Parametric correlations in energy density functionals
aa r X i v : . [ nu c l - t h ] O c t Parametric correlations in energy density functionals.
A. Taninah, S. E. Agbemava,
1, 2
A. V. Afanasjev, and P. Ring Department of Physics and Astronomy, Mississippi State University, MS 39762 Ghana Atomic Energy Commission, National Nuclear Research Institute, P.O. Box LG80, Legon, Ghana Fakultät für Physik, Technische Universität München, D-85748 Garching, Germany (Dated: October 30, 2019)Parametric correlations are studied in several classes of covariant density functional theories(CDFTs) using a statistical analysis in a large parameter hyperspace. In the present manuscript,we investigate such correlations for two specific types of models, namely, for models with densitydependent meson exchange and for point coupling models. Combined with the results obtainedpreviously in Ref. [1] for a non-linear meson exchange model, these results indicate that parametriccorrelations exist in all major classes of CDFTs when the functionals are fitted to the ground stateproperties of finite nuclei and to nuclear matter properties. In particular, for the density depen-dence in the isoscalar channel only one parameter is really independent. Accounting for these factspotentially allows one to reduce the number of free parameters considerably.
PACS numbers: 21.10.Dr, 21.10.Pc, 21.10.Ft, 21.60.Jz, 21.60.Ka
Since the early seventies, analogously to Coulombicquantum mechanical many-body systems, density func-tional theory (DFT) has played an important role innuclear physics. In principle, it corresponds to an ex-act mapping of the complex many-body system to thatof an artificial one-body system and therefore one withrelatively small computational costs. It is universal inthe sense that the form of the energy density functional(EDF) does not depend on the nucleus, nor on the spe-cific region where it is applied, but only on the under-lying interaction. Thus there is only one universal func-tional for the Coulomb interaction in atomic, molecularand condensed matter physics, but another one for nu-clear phenomena determined by the strong interactionand the Coulomb force. In Coulombic systems the den-sity functional can be derived in a microscopic way fromthe Coulomb force. On the contrary in nuclear physics,because of the complexity of the nuclear force such at-tempts are still in their infancy [2, 3]. All the successfulfunctionals are phenomenological. Their various formsobey the symmetries of the system, but in the absolutemajority of the cases the parameters are adjusted to ex-perimental data in finite nuclei and in homogeneous nu-clear matter.Covariant density functional theories (CDFT) [3–7] areparticularly interesting because they obey a basic sym-metries of QCD. In particular, Lorentz invariance whichnot only automatically includes the spin-orbit coupling,but also puts stringent restrictions on the number ofphenomenological parameters without loosing the goodagreement with experimental dataNonetheless, over the years, the number of phenomeno-logical functionals has grown considerably not only fornon-relativistic Skyrme DFTs, but also for CDFTs, sothat in recent years, questions have arisen about the re-liability and predictive power of such functionals [8, 9].Apart from the systematic uncertainties which are con-nected with the analytic forms and the various terms insuch functionals, there are so-called statistical uncertain- ties, connected with the procedures and strategies to ad-just the various parameters to experimental data. Herewe investigate whether the parameters in such CDFTs areindependent. We search for correlations between such pa-rameters in order to reduce their number. This will notonly reduce the numerical efforts for determining newparameter sets, but also decrease the statistical uncer-tainties and, therefore, increase the predictive power ofsuch functionals.The Zagreb group [10, 11] has already tried to reducethe number of parameters in point-coupling models witha density dependence of exponential form, as in the func-tional DD-PC1 [12]. Using the manifold boundary ap-proximation method (Ref. [11]) they showed that it ispossible to reduce the number of parameters for this func-tional from ten to eight without sacrificing the quality ofthe reproduction of empirical data. This method is basedon the behavior of the penalty function in the