuclide[60]Ca: [MeV]  

[fm]  
KIDS0  
K240,1.0,0.82  
K240,0.7,0.82  
K240,0.9,1.00  
K220,1.0,0.82  
K220,0.7,0.82  
GSkI ADK06 ()  
SLy4 CBHMS98 ()  
Results from HF calculations for the input nuclei \nuclide[40,48]Ca, \nuclide[208]Pb and for other (semi)magic nuclei are shown in Fig. 3 along with the available data. Here, all results for \nuclide[16]O, \nuclide[28]O, \nuclide[60]Ca, \nuclide[90]Zr, \nuclide[132]Sn, and \nuclide[218]U are predictions. We compute a mean absolute deviation of the calculated observable ( or ) with respect to data defined as
(8) 
where the sum runs over nuclei considered here for which data exist. Results are shown in Table 1.
We observe that in the scale of the graphs the results for the bulk properties are practically indistinguishable and, apart from \nuclide[16]O, in excellent agreement with available data and on a par with other models as the values of and indicate. From these results, we draw two conclusions: (1) A goodquality Skyrme model can easily be reverseengineered from a goodquality EoS without extensive fitting, while (2) bulk and static quantities are practically independent from the effective mass. The second conclusion is a most unusual, though intuitively unsurprising, result, which would not have been revealed had we ascribed the term fully to the kinetic energy from the outset. Let us note that, in that case, for KIDSad2 we would have obtained and .
The insensitivity is also examined in Table 1 for the exotic nucleus Ca. Some dependence is observed in the much finer cases of the neutron skin thickness, in particular, of \nuclide[208]Pb, \nuclide[218]U, and \nuclide[90]Zr, likely attributable to structural details that should be examined further in subsequent studies. We also find a tendency that the MeV parameterizations perform better than those with MeV. Systematic studies will be reported elasewhere.
The effective mass values can of course affect dynamical properties such as removal and capture energies (singleparticle spectrum) and nuclear collective motion. In Fig. 4 we compare obtained single particle energies with data and other model calculations, UNEDF2 KMNO13 (), GSkI ADK06 (), and SLy4 CBHMS98 (). In the first two models (UNEDF2 and GSkI) single particle energy levels of \nuclide[208]Pb are used in the fitting, so they are expected to give better results. For a more precise comparison we consider a mean absolute deviation,
(9) 
with the sum executed on the whole states shown in Fig. 4. This value is shown under the name of the model in Fig. 4. The accuracy of KIDS models with high effective mass is similar to those of GSkI and UNEDF2 models. Strictly speaking, only the highest occupied state of a manybody system can be considered as an observable Bechstedt (); DV (). The comparison is nonetheless interesting, in confirming that higher values of may be needed to reproduce the singleparticle spectrum of \nuclide[208]Pb.
Summary
We presented and validated a method for extracting a generalized Skyrmetype EDF for nuclei from a given, immutable EoS. The scheme utilizes a natural and versatile EoS for SNM and PNM which is “agnostic” with respect to effective masses. We have shown that 1) a goodquality Skyrme model can easily be reverseengineered from a goodquality EoS without extensive fitting, and that 2) bulk and static quantities are practically independent of effective masses. To our knowledge these are unique results obtained by bridging EDF for finite and homogeneous systems. Future applications abound: Our methodology will allow us to study the effect of the EoS parameters and of the effective mass on nuclear observables while retaining the freedom to vary major EoS parameters and the effective mass independently. An exploration of symmetryenergy parameters is underway Ahn18 (). Finally, it remains feasible to constrain the momentum dependence based on microscopic calculations of the effective mass or polarized matter and the spinorbit coupling from, e.g., relativistic approaches.
Acknowledgments
Acknowledgements.
We thank B. K. Agrawal for providing us with the correct values of GSkI and GSkII parameters. This work was supported by the National Research Foundation of Korea under Grant Nos. NRF2015R1D1A1A01059603 and NRF2017R1D1A1B03029020. The work of P.P. was supported by the Rare Isotope Science Project of the Institute for Basic Science funded by Ministry of Science, ICT and Future Planning and the National Research Foundation (NRF) of Korea (2013M7A1A1075764).References
 (1) A. Sitenko and V. Tartakovskii, Theory of Nucleus: Nuclear Structure and Nuclear Interaction (Kluwer Academic Publishers, 1997).
