Low-momentum interactions with Brown-Rho-Ericson scalings and the density dependence of the nuclear symmetry energy

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by  Huan Dong

Low-momentum interactions with Brown-Rho-Ericson scalings
and the density dependence of the nuclear symmetry energy
Huan Dong and T. T. S. Kuo
Department of Physics and Astronomy, Stony Brook University, New York 11794-3800, USA
R. Machleidt
Department of Physics, University of Idaho, Moscow, Idaho 83844, USA
We have calculated the nuclear symmetry energy Esym(ρ) up to densities of 4 ∼ 5ρ0 with the
effects from the Brown-Rho (BR) and Ericson scalings for the in-medium mesons included. Using
theVlow−k low-momentuminteractionwithandwithoutsuchscalings,theequationsofstate(EOS)
of symmetric and asymmetric nuclear matter havebeen calculated using a ring-diagarm formalism
1 wheretheparticle-particle-hole-holeringdiagramsareincludedtoallorders. TheEOSforsymmetric
1 nuclear matter and neutron matter obtained with linear BR scaling are both overly stiff compared
0 withtheempiricalconstraintsofDanielewiczetal. [9]. Incontrast,satisfactoryresultsareobtained
2 byeitherusingthenonlinearEricsonscalingorbyaddingaSkyrme-typethree-nucleonforce(TNF)
n to the unscaled Vlow−k interaction. Our results for Esym(ρ) obtained with the nonlinear Ericson
scaling are in good agreement with theempirical valuesof Tsang et al. [7] and Li et al. [10], while
a
those with TNF are slightly below these values. For densities below the nuclear saturation density
J
ρ0, the results of the above calculations are nearly equivalent to each other and all in satisfactory
0
agreement with theempirical values.
1