vicinity ofa minimal valley. As designed, this method is not com-pletely general and it still has to be shown that it canreveal all parametric correlations in the full parameterhyperspace.In the present investigation we go two steps further:(i) we consider all major classes of covariant energy den-sity functionals (CEDFs) used at present, and (ii) weuse methods which allow us to search for such correla-tions in the entire parameter hyperspace. Our resultsare closely related to the efforts of the DFT communityfor a microscopic derivation of EDFs and to the searchfor terms which are missing in the present generation ofEDFs [3, 13]. The absence/presence of dependencies be-tween the parameters of the EDFs can indicate whetherthe terms added to the Hamiltonian/Lagrangian haveroots in physics or simply reflect additional functional de-pendencies, introduced either by model approximationsor by the fitting protocol, which do not have a deeperphysical context.There are three types of CEDFs in the literature,(i) those based on meson exchange with non-linear me-son couplings (NLME), (ii) those based on meson ex-change with density dependent meson-nucleon couplings(DDME), and finally (iii) those based on point coupling(PC) models containing various zero-range interactions inthe Lagrangian. In Ref. [1] the (NLME) meson-exchangemodel with non-linear couplings for the σ -mesons intro-duced by Boguta and Bodmer in Ref. [14] has been inves-tigated and it has been found that there is a linear cor-relation between the parameters g and g . Within thispaper we investigate parametric correlations for the tworemaining types of CEDFs, namely, for those with den-sity dependent meson exchange as introduced by Typeland Wolter in Ref. [15] and the point coupling modelintroduced by Bürvenich et al in Ref. [16].The Lagrangians of the three different functionals canbe written as: L = L common + L model − specific where the L common consist of the Lagrangian of the free nucleonsand the electromagnetic interaction. It is identical for allthree classes of functionals and is written as L common = L free + L em (1)with L free = ¯ ψ ( iγ µ ∂ µ − m ) ψ (2)and L em = − F µν F µν − e − τ ψγ µ ψA µ . (3)For each model there is a specific term in the La-grangian: for the DDME models we have L DDME = ( ∂σ ) − m σ σ −
14 Ω µν Ω µν + m ω ω − ⃗ R µν ⃗ R µν + m ρ ⃗ ρ − g σ ( ¯ ψψ ) σ − g ω ( ¯ ψγ µ ψ ) ω µ − g ρ ( ¯ ψ ⃗ τ γ µ ψ )⃗ ρ µ (4)with the density dependence of the coupling constantsgiven by g i ( ρ ) = g i ( ρ ) f i ( x ) for i = σ, ω (5) g ρ ( ρ ) = g ρ ( ρ ) exp [ − a ρ ( x − )] (6)where ρ denotes the saturation density of symmetricnuclear matter and x = ρ / ρ . The functions f i ( x ) aregiven by the Typel-Wolter ansatz [15] f i ( x ) = a i + b i ( x + d i ) + c i ( x + d i ) . (7)Because of the five conditions f i ( ) = , f ′′ i ( ) = , and f ′′ σ ( ) = f ′′ ω ( ) , only three of the eight parameters a i , b i , c i , and d i are independent and we finally have the fourparameters b σ , c σ , c ω , and a ρ characterizing the densitydependence. In addition we have the four parameters ofthe Lagrangian L DDME m σ , g σ , g ω , and g ρ . As usualthe masses of the ω - and the ρ -meson are kept fixed atthe values m ω = MeV and m ρ = MeV [17, 18]. Wetherefore have N par = parameters in the DDME classof the models. The NL5 class of the functionals generated in Ref. [19]has the same model specific Lagrangian as the DDMEclass except that the coupling constants g σ , g ω , and g ρ are constants and there are extra terms for a non-linear σ meson coupling. These couplings are important forthe description of surface properties of finite nuclei, es-pecially the incompressibility [14] and for nuclear defor-mations [20]. L NL = L DDME − X − g σ − g σ (8)For the NL5 class we have N par = parameters m σ , g σ , g ω , g ρ , g , and g .The Lagrangian of the PC models contains three parts:(i) the four-fermion point coupling terms: L f = − α S ( ¯ ψψ )( ¯ ψψ ) − α V ( ¯ ψγ µ ψ )( ¯ ψγ µ ψ ) − α T S ( ¯ ψ ⃗ τ ψ )( ¯ ψ ⃗ τ ψ ) − α T V ( ¯ ψ ⃗ τ γ µ ψ )( ¯ ψ ⃗ τ γ µ ψ ) , (9)(ii) the gradient terms which are important to simulatethe effects of finite range: L der = − δ S ∂ ν ( ¯ ψψ ) ∂ ν ( ¯ ψψ ) − δ V ∂ ν ( ¯ ψγ µ ψ ) ∂ ν ( ¯ ψγ µ ψ ) − δ T S ∂ ν ( ¯ ψ ⃗ τ ψ ) ∂ ν ( ¯ ψ ⃗ τ ψ ) − δ T V ∂ ν ( ¯ ψ ⃗ τ γ µ ψ ) ∂ ν ( ¯ ψ ⃗ τ γ µ ψ ) , (10)(iii) The higher order terms which are responsible for