Isovector properties of the Gogny interaction

PHYSICAL REVIEW C 90, 054327 (2014)

Isovector properties of the Gogny interaction

Roshan Sellahewa and Arnau RiosDepartment of Physics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom

(Received 30 July 2014; published 19 November 2014)

We analyze the properties of the Gogny interaction in homogeneous matter, with special emphasis on theisovector sector. We provide analytical expressions for both the single-particle and the bulk properties ofsymmetric and asymmetric nuclear matter. We perform an extensive analysis of these properties using elevenparametrizations extracted from the literature. We find that most Gogny interactions have low values for theslope of the symmetry energy, outside the range of empirically extracted values. As a test of extreme isospindependence, we also study the neutron star mass-radius relations implied by the different Gogny equations ofstate. Our results call for a more careful fitting procedure of the isovector properties of Gogny functionals.

DOI: 10.1103/PhysRevC.90.054327 PACS number(s): 21.30.Fe, 21.65.Cd, 21.65.Ef, 21.60.Jz

I. INTRODUCTION

The Gogny force is a well-known and extensively usedeffective nuclear interaction [1]. Unlike the Skyrme densityfunctional, which parametrizes the dependence on the relativedistance by contact interactions and derivatives, the Gognyforce has a built-in finite range [2]. This brings the Gognyforce closer in spirit to realistic interactions. Moreover, thenonzero range is essential to avoid spurious truncations inthe pairing channel within HartreeFockBogoliubov (HFB)nuclear structure calculations, and it was the main motivationbehind its inception in the 1980s by the Bruye`res group [3].In the decade that followed, Gogny forces were particularlyused in nuclear fission studies [4]. In this context, Gognyinteractions are still a popular starting point for a varietyof reasons [5]. Heavy nuclei, deformation, and multipolarcollective degrees of freedom have also been studied by usingGogny HFB [68]. Recently, even Gogny time-dependentcalculations have become available [9].

Here, we explore the predictions of the Gogny functionalin a different context. Infinite nuclear matter has usuallybeen taken as a reference in the fitting procedure of Gognyfunctionals [2] (see below for a more detailed discussion).This includes isoscalar properties, such as saturation energiesand saturation density. The compressibility of nuclear matter,of crucial importance for a variety of nuclear structureobservables, has also been extensively studied with the Gognyfunctional [10]. In contrast, the isovector properties of theGogny parametrizations have hardly ever been discussed.Typically, only the symmetry energy is considered, if any-thing [3]. The remaining isovector dependence is expected tobe captured by the fit to finite nuclei. Recent studies withthe Skyrme functional, however, indicate that, in additionto nuclei, neutron-rich infinite matter is also needed toconstrain the isovector sector [11,12]. One could thereforeput into question the predictive power of Gogny interactionsin isovector-dominated properties, such as neutron skins orneutron-rich systems.

The isovector properties of Skyrme functionals have beenextensively studied [13,14], with a very wide variety of criteriato characterize their quality [15]. Here, we aim at providing ageneric description of isospin asymmetric nuclear matter withthe Gogny interaction. Because only eleven functionals are

available in the literature, one might think that this study islimited in the amount of variability, which is often referredto as systematic uncertainty in the context of energy densityfunctionals (EDFs) [16,17]. However, we find that, even withina relatively narrow set of Gogny functionals, there is a largevariation in isospin properties. In particular, we observe thatthe density dependence of the symmetry energy provided byGogny forces is too soft and lies outside of currently acceptedvalues [18,19]. This points to poor constraints in the isovectorsector, which should be improved in future fitting protocols.

We aim at finding general trends and, by providinganalytical expressions, we hope to find specific combinationsof parameters that might be responsible for critical behavior.Where we can, we have compared with existing values ofisoscalar and isovector properties [20,21]. As a specific aspectof the isovector sector, we discuss neutron star properties aspredicted by the present generation of Gogny forces. On theone hand, this might go beyond the scope of applicabilityof part of the Gogny functionals. On the other hand, a newgeneration of observations is starting to put severe constraintson the equation of state of neutron-rich matter [2224].Ideally, these constraints should also be considered in fittingprocedures of energy density functionals [11]. In line with thepoor reproduction of bulk isovector properties, our calculationsindicate that it is difficult to produce sufficiently massiveneutron stars with the present generation of Gogny functionals.

II. GOGNY INTERACTIONS

In terms of its functional form, the Gogny force is a naturalextension of the early Brink and Boeker interaction [25]. Thefinite-range part is modelled by two Gaussians, including avariety of spin-isospin exchange terms, as well as a zero-range density-dependent term that is helpful in reproducingsaturation:

V (r) =i=1,2

(Wi + BiP HiP MiPP )er2i

+i=1,2

t i0(1 + xi0P

)i (r )

+ iW0(1 + 2)[k (r )k]. (1)

0556-2813/2014/90(5)/054327(21) 054327-1 2014 American Physical Society

ROSHAN SELLAHEWA AND ARNAU RIOS PHYSICAL REVIEW C 90, 054327 (2014)

The first contribution includes the finite-range dependence,because r is the relative distance between two nucleons. AllGogny forces contain two terms (denoted by i = 1,2) witheffective ranges 1 0.5 to 0.7 fm and 2 1 to 1.2 fm,which in principle mimic a short- and a long-range compo-nent, respectively.1 The spin-isospin structure of the forceis relatively rich and includes spin and isospin exchangeoperators, P and P , respectively. The second contributionis a zero-range, density-dependent component that accountsfor three-body correlations.2 The force further incorporates aspin-orbit component, proportional to W0. This is a functionof the relative momentum, k = ( 1 2)/2i, acting either onthe bra or the ket ( k) of two-nucleon states. Tensor terms inthe Gogny functional have also been considered for a varietyof applications [2729]. Both terms depend on gradients ofthe density and are therefore irrelevant for nuclear matter bulkproperties. We will not consider them hereafter.

There are about 14 to 17 numerical parameters to be fit in aGogny functional, although more often than not some of theseare fixed at the outset of the fitting procedure. In the following,we give results for the eleven Gogny parametrizations thatwe have been able to find in the literature. The originalforce, D1, was fit to the properties of closed-shell nuclei, 16Oand 90Zr, as well as to nuclear matter saturation properties,including a relatively low saturation symmetry energy ofS = 30.7 MeV [3]. A new parametrization, D1S, was devisedshortly after specifically for the study of fission [4] andhas been used extensively further [5,7,30,31]. In an effortto pin down the bulk isoscalar properties of nuclear matterfrom nuclear data, including pairing correlations, Blaizotand collaborators formulated a series of Gogny interactions(D250, D260, D280, and D300) with a wide range ofcompressibilities [10]. These have not been used extensivelyin the literature but provide an interesting testing ground forisoscalar-dependent properties [32].

Farine and collaborators conceived D1P as an extensionof the usual Gogny functionals, increasing the number ofzero-range terms to two [26]. Among other things, thisextension improves the neutron-matter equation of state byfits to realistic many-body calculations. Not surprisingly,we find that D1P performs well in the isovector sector. Asimilar idea is behind the D1N parametrization of Chappertet al. [33,34], which also reduces the difference betweentheoretical and experimental masses in the actinide region.D1AS is an extension of D1, which has been used in the contextof transport calculations [35]. A major motivation for thisforce was to provide a stiffer symmetry energy, but its nuclearstructure properties have not been explored to our knowledge.GT2, in contrast, was developed to provide realistic nuclearstructure calculations including a tensor term, to account forchanging shell structure in neutron-rich systems [27]. Finally,D1M provides a global fit to masses of comparable quality to

1Note that these are generally fixed at the start of the fitting protocoland hence cannot be considered as fit parameters.

2Note that, in most cases, the force contains a single t0 term. Theexception is the D1P parametrization, which includes two zero-rangeterms to make the fitting procedure more flexible [26].

mass formulas within the HFB approach, including quadrupolecorrelation energies [36,37].

The finite range of the Gogny force is a more realisticfeature that is now customarily used in nuclear structurestudies. In contrast, momentum dependence is incorporatedinto transport studies on a more intermittent basis [38]. Inthat context, one usually employs the so-called momentum-dependent interactions (MDI), which are vaguely related toGogny forces. In particular, the momentum dependence andthe isospin dependence are parametrized differently. In thefollowing, we focus strictly on Gogny functionals, but someof the conclusions can be relevant for MDI-type interactions.

For nuclear systems with different isospin contributions,like isospin-polarized matter, one can group the spin-isospinprefactors in the Gogny matrix elements into different terms.The zero-range contribution has a direct and an exchange partthat, in practice, are computed together. For the finite-rangeterms, however, it is convenient to split the contribution intodirect terms, which will be proportional to densities, andexchange terms, which involve more complicated functionsof Fermi momenta. For the zero-range and direct terms, forinstance, we differentiate between isoscalar (0 subscript):

Ai0 =3/23i

4[4Wi + 2Bi 2Hi Mi], (2)

Ci0 =34t0, (3)

and isovector (1 subscript):

Ai1 =3/23i

4[2Hi Mi], (4)

Ci1 = 14t0(1 + 2x0), (5)

contributions. For the finite-range exchange contribution, it isuseful to consider terms associated with equal and unequalisospin pairs:

Binn = Bipp = 1

[Wi + 2Bi Hi 2Mi], (6)

Binp =1

[Hi + 2Mi]. (7)

In discussing isoscalar and isovector single-particle properties,we also introduce the exchange isoscalar and isovector terms:

Bi0 = Binn + Binp = 1

[Wi + 2Bi 2Hi 4Mi], (8)

Bi1 = Binn Binp = 1

[Wi + 2Bi]. (9)

In infinite matter, and within the HartreeFock approximationemployed in this work, the single-particle (bulk) propertieshave contributions associated with the exchange term whichare proportional to single (double) integrals of Gaussians overthe Fermi surfaces of neutrons and protons. These integrals canbe computed analytically and give rise to a series of polynomialand Gaussian functions. To avoid cluttering our discussionwith equations, we provide all analytical expressions in theAppendixes.