 (2) A. L. Fetter and J. D. Walecka, Quantum Theory of ManyParticle Systems (McGrawHill, 1971).
 (3) J. Piekarewicz, Nuclear astrophysics in the new era of multimessenger astronomy, arXiv:1805.04780.
 (4) P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
 (5) W. Kohn and L. J. Sham, Selfconsistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
 (6) J. P. Perdew and K. Schmidt, Jacob’s ladder of density functional approximations for the exchangecorrelation energy, AIP Conf. Proc. 577, 1 (2001).
 (7) F. Bechstedt, ManyBody Approach to Electronic Excitations. Concepts and Applications, Springer Series in SolidState Sciences Vol. 181 (Springer, 2015).
 (8) J. Engel, Instrinsicdensity functional, Phys. Rev. C 75, 014306 (2007).
 (9) J. Messud, Generalization of internal densityfunctional theory and KohnSham scheme to multicomponent selfbound systems, and link with traditional densityfunctional theory, Phys. Rev. A 84, 052113 (2011).
 (10) J. Messud, Alternate, wellfounded way to treat centerofmass correlations: Proposal of a local centerofmass correlations potentials, Phys. Rev. C 87, 024302 (2013).
 (11) M. Bender, P.H. Heenen, and P.G. Reinhard, Selfconsistent meanfield models for nuclear structure, Rev. Mod. Phys. 75, 121 (2003).
 (12) T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, and K. Yabana, Timedependent densityfunctional description of nuclear dynamics, Rev. Mod. Phys. 88, 045004 (2016).
 (13) B. Friedman and V. R. Pandharipande, Hot and cold, nuclear and neutron matter, Nucl. Phys. A 361, 502 (1981).
 (14) A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, The equation of state of nuclear matter and neutron star structure, Phys. Rev. C 58, 1804 (1998).
 (15) A. Gezerlis and J. Carlson, Lowdensity neutron matter, Phys. Rev. C 81, 025803 (2010).
 (16) E. Epelbaum, H. Krebs, D. Lee, and U.G. Meißner, Groundstate energy of dilute neutron matter at nexttoleading order in lattice chiral effective field theory, Eur. Phys. J. A 40, 199 (2009).
 (17) C. Drischler, V. Somà, and A. Schwenk, Microscopic calculations and energy expansions for neutronrich matter, Phys. Rev. C 89, 025806 (2014).
 (18) C. Drischler, K. Hebeler, and A. Schwenk, Asymmetric nuclear matter based on chiral two and threenucleon interactions, Phys. Rev. C 93, 054314 (2016).
 (19) P. Becker, D. Davesne, J. Meyer, J. Navarro, and A. Pastore, Solution of HartreeFockBogoliubov equations and fitting procedure using the N2LO Skyrme pseudopotential in spherical symmetry, Phys. Rev. C 96, 044330 (2017).
 (20) D. Davesne, J. Navarro, J. Meyer, K. Bennaceur, and A. Pastore, Twobody contributions to the effective mass in nuclear effective interactions, Phys. Rev. C 97, 044304 (2018).
 (21) M. Dutra, O. Lourenço, J. S. Sá Martins, A. Delfino, J. R. Stone, and P. D. Stevenson, Skyrme interaction and nuclear matter constraints, Phys. Rev. C 85, 035201 (2012).
 (22) P. D. Stevenson, P. M. Goddard, J. R. Stone, and M. Dutra, Do Skyrme forces that fit nuclear matter work well in finite nuclei?, AIP Conf. Proc. 1529, 262 (2013).
 (23) P. Papakonstantinou, T.S. Park, Y. Lim, and C. H. Hyun, Density dependence of the nuclear energydensity functional, Phys. Rev. C 97, 014312 (2018).
 (24) H. Gil, P. Papakonstantinou, C. H. Hyun, T.S. Park, and Y. Oh, Nuclear energy density functional for KIDS, Acta Phys. Pol. B 48, 305 (2017).
 (25) H. Gil, Y. Oh, C. H. Hyun, and P. Papakonstantinou, Skyrmetype nuclear force for the KIDS energy density functional, New Physics (Sae Mulli) 67, 456 (2017).