theeffects of medium dependence L hot = − β S ( ¯ ψψ ) − γ S ( ¯ ψψ ) − γ V [( ¯ ψγ µ ψ )( ¯ ψγ µ ψ )] . (11)For the PC models we have N par = parameters α S , α V , α T V , δ S , δ V , δ T V , β S , γ S , γ V . In these calculations weneglect the scalar-isovector channel, i.e. we use α T S = δ T S = , because it has been shown in Ref. [18], that theinformation on masses and radii in finite nuclei does notallow one to distinguish the effects of the two isovectormesons δ and ρ . The particular realizations of the DDMEand PC models used in the present manuscript, whichdepend on the details of fitting protocol, are labeled hereas DDME-X and PC-X, respectively.In order to determine the N = N par parameters p = ( p , p , ..., p N ) of our model we adjust them to a set of N data data points. These data points belong to N type ofdifferent types and for each type, labeled by i , there are n i data points of the same type, which means N data = N type ∑ i = n i . (12)The experimental value of the physical observable j oftype i is given by O expi,j and the corresponding value calcu-lated with our model and the parameter set p is O i,j ( p ) .Adopting for each of the physical observables an error ∆ O i,j which, for the functionals under study, are sum-marized in Table 1 of the supplementary material , weintroduce for each parameter set p the penalty function χ ( p ) = N type ∑ i = n i ∑ j = ( O i,j ( p ) − O expi,j ∆ O i,j ) (13)and the optimal parametrization is found for the param-eter set p corresponding to the minimum of the penaltyfunction χ ( p ) . We measure the overall quality of thecalculated results by defining the normalized objectivefunction χ norm ( p ) = s χ ( p ) (14)where the normalization factor s = χ ( p ) N data − N par (15)is a global scale factor, called the Birge factor [24] anddefined at the optimal parametrization. This leads to anaverage χ ( p ) per degree of freedom equal to one [8].The functional variations under consideration are de-fined by the condition χ norm ( p ) ≤ χ norm ( p ) + ∆ χ max (16)The condition ∆ χ max = . specifies the ’physically rea-sonable’ domain around p in which the parametrization p provides a reasonable fit and thus can be consideredas acceptable [8, 25]. This condition also allows one todefine statistical errors for the physical observables ofinterest (see Refs. [1, 8]). For example, in the CDFTframework, this was done for the NL5(*) functionals inRef. [1].The NL5(*) functionals contain only 6 parameters andthus the volume of the hyperspace is rather modest. Onthe contrary, the DDME-X and PC-X functionals contain8 and 9 parameters, respectively. This leads to a drastic Note that contrary to previous studies all minimizations ofthe functionals are performed within the Relativistic Hartree-Bogoliubov (RHB) framework with separable pairing of Ref. [21]scaled according to Ref. [9]. For the rest, the fitting protocols ofthe DDME-X and PC-X functionals are identical to the fittingprotocols of the functionals DD-ME2 and PCPK-1 functionalsdefined in Refs. [17, 22]. In a similar fashion, the fitting pro-tocol of the NL5(E) functional is very similar to the one of theNL3* one (see Ref. [1] for details). The optimal DDME-X andPC-X functionals (see Tables II and III of the supplementary ma-terial) are defined by the simulating annealing method and bynumerous applications of the simplex method (see Ref. [23] for adescription of the method). Note that DDME-X and PC-X havebetter penalty function as compared to the original parametersets DD-ME2 and PC-PK1. increase of the volume of the hyperspace which makes nu-merical calculations with ∆ χ max = . impossible. Thus,in the present investigation we do not consider statisticalerrors but rather focus on parametric correlations. Asshown in Ref. [1] these correlations between the modelparameters are visible even for higher values of ∆ χ max .Thus, we use ∆ χ max = . for the DDME-X and PC-Xfunctionals.The numerical calculations are performed in the follow-ing way: New parametrizations p = ( p k , k = , N ) are ran-domly generated in the N = N par -dimensional parameterhyperspace and they are accepted if the condition (16)is satisfied. The domain in the N = N par -dimensionalparameter hyperspace, in which the calculations are per-formed, is defined as P space = [ p min − p max , p min − p max , ..., p N min − p N max ] , where p k min and p k max repre-sent the lower and upper boundaries for the variation ofthe k − th parameter. These boundaries are defined insuch a way that their further increase (for p k max ) or de-crease (for p k min ) does not lead to additional points inparameter hyperspace which satisfy Eq. (16).Note that, in the following, instead of the functionalparameters p k ( k = , N ) we are using the ratios (see Ref.