054327-2

ISOVECTOR PROPERTIES OF THE GOGNY INTERACTION PHYSICAL REVIEW C 90, 054327 (2014)

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

A10 A20 A

11 A

21 B

1nn B

2nn B

1np B

2np C

10 C

11

Mat

rixele

men

ts

D1D1SD1ND1MD1PD1AS

D250D260D280D300GT2

FIG. 1. (Color online) Matrix elements of Eqs. (2) to (7) for theeleven Gogny functionals under consideration. The first (second)coefficient of every couple corresponds to i = 1 (i = 2). For thezero-range terms C0 and C1, we only present i = 1. The units areMeV fm3 for Ais; MeV for Bis, and MeV fm3i+3 for Cis.

The numerical parameters appearing in Eq. (1) are obtainedby a fitting procedure of the Gogny functional. With these,one can compute the different matrix elements. We note thatthe parameters can be degenerate, in the sense that onlylinear combinations enter the fitting procedure. In addition, theseparation of matrix elements in zero-range, direct finite-range,and exchange terms is arbitrary. Hence, the independent valuesof these matrix elements are not necessarily meaningful. Somespecific parameters, however, do determine physical properties(see below for how Bnn is entirely responsible for the effectivemass splitting), and hence it can be interesting to find outtheir values. While a detailed analysis is beyond the scope ofthis work, we provide a plot with the values of these matrixelements in Fig. 1. This provides, at a glance, an explanation ofthe different isovector and isoscalar parameters for the elevenfunctionals under consideration.

There are, for instance, common trends that can impactisoscalar and isovector properties of matter. Most forces prefera large positive A11 and a negative A21, suggesting cancellationsin the isovector, direct finite-range part of the functional. Incontrast, the majority of forces prefer negative B1,2xy , whichsuggests that the exchange terms act as overall attractivecontributions. The isospin singlet zero-range term C0 isrepulsive, as expected from the usual density-dependent termsof the functionals. In contrast, all forces, except for D1AS,present attractive C1 contributions. This suggests a dominanceof attractive terms in the isovector channels which, as we shallsee, hampers the development of stiff symmetry energies.

In addition, we find a relatively large spread for mostparameters. This is a sign of large functional dependence orsystematic uncertainty [16]. In particular, all the short-range

parameters A1x and B1x present a much larger variabilitythan their long-range counterparts, A2x and B2x . In terms offunctionals, D1M is an outlier as compared to most otherparametrizations, with extreme values of finite-range exchangeparameters, B1nn and B1np. The specific optimization procedureof this force should most likely account for these largedifferences [36]. Similarly, GT2 also shows a distinct behaviorfor A1,21 and B1,2nn , as already acknowledged in the originalpublication [27]. The isoscalar zero-range matrix elementsC10 are, as expected, all repulsive, whereas their isovectorcounterparts, C11 , are attractive and of a similar order ofmagnitude.3 We do not show the C2x parameters, since theyare zero for all forces except for D1P, where they are repulsiveCx2 192 MeV fm4.

III. MICROSCOPIC PROPERTIES

A. Single-particle potentialsWe start our discussion by looking at a series of single-

particle properties of asymmetric nuclear matter as obtainedby different Gogny functionals at different densities. All theseproperties characterize, in one way or another, the single-particle potential of a neutron or a proton with momentumk, denoted by U (k). The isospin index corresponds toa neutron, = +, or a proton, = . We work withinthe HartreeFock approximation and, in asymmetric infinitematter, the single-particle potential is the result of an integraland spin average over the neutron and proton Fermi surfaces:

U (k) = 12 ,

d3k

(2 )3 k ; k |V |k ; k A,

(10)so that the integral runs only for k < k F . The subscript Adenotes an antisymmetrization in the matrix element.

Using the expression for the Gogny functional in Eq. (1),one can find analytical expressions for U (k). The zero-rangeand the direct terms, for instance, are momentum independentand can be integrated straight away. This gives rise to twoterms proportional to the density, or to i+1, in the case ofthe zero-range contribution. The exchange term, arising fromthe antisymmetrization, is more cumbersome. It involves anintegral over a Gaussian momentum factor, which includes atleast one angular integration. This integral can be computed ina closed form, and we provide the explicit expression in the Ap-pendix [Eq. (A1)]. The isovector contribution is proportionalto the isospin-asymmetry fraction = np

. The result of the

integral in the exchange finite-range contribution depends onthe Fermi momenta of each species, kF = ( 3

2

2 [1 ])1/3.We do not provide the rearrangement term, Eq. (A5), explicitlyhere, but we include it in all figures that need it. The functionu(q,qF ), given in Eq. (A2), involves Gaussians and errorfunctions. This encodes the momentum dependence of the

3All forces have i = 13 except for D250 and D300 which havei = 23 .

054327-3


-160-120-80-40

040

U (k)

[MeV

] D1

0.00.20.4

0.60.81.0

D1S

-160-120-80-40

040

U (k)

[MeV

] D1N D1M

-160-120-80-40

040

U (k)

[MeV

] D1P D250

-160-120-80-40

040

U (k)

[MeV

] D260 D280

-160-120-80-40

040

0 1 2 3 4 5

U (k)

[MeV

]

Momentum, k [fm-1]

D300

0 1 2 3 4 5Momentum, k [fm-1]

GT2

FIG. 2. (Color online) Single-particle potential of neutrons (solidtriangles) and protons (solid circles) as a function of momentum forsix different isospin asymmetries: = 0 (solid line), 0.2 (long dashedline), 0.4 (short dashed line), 0.6 (dotted line), 0.8 (dashed-dottedline), and 1.0 (double-dotted dashed line). The results were obtainedat = 0.16 fm3. The symbols denote the single-particle potentialof a neutron (triangles) or a proton (circles) at the respective Fermimomentum kF .

single-particle potential, as all the other terms are constants asa function of k.

We show the single-particle potentials U (k) as a functionof momentum in Fig. 2. The different panels correspond to 10different Gogny parametrizations.4 The results have all beencomputed at = 0.16 fm3 at different isospin asymmetries(see figure caption for details). We highlight with symbolsthe neutron (triangles) and proton (circles) potentials at the

4We ignore D1AS for the time being, because its momentumdependence is identical to D1.

respective Fermi surfaces. These contributions are relevant forour understanding of the evolution of isospin in single-particleproperties.

The single-particle potential for symmetric matter (solidlines) is rather well constrained at low momenta. All forcespredict values U (0) 70 to 80 MeV at zero momentum and = 0.16 fm3. At low momentum, below about 2 fm1, thesymmetric-matter single-particle potentials are similar. As afunction of momentum, U (k) generally increases with k inthis region. Above k 2 fm1, for some functionals (D1N,D1P, or GT2) the potential saturates, decreases (D1S, D1M,D250), or increases (D260, D280, D300). For symmetricmatter, the momentum dependence is dictated by the sum oftwo terms, governed by Bi0 and the function u(q,qF ) evaluatedat the Fermi momentum, qF = ikF , of symmetric matter.We note that in the k 1 limit, the momentum-dependentexchange term becomes negligible and the direct plus zero-range terms dominate. Hence, the high-momentum value isentirely dominated by momentum-independent terms.

The interplay between asymmetry and momentum is alsorelevant, particularly in transport calculations [21,39,40]. Thedata presented in Fig. 2 shows that, at low momentum, theisospin asymmetry dependence of single-particle potentials isrelatively well constrained, at least around saturation density.In all cases, we observe an increase of the neutron Un(k) as afunction of asymmetry, whereas the proton Up(k) decreases.This corresponds to the physically intuitive idea that neutrons(protons) are less (more) bound in neutron-rich systems. Inthe low-momentum region, below 2 fm1, the dependencein asymmetry is rather monotonic. For an increase of 0.2in asymmetry, we find a steady decrease of Up by around10 MeV. In contrast, the neutron potential increases by about5 to 7 MeV in a pattern that is less linear. This suggests that,in the limit of neutron-rich systems, the nonlinear exchangeterms dominate the isospin dependence of the single-particlemomentum. For neutrons in neutron matter, the single-particlepotential at the Fermi surface is Un 32 MeV, with aspread of around 5 to 10 MeV. In the limit of extremeisospin imbalance ( = 1), we find that most forces predicta similar value for the zero-momentum proton potential,Up(0) 115 MeV, with a spread of around 10 MeV. Aproton impurity is a particularly interesting system, because itsmomentum dependence is entirely governed by Binp [41,42].