 (26) A. Bulgac, M. M. Forbes, S. Jin, R. N. Perez, and N. Schunck, Minimal nuclear energy density functional, Phys. Rev. C 97, 044313 (2018).
 (27) H. K. Lee, B.Y. Park, and M. Rho, HalfSkyrmion, tensor forces and symmetry energy in cold dense matter, Phys. Rev. C 83, 025206 (2011), 84, 059902(E) (2011).
 (28) W.G. Paeng, T. T. S. Kuo, H. K. Lee, Y.L. Ma, and M. Rho, Scaleinvariant hidden local symmetry, topology change, and dense baryonic matter. II, Phys. Rev. D 96, 014031 (2017).
 (29) C. J. Yang, M. Grasso, and D. Lacroix, From dilute matter to the equilibrium point in the energydensityfunctional theory, Phys. Rev. C 94, 031301(R) (2016).
 (30) P.G. Reinhard, The SkyrmeHartreeFock model of the nuclear ground state, in Computational Nuclear Physics 1, edited by K. Langanke, J. A. Maruhn, and S. E. Koonin, Springer, 1991.
 (31) B. Cochet, K. Bennaceur, P. Bonche, T. Duguet, and J. Meyer, Compressibility, effective mass and density dependence in Skyrme forces, Nucl. Phys. A 731, 34 (2004).
 (32) B. K. Agrawal, S. K. Dhiman, and R. Kumar, Exploring the extended densitydependent Skyrme effective forces for normal and isospinrich nuclei to neutron stars, Phys. Rev. C 73, 034319 (2006).
 (33) X. Y. Xiong, J. C. Pei, and W. J. Chen, Extension and parametrization of highorder density dependence in Skyrme forces, Phys. Rev. C 93, 024311 (2016).
 (34) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, A Skyrme parametrization from subnuclear to neutron star densities, Nucl. Phys. A 627, 710 (1997).
 (35) V. Hellemans, A. Pastore, T. Duguet, K. Bennaceur, D. Davesne, J. Meyer, M. Bender, and P.H. Heenen, Spurious finitesize instabilities in nuclear energy density functionals, Phys. Rev. C 88, 064323 (2013).
 (36) M. Kutschera and W. Wójcik, Polarized neutron matter with Skyrme forces, Phys. Lett. B 325, 271 (1994).
 (37) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, A Skyrme parametrization from subnuclear to neutron star densities Part II. Nuclei far from stabilities, Nucl. Phys. A 635, 231 (1998), 643, 441(E) (1998).
 (38) I. Angeli and K. P. Marinova, Table of experimental nuclear ground state charge radii: An update, Atom. Data Nucl. Data Tabl. 99, 69 (2013).
 (39) J. Jastrzȩbski, A. Trzcińska, P. Lubiński, B. Kłos, F. J. Hartmann, T. von Egidy, and S. Wycech, Neutron density distributions from antiprotonic atoms compared with hadron scattering data, Int. J. Mod. Phys. E 13, 343 (2004).
 (40) M. H. Mahzoon, M. C. Atkinson, R. J. Charity, and W. H. Dickhoff, Neutron skin thickness of \nuclide[48]Ca from a nonlocal dispersive opticalmodel analysis, Phys. Rev. Lett. 119, 222503 (2017).
 (41) PREX Collaboration, S. Abrahamyan et al., Measurement of the neutron radius of \nuclide[208]Pb through parity violation in electron scattering, Phys. Rev. Lett. 108, 112502 (2012).
 (42) M. Kortelainen, J. McDonnell, W. Nazarewicz, E. Olsen, P.G. Reinhard, J. Sarich, N. Schunck, S. M. Wild, D. Davesne, J. Erler, and A. Pastore, Nuclear energy density optimization: Shell structure, Phys. Rev. C 89, 054314 (2014).
 (43) W. H. Dickhoff and D. Van Neck, ManyBody Theory Exposed! Propagator Description of Quantum Mechanics in ManyBody Systems (World Scientific, 2005).
 (44) N. Schwierz, I. Wiedenhöver, and A. Volya, Parameterization of the WoodsSaxon potential for shellmodel calculations, arXiv:0709.3525.
 (45) G. Ahn, Master’s thesis, University of Athens, 2018.