[1]) f ( p k ) = p k p optk (17)where p optk is the value of the parameter in the optimalfunctional and k indicates the type of the parameter.This allows one to understand the range of the variationsof the parameters and related parametric correlations inthe functionals.In Fig. 1 we consider the CEDF DDME-X and show,for the randomly generated parameters obeying the con-dition (16), the 2-dimensional distributions of indicatedpairs of the parameters. The parameters vary with re-spect to the central value of the distribution (which aretypically given by the parameters of the optimal func-tional) by at most 0.5% for m σ , 0.6% for g σ , 1% for g ω ,2.5% for g ρ , 10% for a ρ , and 30% for c σ , b σ and c ω . Sim-ilar plots are presented in Fig. 2 for the PC-X functional.One can speak of parametric correlations betweenthese parameters when one parameter p k can, with a rea-sonable degree of accuracy, be expressed as a function ofother parameters, for example, as a function of the pa-rameter p j . The simplest type of the correlations is alinear one as given by f ( p k ) = af ( p j ) + b (18)For example, the following linear relations exist betweenthe parameters of the DDME-X functional (shown bysolid black lines in Figs. 1e and 1f) f ( b σ ) = . f ( c σ ) − . f ( c ω ) = . f ( c σ ) − . (19)and between the parameters of the PC-X functional (a) f( g σ ) f(m σ ) (b) f( g ω ) f(m σ ) (c) f( g ρ ) f(m σ ) (g) f( a ρ ) f(g ρ ) (e) f( b σ ) f(c σ ) (f) f( c ω ) f(c σ ) FIG. 1. Two-dimensional projections of the distribution of the functional variations in the 8-dimensional parameter hyperspaceof the DDME-X functional. The colors indicate the ∆ χ value of the χ norm ( p ) of the functional variation where the latter isexpressed as χ norm ( p ) = χ norm ( p ) + ∆ χ . A color map is used for the functional variations with maximum value of ∆ χ equalto ∆ χ max = . ; there are 200 such variations. The optimal functional is located at the intersection of the lines f ( p k ) = . and f ( p j ) = . . The solid lines in panels (e) and (f) display the parametric correlations between the respective parameters. (a) f( α T V ) f( α S ) (b) f( α V ) f( α S ) (c) f( β S ) f( α S ) (d) f( δ T V ) f( α TV ) (e) f( δ V ) f( δ S ) (f) f( β S ) f( γ S ) -0.6 0.2 1 1.8 2.6 0.68 0.76 0.84 0.92 1 1.08 1.16 (g) f( γ V ) f( γ S ) (h) f( β S ) f( γ V ) FIG. 2. The same as Fig. 1 but for the functional PC-X. (shown by solid black lines in Figs. 2b, 2e, and 2g) f ( α v ) = . f ( α s ) − . f ( δ v ) = . f ( δ s ) + . f ( γ v ) = − . f ( γ s ) + . (20)In the case of non-linear functionals linear relations exitsbetween g and g which define, in Eq. (8), the densitydependence of the functional (see Ref. [1]). Because of the two linear correlations (19) for the func-tionals DDME-X and because of the three linear corre-lations (20) for the functionals PC-X the number of in-dependent parameters can be reduced from 8 to 6 in thefunctional DDME-X and from 9 to 6 in the functionalPC-X. Note that the accounting of the parametric cor-relations in the case of non-linear meson-exchange mod-els leaves only 5 independent parameters (see Ref. [1]). f( g σ ) f(m σ )(a) f( g ω ) f(m σ )(b) f( g ρ ) f(m σ )(c) f( a ρ ) f(g ρ )(d) f( b σ ) f(c σ )(e) f( c ω ) f(c σ )(f) FIG. 3. Two-dimensional projections of the distribution of the parameters corresponding to local minima obtained by simplex-based minimizations for the functional DDME-X. The colors indicate the ∆ χ value of the χ norm ( p ) for the functionals inthese local minima where the latter is expressed as χ norm ( p ) = χ norm ( p ) + ∆ χ . Only local minima with ∆ χ < . are usedhere. There are 200 such minima. The optimal functional corresponding to the global minimum is located at the intersectionof the lines f ( p k ) = . and f ( p j ) = . . f( α T V ) f( α S )(a) f( α V ) f( α S )(b) f( β S ) f( α S )(c) f( δ T V ) f( α TV )(d) f( δ V ) f( δ S ) (e) f( β S ) f( γ S ) (f) f( γ V ) f( γ S ) (g) f( β S ) f( γ V ) (h) FIG. 4. The same as Fig. 3 but for the PC-X functional.