The asymmetry dependence in the high-momentum region,k 2 fm1, is less constrained. One finds a large variety ofresults. For D1, for instance, the single-particle energy ofneutrons for k > 3 fm1 is lower than that of protons. Thisinversion occurs also for D260, D270, D300, and GT2 in aregion ranging between 2 and 3 fm1. The results obtained withD1S, D1P, and D250 suggest that the asymmetry dependenceof the neutron potential is very weak above 2.5 fm1. In starkcontrast, D1M suggests a strong increase (decrease) of Un (Up)with asymmetry in the large-momentum region. As a matterof fact, for this force at large asymmetries, Up decreasesrather steeply as a function of momentum. We will see theconsequences of this behavior in the analysis of the effectivemass that follows. All in all, this figure suggests that the isospinasymmetry dependence of the high-momentum single-particleproperties is not constrained in the Gogny functional. One

054327-4


-80-60-40-20

020406080

Isos

cala

r, U 0

(k) [M

eV]

Density, =0.08 fm-3 Density, =0.16 fm-3

-80-60-40-20020406080

Density, =0.24 fm-3

-80-60-40-20

020406080

0 1 2 3 4 5

Isov

ecto

r, U 1

(k) [M

eV]

Momentum, k [fm-1]

D1D1SD1ND1MD1PD1AS

0 1 2 3 4 5Momentum, k [fm-1]

D250D260D280D300GT2

0 1 2 3 4 5-80-60-40-20020406080

Momentum, k [fm-1]FIG. 3. (Color online) (top panels) Isoscalar and (bottom panels) isovector components of the single-particle potential as a function of

momentum. Results for all Gogny functionals are displayed at three densities: (left panels) = 0.08 fm3, (central panels) = 0.16 fm3,and (right panels) = 0.24 fm3 are displayed. The gray band in the bottom-central panel is the allowed region of saturation isovectorsingle-particle potentials obtained in Ref. [46]. The arrows mark the position of the Fermi momentum at each density.

could foresee improvements in this direction by using fittingprotocols that take into account the information availablefrom realistic many-body calculations in isospin asymmetricnuclear matter [4345].

Up to this point, we have only displayed results computedat a single density = 0.16 fm3. One would like toknow whether similar issues are found at different densities.Rather than showing the whole asymmetry dependence of thesingle-particle potentials for different densities, we opt fordisplaying separately the isovector and isoscalar componentsof the single-particle potential. As described in Appendix A,one can introduce an isoscalar potential, U0(k), which isessentially the average of the neutron and proton potentials.This basically corresponds to the single-particle potentialof symmetric matter, which one would expect to be wellconstrained at subsaturation densities by nuclear data. Theisovector component, U sym1 (k) [see Eq. (A11) for a definition],is the first derivative with respect to of the single-particlepotentials. It therefore encodes information on how thepotentials evolve with asymmetry. We note that this isovectorpotential is essentially equivalent to the Lane potential in awide density, asymmetry, and momentum range [46,47].

We show the isoscalar potential for three characteristicdensities in the top panels of Fig. 3. At half saturation (top-leftpanel), the low-momentum part of the single-particle potentialis very well constrained, in the sense that all functionalspredict very similar values. At k = 0, for instance, one findsU0(0) 50 MeV. The increase of U0(k) with momentum israther mild and, at k = 2 fm1, most potentials are around

30 MeV. As momentum increases past this point, though, arelatively large spread of values develops.

At saturation (central panels), one finds qualitatively equiv-alent results. The low-momentum region of the potential is wellconstrained, although larger differences between functionalsare observed. In the region k 1 fm1, all isoscalar potentialsare close to U0 60 MeV. Above this momentum region,large divergences appear. Whereas some functionals increaseindefinitely with momentum, others saturate or even decrease.For densities above saturation, in contrast, there is a relativelywide spread of potentials both at low and at high momenta.For instance, the zero-momentum single-particle potentialis predicted to range between 90 and 70 MeV. Aswith other densities, the spread increases substantially abovek > 2 fm1.

Regarding the momentum dependence of the differentfunctionals, it is interesting to note that it is (up to density-dependent normalizations) essentially the same at differentdensities. D1S, for instance, predicts at all density an isoscalarpotential that increases, then saturates and further on decreaseswith momentum. In contrast, the D1N isoscalar potentialincreases monotonically at low k and saturates rather quicklyat all densities. D280 is extreme, in that its momentumdependence is the steepest, with a large increase in U as kbecomes larger.

We note that the functionals with largest U0 at highmomentum (D280, D260, and D300) are those that displayedlarge A10 values in Fig. 1. At large k, one expects the exchangeterm in U0 to become negligible. The isoscalar single-particle

054327-5


potential hence tends to the momentum-independent value:

U0(k 1) i=1,2

[Ai0 + Ci0i

]. (11)

Forces with large Ax0 and Cx0 matrix elements will eventuallyshow large momentum-independent contributions in the high-momentum region of U0(k). Moreover, since most zero-rangeisoscalar couplings C10 are positive, those functionals that havelarge and positive A10 will develop strongly repulsive single-particle potentials at high momenta and large densities. Notethat D1P was precisely fit to have U0(k) cross zero aroundk 3.2 fm1 and tend to an asymptotic value of 30 MeV fork 1 [26].

The bottom panels of Fig. 3 show a strikingly differentpattern. Even at subsaturation densities (bottom left panel),the low-momentum components of the isovector potentialscoming from different functionals are rather different. Atzero momentum, we find values ranging from U1(0) 19 to35 MeV, with significant divergences. At saturation (bottomcentral panel), most functionals predict U1(0) 33 MeV, ex-cept for GT2, D1M, and D1N. Between k = 1.4 and 1.6 fm1,there is an area of overall agreement between functionals, withvalues of U1 25 MeV, but the results diverge even more thanin the isoscalar sector as momentum increases. The isovectorpotentials above saturation (bottom-right panel) cover a widerange of values, which indicates that they are not constrainedby the parameter-fitting procedure.

A recent analysis of the isovector optical potential andits connection to bulk properties suggests that U1 shoulddecrease with momentum [39]. We show with a gray bandthe allowed region of the saturation density isovector single-particle potential as obtained from the optical potential fits ofRef. [46]. At low momentum, the allowed region is above allsingle-particle potentials. The steep decrease as a function ofmomentum suggested by this analysis is only reproduced bya minority (D260, D280, and GT2) of extreme functionals.Empirical values and theoretical predictions seem to havesomewhat similar momentum dependencies, but the absolutevalues of U0(k) are somewhat too low. This shift in absolutevalues could be due to the lack of nonlocality correctionsin U0(k) [46,47]. We note that our results for U0 and U1agree with those presented in Ref. [21] for the correspondingparametrizations.

The high-momentum asymmetry of U1 is also determinedentirely by the direct and zero-range matrix elements:

U1(k 1) i=1,2

[Ai1 + Ci1i

]. (12)

Hence, the extremely large and positive A11 values of D1M andD1N dictate limiting values of U1 which are large, positive, andincreasing with density. In contrast, because A11 0 for GT2,its isovector potential becomes very negative as momentumincreases.

To some extent, the large differences among functionalsare not surprising. The fitting procedure includes only aseries of points of finite nuclei, which are typically subsat-uration systems, and bulk, zero-temperature saturation matterproperties. All of these data are essentially determined by

single-particle properties at (a) densities below saturations and(b) momenta below the Fermi momentum, which is typicallyof the order kF 1 to 1.3 fm1. In addition, since most ofthese systems are almost isospin symmetric, the single-particleisovector properties are rather poorly constrained. Conse-quently, the single-particle potential is only well constrainedat low momentum, below saturation and near symmetricsystems. A covariance analysis that propagates the statisticaluncertainties of the fitting procedure would further quantifythis statement [16].

B. Effective massesThe effective mass provides a sensitive characterization of

the momentum dependence of the single-particle potential. Wenote again that the momentum dependence of the mean-field isexclusively due to the exchange term, because both the directand zero-range contributions are constants as a function ofk. In other words, the function u(q,qF ), given in Eq. (A2)is entirely responsible for the nontrivial effective mass. As aconsequence, the effective mass, m , is only proportional tothe matrix elements Bnn and Bnp:

mN

m= 1 + mN

2k

U (k)k

= 1 + mN2

i=1,2

{Binnm

(ik,ik

F

)

+Binpm(ik,ik

F

)}. (13)

The function m(q,qF ), given in Eq. (A7), is essentially amomentum derivative of the u(q,qF ) appearing in the single-particle potential.

In analogy to Fig. 2, we present in the ten panels of Fig. 4the results for the effective mass as a function of momentum at = 0.16 fm3 for different functionals. We explore differentasymmetries and show the corresponding Fermi momentumeffective masses of neutrons (protons) with solid triangles(solid circles). D1AS is a momentum-independent extensionof D1, so their effective masses are the same.

For isospin symmetric matter, = 0, and the correspondingsaturation density, we provide the values of the effective massof the eleven functionals in column one of Table I. In thiscase, the results of Fig. 4 indicate that all effective massesat the Fermi surface are similar. We note, however, that themomentum dependence of m (k) is relatively disparate for thedifferent functionals. D1 shows a steady increase, saturatingto m m at large momenta. Qualitatively similar results arefound for D1N, D1P, D260, D300, and GT2. In contrast, D1S,D1M, and D250 go through a maximum at k 3.5 fm1before settling down at m m. D280 is unique in that theeffective mass at low momentum is a decreasing function ofmomentum.

The asymmetry dependence of the effective masses is alsorather heterogeneous. For most forces, the neutron effectivemass increases with asymmetry at low momentum. For D1P,for instance, an increase of 0.2 in asymmetry results in anincrease in the effective mass of around 0.02. The decreasein proton effective mass is slightly smaller. If we excludeD1N, the effective mass of a proton impurity in neutron matter

054327-6


0.4

0.6

0.8

1.0

Eff.

mas

s, m

*

/m

D1

0.00.20.4

0.60.81.0

D1S

0.4

0.6

0.8

1.0

Eff.

mas

s, m

*

/m

D1N D1M

0.4

0.6

0.8

1.0

Eff.

mas

s, m

*

/m

D1P D250

0.4

0.6

0.8

1.0

Eff.

mas

s, m

*

/m

D260 D280

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Eff.

mas

s, m

*

/m

Momentum, k [fm-1]

D300

0 1 2 3 4 5Momentum, k [fm-1]

GT2

FIG. 4. (Color online) Effective mass of neutrons (solid circles)and protons (solid triangles) as a function of momentum for sixdifferent isospin asymmetries. The results were obtained at =0.16 fm3. See Fig. 2 for further explanations.

at k = 0 falls within mp(k = 0) 0.48 to 0.62. In contrast,the neutron effective mass in neutron matter changes frommn(knF ) 0.68 to 1.26. Again, we point out that the impuritysystem could be used to constrain the value of Binp [41,42].

D1N and D250 are exceptional in the sense that theirneutron effective masses at low momenta hardly depend onasymmetry. The momentum dependence of the neutron effec-tive mass for D1M is striking because it changes drasticallyfrom low to large asymmetries. In particular, the protoneffective mass at large momenta increases substantially tovalues above mp 1.7mN in neutron matter. GT2 (bottom-right panel) shows a similarly large dependence on asymmetry,but in this case it is the neutron effective mass that growssubstantially with asymmetry.

Note also that D1S, D1M, and D250 show a crossing point,above which the neutron effective mass is smaller than theproton effective mass. Again, because this high-momentum,high-asymmetry region is relatively unconstrained, it is notsurprising to find this variety of results. It would be interestingto explore whether this reordering with momentum has anyimpact on transport calculations [38] and observables of low-energy nuclear collisions [40].

The effective mass is often characterized by its value at theFermi surface, m (k = kF ), rather than by its full momentumdependence. This allows for a simple characterization of thevariation of momentum dependence with asymmetry, at leastclose to the respective Fermi surfaces. Since m is symmetricin its two arguments, the isovector effective-mass splitting isproportional toBinn only and to the difference of two symmetricm functions evaluated at the two Fermi surfaces:mN

mn mN

mp=

i=1,2

Binn[m(ik

nF ,ik

nF

) m(ikpF ,ikpF )].(14)

Because of its simplicity, this splitting can be analyzedrather straightforwardly. For symmetric arguments, m(qF ,qF )is a decreasing function of its argument up to qF 2.38.Consequently, as long as ikF 2.378 and for knF > k

pF , the

difference of m functions will be positive. In these conditions,the sign of the isovector splitting will be proportional to thatof Binn.

TABLE I. Isoscalar and isovector properties of nuclear matter as predicted by Gogny functional. All properties have units of MeV, exceptfor the dimensionless effective mass m

m(column two) and the saturation density 0 in fm3 (column three).

Force mm

0 e0 K0 Q0 S L Ksym Qsym K

D1 0.670 0.166 16.30 229.4 460.7 30.70 18.36 274.6 616.7 347.9D1S 0.697 0.163 16.01 202.9 515.8 31.13 22.43 241.5 644.2 319.1D1N 0.747 0.161 15.96 225.6 438.2 29.60 33.58 168.5 440.1 304.8D1M 0.746 0.165 16.02 225.0 459.0 28.55 24.83 133.2 735.7 231.6D1P 0.671 0.170 15.25 254.1 340.6 32.75 50.28 159.3 408.3 393.6D1AS 0.670 0.166 16.30 229.4 460.7 31.30 66.55 89.1 245.8 354.7D250 0.702 0.158 15.84 249.9 362.0 31.57 24.82 289.4 484.4 402.4D260 0.615 0.160 16.25 259.5 373.1 30.11 17.57 298.7 539.4 378.9D280 0.575 0.153 16.33 285.2 298.5 33.14 46.53 211.9 326.2 442.4D300 0.681 0.156 16.22 299.1 237.5 31.22 25.84 315.1 359.6 449.6GT2 0.672 0.161 16.02 228.1 444.0 33.94 5.02 445.9 741.2 466.2

054327-7


-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

Effe

ctive

mas

s sp

littin

g, (m

*

n - m

*

p)/m N

Asymmetry,

D1D1SD1ND1MD1P

D250D260D280D300GT2

FIG. 5. (Color online) Isovector mass splitting at the Fermi sur-face, Eq. (14), as a function of isospin asymmetry at = 0.16 fm3.Most functionals show a linear dependence at small asymmetries.

These findings are confirmed by the results displayed inFig. 5, where we show the isovector effective mass splittingas a function of isospin asymmetry at = 0.16 fm3. Tenfunctionals have positive splittings, with a neutron effectivemass larger than a proton effective mass, mn > mp. Theexception is D1N, which has a very small but negativeisovector splitting [34]. In contrast, since GT2 has a largeand positive B1nn, its isovector mass splitting is almost twiceas large as the results from other functionals. For the typicalisospin asymmetry of heavy nuclei, = 0.2, the isovectormass splitting predicted by most forces is of the ordermnmp

mN 0.04 to 0.06. If we identify this as the isovector

k-mass splitting, this result is consistent with the findings ofRef. [46].

It is also interesting to note that most functionals showa linear dependence on for small asymmetries. This weakasymmetry dependence is entirely due to the shift in Fermimomenta within the m functions in Eq. (14). The slope of thecurve could potentially be used to constrain Bxnn, if accuratedata for the isovector effective mass splitting were available.Efforts in this direction using optical model potentials arealready underway [46,47]. A stronger dependence in asym-metry appears as the neutron-matter limit is approached. Atthat point, the functional form of m becomes relevant and anonlinear shape is expected.

IV. MACROSCOPIC PROPERTIES

A. Isoscalar propertiesIn the previous section, we analyzed the microscopic

single-particle properties predicted by a variety of Gognyfunctionals. From now on, we concentrate on the resultsthat these functionals predict for the bulk, thermodynamicalproperties. We start by analyzing isoscalar and isovector

properties at densities close to saturation. We present in Table Ia series of properties of nuclear matter obtained by all theGogny parametrizations of interest. We note that some ofthese properties, particularly in the isoscalar sector, are inputsin the fitting procedure. As such, they do not yield any newinformation. We provide the analytical expressions for thesequantities in Appendix B. We note that we have also employednumerical stencils to test our expressions. We cross checkedthese data with the original publications as well as withother modern sources [34,48,49] and found a good generalagreement.

The second column of Table I shows the symmetric mattereffective masses obtained at the corresponding saturationdensities (column three). Note that the latter were obtainedfor each functional by using the same routines and underlyingnumerical constants, and hence the variability is entirely dueto the Gogny functional parameters. The effective masses arecomputed at the Fermi surface, and hence they also correspondto the circles in the continuous lines of Fig. 4. In general, alleffective masses fall within the range m 0.67m to 0.75m,which is slightly lower than modern Skyrme forces [50] but inagreement with many-body estimates [51]. The exceptions areD260 (m = 0.615m) and D280 (m = 0.575m), which havelower effective masses. We note that these two functionalshave = 13 , unlike D250 and D300 which were purposelydesigned with = 23 [10]. The range in compressibilities fora fixed can only be achieved with substantial variations onthe effective masses.

The equation of state (EoS) of symmetric nuclear matter ischaracterized by a few, relatively well-known parameters. Theenergy per particle presents a minimum around the saturationdensity 0 0.16 fm3, at a value of e0 16 MeV. Thesaturation densities and energies of the third and fourthcolumns of Table I mostly reflect the biases of the fittinginput data, as reported in the original publications. In general,they are all close to the empirical values. We note, however,the relatively repulsive saturation energy and relatively largesaturation density of D1P.5 By construction, D1AS has thesame isoscalar properties as D1 [35].

The curvature of the energy per particle around the satura-tion point can be described in terms of the incompressibilityK0. This determines to a large extent the EoS of symmetricnuclear matter and is shown in the fifth column. An explicitexpression for the incompressibility is given in Eq. (B6).Most values are centered around the empirical estimateK 230 MeV. The exceptions are, of course, the forcesD250D300, which were specifically obtained to match agiven compressibility. We note, in particular, that D280 isslightly off its expected value. A further characterization ofthe saturation point is obtained from the skewness Q0 [seeEq. (B7) for an explicit expression]. In contrast to 0, e0, and

5Our values for the effective mass, saturation density, energy,compressibility, and symmetry energy for D1P differ slightly fromthe values in the original publication [26]. This might be caused by asmall difference in the value of 1. We use 1 = 13 , but the results ofthe original publication are better reproduced with 1 0.345. Thesedifferences do not affect the results discussed in the following.

054327-8


1

10

100

1 1.5 2 2.5 3 3.5 4

Pres

sure

, P [M

eV fm

-3 ]

Density, /0

D1D1SD1ND1MD1PD1AS

D250D260D280D300GT2

FIG. 6. (Color online) EoS (pressure as a function of density)for symmetric nuclear matter for all Gogny functionals. The densityis divided by the respective saturation density of each functional.The shaded region corresponds to the EoS consistent with theexperimental flow data of Ref. [53].

K0, the skewness is not generally included in functional fits.Consequently, its value is not as constrained. We find thatGogny functionals predict a negative skewness, in the rangebetween Q0 237 and Q0 516 MeV. This agrees witha recent study based on a combined analysis of flow data andneutron star masses in the relativistic mean-field picture, whichsuggests 494 < Q0 < 10 MeV [52]. Note that neitherthe compressibility nor the skewness receive a contributionfrom Ai0. K0 and Q0 are therefore entirely determined bya competition between the zero-range and the finite-rangeexchange terms (in addition to the sizable kinetic terms, ofcourse).