Thus, one can conclude that the ground state and nu-clear matter properties usually used in the fitting pro-tocols allow one to define only 5-6 (dependent on themodel structure) independent parameters in the case ofCDFT. Models with a larger number of parameters aremost likely over-parametrized.These results are consistent for the three models. For the NLME model we have only a density dependence inthe isoscalar channel. Originally it is determined by 2 pa-rameters g and g . The parametric correlations lead toa reduction to only one parameter for the density depen-dence in the isoscalar channel. The density dependencein the isovector channel is neglected and this obviouslyleads to unphysically large values of the slope of the sym-metry energy L (see Ref. [26]). In the DDME model,we have originally 3 parameters in the isoscalar channeland one parameter in the isovector channel. We foundno parametric correlations in the isovector channel, butthe number of parameters in the isoscalar channel is re-duced by parametric correlations from 3 to 1. In thePC-models we have also one parameter in the isovectorchannel, but the number of parameters in the isoscalarchannel is reduced from 4 to 1. Finally we have in allcases one parameter in the isoscalar channel and one pa-rameter in the isovector channel.This result can be understood qualitatively also ona microscopic basis. Starting from the bare nucleon-nucleon interaction adjusted to the nucleon-nucleon scat-tering data [27] and using relativistic Brueckner-Hartree-Fock theory in symmetric and asymmetric nuclear matterat various densities one is able to derive the relativisticself-energies of nucleons in nuclear matter without anyphenomenological parameters [28–32]. By adjusting theself-energies obtained from CDFT in nuclear matter atthe same density one is able to derive the density depen-dence of the coupling constants in a microscopic way [30].However, in the Brueckner calculations, a number of ap-proximations have been used and therefore this mappingis not unique. At present, the results obtained from suchcalculations in finite nuclei are rather different and, so far,their quality is far from that obtained with phenomeno-logical CDFTs (see, for instance, Fig. 11 in Ref. [33]).However, they all show in the isoscalar channel a densitydependence in the relevant density interval between 0.5and 1.1 of the saturation density, which is close to a lin-ear density dependence (see, for instance, Refs. [34–36]).This fact gives at least a qualitative explanation, why theparametric correlations discussed here allow a reductionto one parameter in the isoscalar channel.In the isovector channel, there is no reduction of thenumber of parameters describing the density dependence,because, from the beginning, we have no density depen-dence in the PC-X CEDF and in the non-linear mesoncoupling models (such as the NL5 family of CEDFs) andonly one parameter for the density dependence in theDDME-X functional. This is easy to understand becausethe effects in the isovector channel are much smaller thanthose in the isoscalar channel in which two huge scalarand vector fields S and V cancel in the nucleonic poten-tial and add up in the spin-orbit one. As it has beenshown in Ref. [18], present data for ground states offinite nuclei do not allow to distinguish correspondingscalar and vector potentials in the isovector channel.It is necessary to recognize that the search for paramet-ric correlations in the multidimensional parameter hyper-space by the method described above is extremely time-consuming even with modern high performance comput-ers. Thus, we looked for alternative methods for sucha search. The simplest method we found is based onthe minimization by the simplex method (see Ref. [23]).However, minimizations by the simplex method are proneto stack in local minima and that is a reason why it is not recommended for the search for global minimum. How-ever, in the context of the search of parametric corre-lations the drawback becomes an advantage. Startingfrom different randomly defined parameter vectors weperform a number of trial minimizations with the sim-plex method. They lead to different local minima in theparameter hyperspace. The distributions of the param-eters corresponding to these local minima are shown inFigs. 3 and 4 for the functionals DDME-X and PC-X, re-spectively. One can see that the parametric correlationsseen in Figs. 1 and 2 are also clearly visible in these twofigures. It is important to note that the search of para-metric correlations via the simplex-based minimizationmethod is at least by an order of magnitude less time-consuming than a fully statistical search based on Eq.