The saturation energy minimum is reflected in a zero for thepressure of symmetric matter, PSNM. We provide an explicitexpression for this function in Eq. (B5). Because of thepositive compressibility, one expects a positive and increasingpressure as a function of density. We confirm that this isindeed the case in Fig. 6, which shows the symmetric matterEoS for the eleven Gogny functionals. We concentrated ondensities well above saturation, where the constraints comingfrom experimental flow data in intermediate-energy heavy-ioncollisions are relevant [53]. Some of the Gogny predictionshad already been compared to flow results in Ref. [49], butours is a more comprehensive analysis.

First, we stress the fact that all Gogny parametrizationsproduce relatively similar results for the pressure at highdensities. This is in stark contrast to the isovector propertiespresented in the following subsection. Together with the resultsof Table I, this indicates that systematic uncertainties, i.e.,those due to differences in the functionals, are rather smallin the isoscalar sector. In other words, the EoS of symmetricmatter is under good control, as one might expect from the

fitting procedure of the functionals. Note that the combinationsof isoscalar A0 and C0 matrix elements are relatively wellconstrained (see Fig. 1) compared to the finite-range exchangeelements B0. All these elements enter the expressions for theenergy [Eq. (B4)], pressure [Eq. (B5)], and compressibility[Eq. (B6)] and hence cannot be disentangled easily in a fittingprocedure that includes these as input.

Second, most functionals fall within the experimentalconstraints obtained from flow data [53]. The differencesbetween equations of state can be succinctly explained bythe compressibility of each functional. If we order thefunctionals in terms of the value of K0, the functional witha largest compressibility, D300, produces the largest pressureat high densities. Consequently, the EoS falls above theempirical constraints at all densities. The following subset,with K0 250 to 280 MeV, is formed of D280, D260, D1P,and D250. Their pressures are only consistent with the flowdata above 2.50. Most other functionals were fit withK0 230 MeV and hence their equations of state are veryclose to each other. D1S, which has the lowest compressibility,yields the lowest pressures but is still well within the flowdata. We note that, as long as K0 < 240 MeV, the EoS iscompatible with the experimental flow data. We note, however,that the flow data of Ref. [53] was obtained with a mean-fieldwith a momentum dependence characterized by m 0.7mand a medium-modified nucleon-nucleon cross section. Thefull momentum dependence of Gogny functionals might inprinciple affect these results. A further characterization of thepressure in terms of skewness lies beyond the scope of thiswork.

B. Isovector properties at saturationThe characterization of isovector bulk properties for Gogny

functionals is one of the main aims of this work. We provide thevalues for some macroscopic isovector properties, computedat saturation, in columns seven to eleven of Table I. Thesymmetry energy, S, was fit in some of the functionals (D1 [3],D1AS [35], D1M [36]). In others, it is indirectly constrainedby fits to the neutron-matter equation of state (D1P [26],D1N [33]). In contrast, fitting procedures for functionals likethe D250D300 family [10] have not explicitly consideredisovector bulk properties. In spite of these differences, we findthat the values of the saturation symmetry energy are generallywithin the range S 30 to 34 MeV. We note that D1M hasa particularly low value for the symmetry energy, imposedduring the fitting procedure [36].

The slope parameter of the symmetry energy,

L() = 3 S()

, (15)

quantifies the density dependence of the symmetry energy. Itis particularly relevant for neutron star physics, since it deter-mines the pressure of neutron matter close to saturation [54].We provide the values of L at saturation in column eight ofTable I. The variation in values of L is extremely wide, and wefind that most Gogny functionals are outside of the empiricallyconstrained ranges. More details are provided below, but let ushighlight the extremely low values of L 5 MeV for GT2 and

054327-9


-500

-400

-300

-200

-100

0

100

200

-20 0 20 40 60 80 100120140

K sym

[M

eV]

Slope parameter, L [MeV]-20 0 20 40 60 80 100120140

-100

0

100

200

300

400

500

600

700

800

900

1000

Q sym

[M

eV]

Slope parameter, L [MeV]

SkyrmeRMFD1D1SD1ND1MD1PD1ASD250D260D280D300GT2

FIG. 7. (Color online) (left panel) Ksym and (right panel) Qsymas a function of the slope parameter L for all Gogny functionals.Small black squares (small triangles) are representative of Skyrme(relativistic mean-field) functionals. The shaded region correspondsto the constraints obtained by Lattimer and Steiner in Ref. [55].

L 18 MeV for both D1 and D260. Inevitably, this will have anegative impact on the neutron star properties associated withthese functionals.

Columns nine and ten of Table I show the values of Ksymand Qsym at saturation for all the functionals. These propertiesare less constrained by empirical data. In general, we findlarge negative values of Ksym, in the range 90 to 446 MeV.These are far more negative than the values predicted bymicroscopic calculations [45]. Similarly, Gogny functionalspredict positive and relatively large values of Qsym, in contrastto the negative and small values obtained by BruecknerHartreeFock calculations [45]. We find the expected generaltrends obtained in other mean-field approaches [56,57]. Weshow in Fig. 7 the values of Ksym (left panel) and Qsym (rightpanel) as a function of the slope parameter. Gogny functionalswith lower values of L, such as GT2, D300, D260, or D1, tendto have more negative values of Ksym and more positive valuesof Qsym. These follow the generic correlations of Skyrme andrelativistic mean-field calculations, which we show in smallsquares and triangles in the figure [45]. All in all, this suggeststhat the correlations arise from basic underlying isovectorphysics. For the Gogny functionals, the correlations mustarise from a few of the underlying parameters, as shown inthe expressions of Eqs. (B12)(B14). Note, in particular, thatboth Ksym and Qsym are independent of the isovector directfinite-range matrix elements Ai1.

The asymmetry dependence of the compressibility isdetermined by a parameter K which, to lowest order, is givenby the combination K Ksym 6L (Q0/K0)L [54,56].This asymmetry dependence can potentially be extractedfrom giant monopole resonance experiments [58]. The valueK = 550 100 MeV is nowadays generally accepted [59].Most Gogny forces, as found in column eleven of Table I, sitsomewhat above the less-negative end, with values betweenK 450 and 300 MeV. As a matter of fact, taking thisexperimental value seriously, only GT2 and D300 would bevalid, in spite of their very low slope parameters. We note

Slop

e pa

ram

eter

, L [M

eV]

Symmetry energy, S [MeV] 0

10

20

30

40

50

60

70

80

26 28 30 32 34

Sn neutron skinHIC

Polarizab

ility

Mass

es

Neutron stars

IAS+

r np

FIG. 8. (Color online) Slope parameter as a function of symme-try energy for all Gogny functionals. The symbols show the values atsaturation, whereas lines correspond to the evolution with density ofboth parameters (see Fig. 9) for a legend. The regions corresponds toempirical constraints, following Refs. [55,60].

that Gogny functionals indicate a preference for less-negativevalues of K , in agreement with the theoretical analysispresented in Ref. [57] as well as with the correlation analysisof Ref. [56]. We also indicate that K is only the lowest-orderterm in the asymmetry dependence of the compressibility,and higher-order contributions can be relevant [56]. Furtherresearch on finite-nuclei properties, particularly resonances,with Gogny interactions will provide a further consistencycheck [8].

In contrast to the relatively poorly constrained K , thereis an abundance of empirical evidence that helps restrictthe values of the saturation symmetry energy and its slope.These two properties are generally tightly correlated, so theirindependent determination is difficult [16,17]. We show inFig. 8 the values of S and L for all Gogny functionals underconsideration. In addition, following Refs. [55,60], we showsome of the empirical constraints obtained from a variety ofmethods. The 68% confidence ellipse labeled Masses isobtained by propagating the fit errors in a density functionalcalculation using the UNEDF0 functional [61], with the choice = 1 MeV [55]. The Sn neutron-skin thickness results comefrom the analysis of Chen and coauthors [62]. From thisanalysis, one can also fit a relation between the skin thicknessin lead and the corresponding symmetry energy and slopeparameters of a variety of functionals. In addition, there is atight linear correlation between the dipole polarizability andthe skin thickness [63]. If we use the fit parameters of Ref. [60]for the former and of Ref. [63] for the latter, we find thepolarizability band in the figure. Note that the position andthe width of the band can change if different fits are used foreither correlation.

054327-10


Isospin diffusion studies in heavy-ion collisions, labeledHIC, provide additional constraints in the region of smallsymmetry energies [18]. The band labeled neutron starsis obtained by considering the 68% confidence values forL obtained from a Bayesian analysis of simultaneous massand radius measurements of neutron stars [64]. Finally, anarrow and small diagonal region above S > 30 MeV isobtained from simultaneous constraints of SkyrmeHartreeFock calculations of isobaric analog states (IAS) and the 208Pbneutron-skin thickness [65].

Strikingly, six of the eleven Gogny parametrizations falloutside of all empirical determinations of S and L. In allcases, the value of L is below the expected results. D1N andD1M sit within the polarizability and the mass constraints,but their slopes are still small compared with the neutronstars constraints. Only D1P, D1AS, and D280 have largeenough slopes to fit within the polarizability, neutron-skin, andastrophysical constraints. As for S, these functionals predictvalues within 1 MeV of the average value of all observations,S 31 MeV. We note that no parametrization falls within theIAS + rnp region or within the joint constraint region.

Figure 8 demonstrates graphically one of the main con-clusions of this work. Few, if any, Gogny functionals showa good simultaneous reproduction of the symmetry energyS and its slope L at saturation as of the present empiricalconstraints. An analysis of the density dependence of thesequantities in the following subsection will demonstrate thatthis is largely a consequence of the poor constraints on theisovector finite-range part of the functional. Future fits ofGogny functionals risk lacking quality in the isovector sectorunless these properties are fit consistently. We point out,however, that the comparison with empirical constraints shouldbe performed with some caution. Most of these constraintswere obtained by using zero-range functionals of the Skyrmetype [6163,65]. A consistent determination of the correlationsdepicted in the figure using finite-range functionals is stillmissing. The last section of this paper, where neutron stars areanalyzed, is an effort in this direction.

C. Density dependence of isovector propertiesIn addition to the values at saturation, there have been a

wide range of attempts to determine the density dependence ofthe symmetry energy and, generally speaking, of all isovectorproperties. With Gogny functionals, an analytic expression ofthe symmetry energy S() is provided in Eq. (B11). Notethat, in addition to the standard isovector zero-range anddirect finite-range terms, the exchange finite-range contribu-tion provides a nontrivial density dependence. The densitydependence of the symmetry energy is explored in Fig. 9.We show results for all Gogny functionals and highlightthe value of S at saturation. These results are comparedwith the constraints from the IAS analysis with Skyrmefunctionals [65].

The subsaturation region has a relatively small systematicerror. Most functionals fall within or very close to the empiri-cally constrained region. GT2, however, has a relatively largesymmetry energy below 0 and overshoots the constraints inthis region. In contrast, D1AS predicts relatively low symmetry

0

5

10

15

20

25

30

35

40

0 0.08 0.16 0.24 0.32 0.4

Sym

met

ry e

nerg

y, S

[MeV

]

Density, [fm-3]

D1D1SD1ND1MD1PD1AS

D250D260D280D300GT2

FIG. 9. (Color online) Symmetry energy as a function of densityfor all Gogny functionals. The shaded region corresponds to theconstraints arising from IAS of Ref. [65].

energies for < 0.14 fm3, but it has a much stiffer densitydependence above 0. The fact that most Gogny functionalsreproduce the low-density symmetry energy indicates that it iswell constrained by finite-nuclei data.

The largest discrepancies between functionals are observedabove saturation density, as expected. We stress the fact thatmost functionals show either a maximum or a plateau in thesymmetry energy as a function of density and that this occurs,in most cases, right above saturation. As a consequence, thevalues of the slope, L, are relatively small for most functionals.The exceptions are D1AS and D1P, which have a ratherstiff character. Above saturation, several functionals displaya sharp decrease in density, which will eventually lead tonegative values of S at densities beyond the range displayedin Fig. 9. For GT2, the change in sign of the symmetry energyhappens already around 0.38 fm3. The presence of thisisospin instability would have consequences in both neutronstars [48,66,67] and heavy-ion collisions [68]. Note, however,that microscopic calculations do not predict any signs of sucha transition [45].

A complementary perspective of these results is obtainedby analyzing the density dependence of the slope parameterL. We provide an expression for this quantity in Eq. (B12).Figure 10 shows the density dependence of the slope parameterfor all Gogny functionals. We also display the neutron starband, L 44 to 66 MeV, discussed already in the context ofFig. 8. Most Gogny functionals show a peak and a subsequentdecrease in L around half saturation density. Consequently, thevalue of L at saturation becomes relatively small, falling belowthe empirical estimates. One therefore expects Gogny forcesto predict rather small neutron-skin thicknesses in 208Pb. Thisis the case for both D1S and D1N, as observed in Ref. [31].More importantly, the pressures for isospin asymmetric matter

054327-11


0

10

20

30

40

50

60

70

0 0.08 0.16 0.24 0.32 0.4

Slop

e pa

ram

eter

, L [M

eV]

Density, [fm-3]

D1D1SD1ND1MD1PD1AS

D250D260D280D300GT2

FIG. 10. (Color online) Slope parameter as a function of densityfor all Gogny functionals. The shaded region corresponds to thecombined constraints obtained by Lattimer and Steiner in Ref. [55].

as predicted by these functionals will be smaller than empiricalestimates suggest. The structure of neutron-rich isotopes willbe affected by the unrealistic prediction of slopes. Note alsothat about half of the functionals predict negative slopes within0.05 fm3 of around saturation, in accordance with the verysoft symmetry energies observed around saturation in Fig. 9.It is worth mentioning that D1M shows a rather unique densitydependence for the slope, with a nuanced decrease in densityabove saturation. For this functional, L does not becomenegative below > 0.42 fm3.

Three functionals fall within the empirical range pre-dicted by neutron star physics. For D280 and D1P, whichhave large enough values of L at saturation, the slope isclose to a maximum at saturation. The decrease of L withdensity eventually leads to decreasing symmetry energiesat relatively large densities (0.26 fm3 for D280 and 0.42fm3 for D1S). In contrast, D1AS is the only functionalpredicting a monotonically increasing slope parameteranda consequently stiff symmetry energy. We note that thiswas achieved by construction, adding a density-dependentzero-range term to the functional [35]. Because of their largepressures for neutron-rich matter around saturation, one wouldexpect relatively large neutron stars associated with thesefunctionals. We will confirm this tendency in the followingsection.

What is the underlying cause for the rapid decrease of thesymmetry energy with density in most Gogny functionals?One can attempt to answer this question by separatingdifferent contributions to both the symmetry energy and theslope parameter. In Fig. 11, we provide the contributionsto both quantities arising from the direct finite-range term,the exchange finite-range term, and the zero-range term(which includes both direct and exchange at once). The direct

contribution to both quantities is particularly simple, as seenfrom Eqs. (B11) and (B12):

Sdirect() = 12i=1,2

Ai1.

Note, in particular, that the term is linear in density andthat only the sum of all isovector matrix elements playsa role. Because the values of these matrix elements arerelatively widely spread (see Fig. 1), we expect a relativelylarge systematic uncertainty. This is precisely what we findin the left panels of Fig. 11. D1M has extremely large andpositive contributions. In contrast, GT2 provides a negativecontribution as density increases. Because the combination

i Ai1 is very similar for some functionals, we find that

D1P (and D250 to a lesser extent) and D1 have very similarvalues of Sdirect and Ldirect. The direct contribution to theslope parameter (bottom-left panel) is three times that of thesymmetry energy (top-left panel),Ldirect() = 3Sdirect(). Notealso that, as expected, D1 and D1AS have the same directfinite-range contributions.

The exchange contribution to the isovector properties(central panels of Fig. 11) shows a density dependence thatis, to a certain extent, the opposite of the direct term. GT2, forinstance, is now large and positive, whereas D1M is large andnegative. In general, these terms increase substantially withdensity, even though the dependence is not linear anymore. Infact, the density dependence is dictated by nontrivial functionsof the Fermi momenta; see Eqs. (B15) and (B16). With theseequations, it is easy to show that, in the low-density limit, theexchange contributions to S and L are (a) linear in density,(b) proportional to B1i only, and (c) related by Lexc() =3Sexc(). The exchange contributions are of the same orderas the direct term, which indicates that large cancellations arenecessary to get isovector properties of a natural size.

The right panels of Fig. 11 show the density dependenceof the zero-range contributions to the isovector properties.First, we note that the dependence on the functional is largelyreduced. The fitting procedure of Gogny functionals is suchthat part of the zero-range, density-dependent term is oftenfixed from the start.6 Consequently, the parameter space forthis term is substantially reduced and SZR(),

SZR() = 12i=1,2

Ci1i+1, (16)

is very similar for all functionals. Also, because in mostparametrizations C21 = 0, the relation

LZR() = 3(i + 1)SZR() (17)

holds.

6For all functionals, 1 is chosen to be 13 or23 . x0 is also often fixed,

which leaves very little room for t10 to change substantially.

054327-12


-50

0

50

100

150

200

Sym

met

ry e

nerg

y, S

[MeV

]

Direct Exchange

-250

-200

-150

-100

-50

0Zero-range

D1D1SD1ND1MD1PD1AS

D250D260D280D300GT2

-100-50

050

100150200250300

0.00 0.08 0.16 0.24 0.32

Slop

e pa

ram

eter

, L [M

eV]

Density, [fm-3]0.00 0.08 0.16 0.24 0.32

Density, [fm-3]0.00 0.08 0.16 0.24 0.32

-450-400-350-300-250-200-150-100-500

Density, [fm-3]

FIG. 11. (Color online) (left panels) Direct, (central panels) exchange, and (right panels) zero-range contributions of the (top panels)symmetry energy and (bottom panels) the slope parameter. Results for eleven functionals are shown. The symbols correspond to the respectivesaturation points. Note that the left and central panels share the same y axis.

Second, and most important, the zero-range density-dependent term is negative at all densities. In the low-densitylimit, the linear density dependence of both the direct and theexchange dominates (although absolute values are determinedby large cancellations between terms). In the limit of largedensities, in contrast, Eqs. (B15) and (B16) suggest that theexchange term is proportional to kF 1/3. In addition, formost functionals the zero-range term has a power 1 = 13 (and2 = 0) which therefore involves SZR

i C

i1

4/3. Together

with the linear dependence on the direct term, it appears that thezero-range term inevitably dominates the density dependenceof isovector properties at high densities. As a consequence,the symmetry energy of most parametrizations becomes soft atrelatively low densities. We note that D1P is the only functionalwith a second zero-range term, that is, t20 = 0. It performswell in the isovector sector. Future parametrizations could usethis freedom to improve the density dependence of isovectorproperties. Specifically, the fact that Sexc is not proportional toLexc for t20 = 0 can help break the tension between these twoisovector parameters.