(16) as it is shown in Figs. 1 and 2.It is also important that the simplex-based minimiza-tion method allows one to find a fine structure of suchcorrelations which can be hidden in a fully statistical ap-proach. This is illustrated in Fig. 3. Figs. 3a-d showthe coexistence of two long-range structures correspond-ing to a global and a sub-global minima; the respectiveparameter ranges are enclosed by the rectangles in pan-els (a-d). While the parametric correlations between theparameters b σ and c σ are the same in both structures[which is not surprising considering that these two pa-rameters describe the same type of meson] (see Fig. 3e),they are different between the c ω and c σ parameters forthese long-range structures (see Fig. 3e). This is also areason why the correlations between the latter two pa-rameters are broader (in width) in the fully statisticalanalysis presented in Fig. 1e; this is because ∆ χ max = . used in this analysis covers both long-range structures.The linear correlations (shown by black lines in Figs.3 and 4) defined via the simplex-based minimizationmethod are given by f ( b σ ) = . f ( c σ ) − . f ( c ω ) = . f ( c σ ) − . f ( c ω ) = . f ( c σ ) − . (21)for the DDME-X functional. Note that the values givenin the second and third lines of Eq. (21) correspond tothe global and sub-global minima of the χ function, re-spectively. The equations f ( α v ) = . f ( α s ) − . f ( δ v ) = . f ( δ s ) + . f ( γ v ) = − . f ( γ s ) + . (22)define similar correlations between the parameters of thePC-X functional. One finds an extreme similarity ofthe parametric correlations for PC-X obtained via thesimplex-based minimization method (Eq. (22)) and thosedefined from full statistical analysis (Eq. (20)). The sameis true for the correlations between the parameters b σ and c σ of the DDME-X functional (compare the upper linesof Eqs. (19) and (21)). However, the results for the para-metric correlations between the parameters c ω and c σ ofthe DDME-X functional obtained by full statistical anal-ysis are located in between those defined by means of thesimplex-based minimization method (compare Eqs. (19)and (21)). This is due to the fact that because of theselection of the ∆ χ max value the results obtained withformer method are an "envelope" of those obtained withlatter method.In the context of the analysis of theoretical uncertain-ties there is one clear advantage in the reduction of thedimensionality of the parameter hyperspace via the re-moval of parametric correlations: such a reduction leadsto a decrease of the statistical errors [1, 37].In conclusion, density functional theories (DFT) aredefined by underlying functionals. Some of those func-tionals depend on a substantial number of parame-ters. However, with the exception of non-linear meson-exchange CEDFs [1] the parametric correlations betweenthem have not been studied before. Using covariant DFTas an example and statistical tools, we have investigatedsuch correlations for major classes of covariant energydensity functionals for the first time. These include thenon-linear meson-exchange functionals (NLME) studiedin Ref. [1] and the functionals DDME-X and PC-X stud-ied for the first time in the present manuscript. Thesefunctionals are defined by the ground state propertiesof spherical nuclei and with exception of PC-X by thepseudodata on nuclear matter. It turns out that para-metric correlations exist between a number of parametersin all of those functionals. For example, linear paramet-ric correlations exist between the parameters g and g which are responsible for the density dependence in theisoscalar channel of the NLME model [1]. For the DDMEfunctionals, the parameters b σ and c ω vary linearly with c σ . Similarly, linear correlations are visible in the pa-rameter pairs ( α V , α S ), ( δ V , δ S ), and ( γ V , γ S ) of the PC-X functionals. The observation of correlations effectivelyreduces the number of independent parameters to five orsix dependent on the structure and the underlying func- tional. In particular, the difference between the numberof independent parameters depends on whether there isa density dependence in the isovector channel. Thus,these numbers represent a limit of how many indepen-dent parameters could be defined in the CDFT using fit-ting protocols based on ground state and nuclear matterproperties. Of course, at this stage, we cannot confirmthat these correlations will also show up also for otherfitting protocols, in particular, for those containing othertypes of data. However, the presently obtained resultsseem to be rather general.It is reasonable to expect that similar parametric corre-lations also exist in non-relativistic energy density func-tionals. In fact, in this case one should expect even moresuch parametric correlations because as it is known anon-relativistic approximation of covariant functionals interms of a p / M -expansion leads to a non-relativistic func-tionals with a large number of terms [38–40]. 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