The decomposition in terms of direct, zero-range, andexchange terms of the isovector properties is arbitrary to alarge extent. Other decompositions can be used to pinpointsimilar issues. In Ref. [21], Chen and coauthors propose a splitof the symmetry energy and the slope in terms of contributionsarising from the single-particle potential at the Fermi surface,based on the Hugenholtzvan Hove theorem. We performed anextensive study of this decomposition, but we present here onlya summarized version. Within this approach, the symmetry

energy can be decomposed into two terms:

S1() = 13

2k2F2m0()

, (18)

S2() = 12Usym1 (kF ). (19)

The first term is a nonlocality-corrected kinetic contribution.The nonlocal correction, evidenced by the effective mass, isof the order of 30% at saturation and usually increases withdensity. The competition between the Fermi momentum andthe effective mass can give rise to S1 terms that decrease withdensity. The second term S2 is proportional to the value ofthe isovector single-particle potential at the Fermi surface. Asobserved in Fig. 3, the value of U1 at the Fermi surface can varysubstantially. For quite a few functionals, this has a tendencyto become negative slightly above saturation and drives thedecrease of S with density [21]. A similar decomposition existsfor the slope parameter, but it involves five different terms.There is in general a substantial Gogny functional dependenceof these terms. This is particularly important for L4 and L5,which are associated with higher orders in the single-particlepotential expansion [21]. L4 and L5 are entirely due to thefinite-range exchange part of the functional.7

7L5 also receives a contribution from the rearrangement potential,associated with the density-dependent zero-range term.

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V. NEUTRON MATTER AND NEUTRON STARS

Neutron matter provides a particularly sensitive test to theisovector behavior of nuclear energy density functionals [23],which is interesting because it can be straightforwardlyconnected to neutron stars [11]. Present- and future-generationobservations will eventually constrain the mass-radius relationfor astrophysical compact objects, and this information canbe fed back into the neutron-matter EoS [22]. Even beforethat happens, essential tests regarding the consistency andisospin dependence of energy density functionals should beperformed. We note that this task has been already carried outin extensive studies of Skyrme functionals [13,15].

As we have seen so far, Gogny functionals are particularlypoorly determined in the isovector sector. We therefore startour discussion by looking at the energy per particle of neutronmatter,

ePNM() = 35

2kn,2F2mn

+ 12i=1,2

{Ai1 + Ci1i

}

+ 12i=1,2

Binng(ik

nF

), (20)

where the Fermi momentum is that of neutron matter,knF = (32)1/3. We show in Fig. 12 the Gogny functionalpredictions for the energy per particle of neutron matter.The left panel concentrates on the low-density regime. Inaddition to the Gogny data, we show results for microscopicdeterminations of the energy per particle of neutron matterat low densities. In the left panel, we show a band obtainedfrom a recent auxiliary field diffusion Monte Carlo calculationusing N2LO local chiral interactions [69]. The width of theband reflects the uncertainty in the original interaction, butwe note that missing three nucleon forces would increase theaverage value at higher densities. The three points with errorbands correspond to the representative data used in Ref. [12].This is obtained as an educated guess, based on a variety ofmany-body calculations in this low-density regime.

At subsaturation densities, we find relatively similar re-sults for the different parametrizations. None of the Gognyfunctionals reproduce the two lowest density points of theBrownSchwenk analysis. A few are able to fit within therelatively large error band of the third. Fitting Skyrme densityfunctionals to nuclei and to these three subsaturation points inneutron matter yields a good quality density functional in theisovector sector [12]. This could be a good starting point forfuture fitting protocols of Gogny functionals. D1AS results gobelow the many-body data, which indicates that the functionalform of Gogny interactions can in principle be amenable toreproduce such low-density points.

The right panel of Fig. 12 concentrates on a wider densityregime, up to slightly above twice the saturation density. Asexpected, the differences between functionals here are quitemarked. While around saturation the variations among func-tionals are accommodated within about 5 MeV, at 0.24 fm3one finds quite larger discrepancies. As a matter of fact, abovesaturations one can distinguish between two classes of Gognyfunctionals. On the one hand, a majority of functionals have arather soft density dependence and predict energies which are

0

2

4

6

8

0.00 0.02 0.04 0.06

Ener

gy p

er p

artic

le, E

/A [M

eV]

Density, [fm-3]0.00 0.08 0.16 0.24 0.32

0

5

10

15

20

25

30

35

40

45

50

Ener

gy p

er p

artic

le, E

/A [M

eV]

Density, [fm-3]

D1D1SD1ND1MD1PD1ASD250D260D280D300GT2

FIG. 12. (Color online) Energy per particle as a function ofdensity for pure neutron matter for all the Gogny functionals. Theshaded region enclosed by a dotted (dashed) line corresponds toquantum Monte Carlo [69] (self-consistent Greens functions [70])calculations based on chiral potentials. The points with error barson the left panel correspond to three representative points used inRef. [12].

well below 30 MeV for 0.32 fm3. On the other, D280,D1AS, and D1P increase far more quickly with density. Thisis important, because the density increase will be reflected ina substantially larger pressure which, in turn, will allow forlarger and heavier neutron stars.

Interestingly, the subgroup of Gogny functionals that havea stiff neutron-matter energy per particle follows closely theband enclosed by dashed lines. This band is the result ofa realistic many-body calculation, obtained within the self-consistent Greens functions approach using an N3LO chiraltwo-body interaction supplemented with a three-body forceat N2LO [70]. Because of the nonperturbative nature of theresummation scheme, these calculations have been performedwith unrenormalized interactions and up to twice the saturationdensity. Again, the width of the band is a reflection of theuncertainties in the underlying low-energy constants of thechiral interactions. We find that the stiff Gogny functionalsare indeed well within this band up to 0.32 fm3. Note,however, that these very same functionals do not reproducethe many-body results at low densities.

GT2 shows a maximum of the energy per particle aroundsaturation and subsequently decreases as density increases.The decrease of energy as a function of density is a telltalesign of a mechanical instability, which is reflected in a negativepressure. We confirm this fact in Fig. 13, where we show thepressure in neutron matter for all the Gogny functionals. GT2indeed shows a negative pressure slightly above saturation.We also find that a few other functionals (D1, D1S, D250)predict negative pressures, although at much larger densities; 0.64 fm3 and above. At this point, of course, one shoulddiscuss the predictive power of these functionals at such highdensities. We do not expect Gogny functionals to describe thephysics of the very-high-density regime, but they might be ableto extrapolate some of the low-density nuclear physics into the

054327-14


0.1

1

10

100

0.5 1 1.5 2 2.5 3 3.5 4

Pres

sure

, P [M

eV fm

-3 ]

Density, /0

D1D1SD1ND1MD1PD1AS

D250D260D280D300GT2

FIG. 13. (Color online) EoS (pressure as a function of density)for pure neutron matter. The shaded region enclosed by a dotted(dashed) line corresponds to the stiff (soft) symmetry energy obtainedwith the experimental flow data of Ref. [53]. The shaded regionenclosed by a full line is obtained from quiescent low-mass x-raybinary mass and radius observations [71].

neutron star region. In this sense, we use Gogny functionals aseffective extrapolation guesses into high densities.

The pressures of Fig. 13 are validated against three setsof constraints. The area enclosed by a solid line is obtainedfrom the data in Fig. 9 of Ref. [71]. This delineates theEoS probability distribution for neutron stars assuming abaseline EoS and column densities that have atmospherescontaining both hydrogen and helium, as preferred by someobservations [71]. At densities above twice saturation, wecompare the Gogny pressures to the flow data obtainedassuming a stiff (gray dotted region) or soft (blue dashedregion) symmetry energy [53]. As expected, the functionalsfalling close to these constraints are those with the highestvalues of L: D1AS, D1P, and D280. The differences betweenmodels in the high-density regime are extreme. Other than thealready-discussed mechanically unstable models, we find thatD260 and D300 have equations of state that are almost an orderof magnitude below the empirical constraints. This anomalousbehavior will impact the structure of the corresponding neutronstars, as we shall see next.

We solved the TolmanOppenheimerVolkov equations tostudy the structure of pure neutron stars [72]. For simplicity,we considered stars formed only of neutrons.8 We stressonce again that the results obtained with Gogny functionalsat very high densities should be handled with care. We donot take them as realistic estimates, but rather as indications

8We also performed calculations in equilibrium, which yield verysimilar results.

Mas

s, M

[Sola

r Mas

ses]

Radius, R [km]

D1ND1MD1PD1ASD260D280D300

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

6 8 10 12 14 16

GRP

<

Caus

ality

FIG. 14. (Color online) Mass-radius relation for pure neutronstars obtained with eleven Gogny functionals. The shaded regionenclosed by a full line is obtained from quiescent low-mass x-raybinary mass and radius observations using atmosphere models thatinclude both hydrogen and helium [71]. The shaded region enclosedby a dashed line corresponds to the 90% confidence region for themass-radius relation of NGC 6397, as obtained in Ref. [24].

of the strengths and weaknesses of the functionals at theextremes of isospin asymmetry. Also, note that we use theGogny functional throughout the star, without any low-densitymatching to a crust EoS. In consequence, our less-massiveneutron stars have rather small radii.

We show the resulting mass-radius relations in Fig. 14.We compare once again with the region (red continuousregion) of the most probable mass-radius of Ref. [71]. Theblue area enclosed by a dashed line corresponds to the90% confidence region probability contours for the mass andradius of the object NGC 6397 qLMXB, using a heliumatmosphere (see right panel of Fig. 9 in Ref. [24]). In addition,we show in Fig. 14 the excluded region by causality [23]and the two measurements of neutron-star masses aroundM 2M [73,74].

Very few functionals predict maximum solar masses closeto this limit, and only four achieve a maximum mass above1.4M. Those that do are essentially the parametrizations thatfell within the constraints in the energy and the pressure ofpure neutron matter. D1AS has the largest maximum mass,just about MD1ASmax 2M at RD1ASmax 10 km. The maximummass for D1P raises up to MD1Pmax 1.98M at RD1Pmax 9 km.The mass-radius relation for these two functionals is relativelyclose to the probability distribution of Lattimer et al. [71] andfalls right through the constraints of Heinke and coauthors [24].D280, in contrast, falls at the edge of the Lattimer constraints,because its EoS runs just below D1P and D1AS. Above 0.5M,these three forces predict essentially constant radii, between11 and 12 km. At the high-mass end, however, D280 falls shortand yields a maximum mass of MD280max 1.74M.

054327-15


In addition to these three functionals, D1M also producesa sizable maximum mass, MD1Mmax 1.83M. Since D1M hasa substantially softer EoS, though, it reaches this mass byexploring far smaller radii, below R 10 km. The onlyother parametrization that is able to support a neutron starabove 1M is D1N. The maximum mass thus obtainedis MD1Mmax 1.29M. Note, however, that the mass-radiusrelation is biased towards small radii and falls well belowthe Lattimer predictions.

We show two more mass-radius relations in Fig. 14. Wealready discussed the fact that D300 and D260 have EoSswhich fall well below the constraints obtained by a variety ofempirical analyses. As expected, these two functionals cannotsupport heavy neutron stars. They also predict rather smallobjects, with minimum radii of the order of 6 to 8 km. Finally,we note that none of the remaining four Gogny functionals (D1,D1S, D250, GT2) are able to sustain sizable neutron stars. Onecan in principle trust these functionals up to the mechanicalinstability region, but their maximum masses are all below0.16M and their radii lie above 17 km. These unrealisticneutron star properties are a further indication of poor isovectorproperties in the Gogny functionals.

VI. CONCLUSIONS

We analyzed the Gogny functional predictions for bulksymmetric and isospin asymmetric nuclear matter. We usedeleven parametrizations from the literature. This is an extensiveset and has allowed for a comprehensive analysis. We provideanalytical expressions for all microscopic and macroscopicproperties. We note that the exchange term is responsible fornontrivial density and asymmetry dependencies.

There is a different variability of predictions depending onwhether one considers the isoscalar or the isovector sector.In general, we find that bulk properties are rather wellconstrained in symmetric matter. For single-particle potentials,the isoscalar components are very similar for all functionals inthe low-momentum region, below the Fermi surface. To someextent, these results are to be expected, since symmetric matterproperties as well as (low-momentum) finite-nucleus data areoften inputs in the fitting procedure of the functionals.

In contrast, we find a wide range of results in the isovectorchannels. At the single-particle level, isovector potentialsabove and below the Fermi surface can be extremely different,depending on the parametrization under consideration. As aconsequence, the effective masses and their isospin asymmetrydependence is somewhat unrestricted. When analyzed as afunction of momenta, effective masses show a wide rangeof values. Looking specifically at the isovector splitting ofthe effective mass at the Fermi surface, we find that mostfunctionals predict mn > mp. Having said that, the exactamount and asymmetry dependence is still uncontrolled.

At the bulk level, we analyzed the isovector propertiespredicted by Gogny functionals. In general, we find that thesaturation symmetry energy of these functionals falls withinempirical estimates. The density dependence, however, israther soft in most cases, with slope parameters that fall belowmost empirical estimates. We find that the slope parameter L isin most cases at its maximum (or decreasing) at saturation. The

three functionals with acceptable L parameters (D1P, D1AS,D280) have somewhat extremes values of S. This is an artificialtension, and we believe updated fitting protocols could helpimprove this behavior.

An analysis of the different contributions in the isovectorchannel shows that, by and large, the variability of isovectorproperties comes from the finite-range matrix elements. Forpure neutron matter, we also find a variety of results. Atsubsaturation, most functionals lie above the region predictedby microscopic many-body calculations. Above saturation, asubgroup of functionals (D1, D1S, D250, and GT2) predicts amechanical instability. Another subgroup (D260, D300, D1N,D1M) avoids the instability but predicts rather low pressuresas a function of density. Finally, a third subgroup (D1P, D1AS,and D280) have stiffer pressures. As a consequence, the lastsubgroup can sustain sizable neutron stars, with radii aroundR 11 to 12 km and maximum masses close to 2M.

These findings point out a poor fitting strategy, particularlyin the isovector sector. The present generation of functionalsshow a wide variability in the underlying matrix elementsof the Gogny interaction, even when they are separated inisospin contributions. Having access to analytical expressions,we identified a few key observables that could be used toimprove fitting procedures. The isovector mass splitting atthe Fermi surface, for instance, is proportional to Binn (anddepends mildly on i). The information from optical potentialanalysis [46,47] or microscopic calculations [45] can be used toconstrain this matrix element. Similarly, the high-momentumlimit of the isoscalar and isovector single-particle potentials isentirely set by the density-dependent zero-range terms of thefunctional. By imposing a realistic Lane isovector potential,one could in principle constrain these values. Finally, andpossibly more importantly, the bulk isovector properties cannowadays be used to fit the functionals [11]. By imposingmore realistic values of L, K , and, if necessary, neutron starproperties, the resulting Gogny functionals would provide abetter description of isospin-rich systems.

Because we do not have access to Gogny fitting protocols ordata, it is difficult to quantify how much of the variability thatwe have found is of a statistical or a systematic nature [16].We suspect, though, that it is the systematic uncertainty thatdominates. Most functionals have been fit from the startwith rather low symmetry energies, and the correspondingslopes turn out to be unphysically small. Similarly, the densityexponents i and the ranges i are often guessed initiallyand not allowed to change. It would be interesting to analyzewhether the data prefer other choices of these values. Wetherefore call for the publication of clearer-fitting protocolsand the associated correlation matrices. Similarly to what hasalready been done in Skyrme-like functionals, a covariancematrix analysis will help quantify, at least, the statisticaluncertainty of the data [61,75]. In addition, covariance analysisidentifies at once which parameters of the fit, if any, are overconstrained.

The results that we presented could be easily extended tostudy other bulk systems of interest. Having access to theanalytical formulas for the single and double Fermi-sphereintegrals of Eqs. (A2) and (B3), it is straightforward to extendthis analysis to spin-polarized systems. Preliminary studies

054327-16


have analyzed the properties of Gogny functionals in extremespin-polarized neutron matter [30,76,77]. Fitting these prop-erties to realistic interactions could be a way to constrain thespin sector of the functional. This would be complementary tothe traditional analysis based on decomposing the total energyof symmetric matter into spin and isospin channels [3,78].

Another extreme of spin and isospin can be achieved byanalyzing the case of a single impurity in a Fermi sea of amajority of a different species. In nuclear physics, the case ofa spin-up proton embedded in neutron matter has been of recentinterest. The quantum many-body problem can be solved inthis instance to quite a good degree of accuracy by using MonteCarlo techniques [41,42]. Comparing these results with theanalytical expressions predicted by the Gogny functional willprovide further means of fitting the spin and the isospin depen-dence of the effective interaction at the single-particle level.

We found that some Gogny functionals, both traditional andmodern, can lead to isospin or mechanical instabilitiesor, inthe case of D1, to both. Although this happens at densitiesbeyond those of interest for nuclear physics, the existenceand characterization of these instabilities has only beenattempted in a rather generic way [48,66,79]. In particular,the momentum dependence of these instabilities is relevantfor Gogny functionals [80]. An analysis that goes beyond thelow-q Landau-parameter approximation in isospin asymmetricmatter could shed further light into the isovector sector of theGogny interaction.

Regarding neutron stars, most Gogny functionals are unableto sustain sufficiently large stars that agree with present astro-physical observations. Mass and radii, however, are just twoof the many observational probes of neutron stars. It would beinteresting to study further properties, such as cooling, with thefew semirealistic density functionals that sustain sufficientlylarge neutron stars. In particular, the study of pairing in isospinasymmetric matter with Gogny forces has not yet been exten-sively undertaken [81] and could provide further constraintsto the parametrizations. Along similar lines, the extension tofinite temperature [32,82] of some of the present results couldprovide an insight into isospin transport properties.

Finally, we note that we have centered our analysis aroundinfinite bulk matter. As we have seen, this allows for a relativelysimple and analytical characterization of both microscopicand thermodynamical properties. Gogny functionals, however,are specifically designed to reproduce nuclear properties andhence the isovector properties of finite nuclei should also beprobed. Closed shell, isospin-rich nuclei, like 48Ca, 132Sn, or208Pb would be excellent potential starting points for such ananalysis [33]. Open-shell isotopes will explore pairing prop-erties, and the combination of asymmetry and pairing mightprovide insightful results [37]. Going further, one might wantto explore extensions of the Gogny functional, particularly to-wards more realistic finite-range density-dependent terms [34].

ACKNOWLEDGMENTS

This work is supported by STFC through Grants No.ST/I005528/1, No. ST/J000051/1, and No. ST/L005816/1.Partial support comes from NewCompStar, COST ActionMP1304.

APPENDIX A: ANALYTICAL EXPRESSIONS FORSINGLE-PARTICLE PROPERTIES

Single-particle potentials with the Gogny force are obtainedby integrating the effective interaction over one Fermi seain the HartreeFock approximation. We express our resultsin terms of the total density and the isospin asymmetryparameter = np

. The Fermi momenta are defined as

usual, with knF = kF

Documents

Isovector properties of the Gogny interaction