PACSnumbers: 21.65.Jk,21.65.Mn,13.75.Cs
]
h
t
- I. INTRODUCTION plied to the study of E . As it is well known, most
l sym
c realistic V contain hard cores, or strong short-range
u NN
repulsions. This feature makes these interactions not
n The nuclear matter symmetry energy is an important
[ as well as very interesting subject in nuclear and astro- suitableforbeingdirectlyusedinnuclearmany-bodycal-
culations;theyneedtobe‘tamed’beforehand. Formany
nuclear physics. As reviewed extensively in the litera-
1 ture [1–8], it plays a crucial role in determining many years, this taming is enacted by way of the BHF theory
v
whereV is convertedinto the BruecknerG-matrix. A
0 important nuclear properties, such as the neutron skin NN
1 of nuclear systems,structure ofnuclei near the drip line, complication of the G-matrix is its energy dependence
(see e.g. [26]), making it rather inconvenientfor calcula-
9 and neutron stars’ masses and radii. It is especially of
1 importance that constraintson the nuclear matter equa- tions. In the Vlow−k approach, a different ‘taming’ pro-
1. tionofstate(EOS)[9]andthedensity(ρ)dependenceof cedure is employed; it is performed by ‘integrating out’
the high-momentum components of V beyond a deci-
0 the symmetry energy Esym(ρ) [7,10] up to ρ≃4ρ0 have NN
1 been experimentally extracted from heavy-ioncollisions, mation scale Λ. In this way, the resulting Vlow−k is en-
1 ρ0 being the saturation density of symmetric nuclear ergy independent. Furthermore Vlow−k is nearly unique,
v: matter. There have been a large number of theoretical namely the Vlow−ks deduced from various realistic VNN
potentials (such as [27–30]) are nearly identical to each
i derivationsofE (ρ)using,forexample,theBrueckner
X Hartree-Fock(BsHymF)[11–13],DiracBHF[6,14–16],varia- other for decimation scale Λ≃2fm−1 [23, 24].
ar tional[17],relativisticmeanfield(RMF)[18]andSkyrme Using this Vlow−k interaction, we shall first calculate
HF [19] many-body methods. The results of these theo- the equations of state (EOS) E(ρ,α) for asymmetric nu-
reticalinvestigationshaveexhibited, however,largevari- clearmatter,fromwhichE (ρ)canbe obtained. Here
sym
ations for E (ρ). Depending on the interactions and E is the ground-state energy per nucleon and ρ is the
sym
many-body methods used, they can give either a ‘hard’ totalbaryondensity. α is the isospinasymmetry param-
E (ρ),inthesensethatitincreasesmonotonicallywith eter defined as α = (ρ −ρ )/ρ, where ρ and ρ de-
sym n p n p
ρ up to ∼ 5ρ , or a ‘soft’ one where E (ρ) arises to a note, repspectively, the neutron and proton density and
0 sym
maximum value at ρ ≃ 1.5ρ and then descends to zero ρ = ρ +ρ . Our EOS will be calculated using a ring-
0 n p
at ∼ 3ρ [4, 8]. It appears that the predicted behav- diagrammany-bodymethod[31–33]. Asweshalldiscuss
0
ior of E (ρ) may depend importantly on the nucleon- later,thismethodincludestheparticle-particlehole-hole
sym
nucleon (NN) interactions and the many-body methods (pphh) ring diagrams to all orders. In comparison, only
employed. thediagramswithtwoholelinesareincludedinthefamil-
In the present work, we shall calculate the nuclear iar HF, BHF andDBHF calculations. In other words,in
symmetry energy using the low-momentum interaction theseHFmethodsaclosedFermiseaisemployedwhilein
Vlow−k derivedfromrealistic NNinteractionsVNN using the ring-diagramframeworkthe effects fromthe fluctua-
arenormalizationgroupapproach[21–25]. Toourknowl- tionsoftheFermiseaaretakenintoaccountbyincluding
edge, this renormalized interaction has not yet been ap- the pphh ring diagrams to all orders.
2
ThenuclearsymmetryenergyE (ρ)isrelatedtothe matrix equivalence equations:
sym
asymmetric nuclear matter EOS by
T(k′,k,k2)=V (k′,k)
E(ρ,α)=E(ρ,α=0)+E (ρ)α2+O(α4). (1) NN
sym 2 ∞ V (k′,q)T(q,k,k2)
+ P NN q2dq, (3)
The contributions from terms of orders higher than α2 π Z0 k2−q2
are usually negligibly small, as illustrated by our results
insectionIII.Withsuchcontributionsneglected,wehave
Tlow−k(k′,k,k2)=Vlow−k(k′,k)
E (ρ)=E(ρ,1)−E(ρ,0). (2)
sym + 2P Λ Vlow−k(k′,q)Tlow−k(q,k,k2)q2dq, (4)
Thenthesymmetryenergyisjustgivenbytheenergydif- π Z0 k2−q2
ference between neutron and symmetric nuclear matter.
In calculating E (ρ), the above EOS clearly play an
sym
important role. In our calculation, we shall require that T(k′,k,k2)=Tlow−k(k′,k,k2);(k′,k)≤Λ. (5)
the NN interaction and many-body methods employed
should give satisfactory results for E(ρ,1) and E(ρ,0)
In the above V represents a realistic NN interaction
of, respectively, neutron and symmetric nuclear matter. NN
suchas the CDBonn potential [27]. P denotes principal-
TheuseofVlow−k alone,however,hasnotbeenabletore- valueintegrationandtheintermediatestatemomentumq
produce the empirical nuclear saturation properties, the
isintegratedfrom0to∞forthewhole-spaceT andfrom
predictedsaturationdensityandbindingenergyperpar-
ticle being both too large compared with the empirical 0 to Λ for Tlow−k. The above Vlow−k preserves the low-
valuesofρ ≃0.16fm−3andE ≃−16MeVforsymmetric energy phase shifts (up to energy Λ2) and the deuteron
nuclearma0tter[31,32]. Toimprovethe situation,itmay bindingenergyofVNN. SinceVlow−k isobtainedbyinte-
grating out the high-momentum components of V , it
benecessarytoincludetheeffectsfromBrown-Rho(BR) NN
is a smooth ‘tamed’ potential which is suitable for being
scaling [34–36] for the in-medium mesons, or a three-
used directly in many-body calculations.
nucleon force (TNF) [40]. BR scaling is suitable only for
thelowdensityregion;itsuggeststhatthemassesoflight
vector mesons in medium are reduced ‘linearly’ with the
density. We consider here the EOS up to about ∼ 5ρ
0
and at such high density the linear BR scaling is clearly
not applicable. In the present work we shall adopt the
nonlinear Ericson scaling [41] for the in-medium mesons
and apply it to our E (ρ) calculations. The effects
sym
fromthe linearBRandnonlinearEricsonscalingsonthe
nuclear EOS and symmetry energy will be studied.
The organization of this paper is as follows. In sec-
tionIIweshallbrieflydescribeourderivationofthelow-
momentum interaction Vlow−k using a T-matrix equiv-
alence approach. Some details about the calculation of
the EOS for asymmetric nuclear matter from this inter-
actionwiththepphhringdiagramssummedtoallorders FIG.1: Sampleringdiagramincludedintheequationofstate
willalsobepresented. The Ericsonscalingis anonlinear E(ρ,α). Each wave line represents a Vlow−k vertex. The HF
one-bubbleinsertions totheFermion lines areincludedtoall
extension of linear BR scaling. The difference between
orders.
themwillbeaddressedinthissection. Ourresultswillbe
presented and discussed in section III. A summary and
conclsion is contained in section IV. Weusearing-diagrammethod[31–33]tocalculatethe
nuclear matter EOS. In this method, the ground-state
energy is expressed as E(ρ,α) =Efree(ρ,α)+∆E(ρ,α)
where Efree denotes the free (non-interacting) EOS and
II. FORMALISM
∆E is the energy shift due to the NN interaction. In
our ring-diagram approach, ∆E is is given by the all-
We shall calculate E (ρ) using a low-momentum ordersumofthepphh ringdiagramsasillustratedinFig.
sym
ring-diagramapproach[31–33], where the pphh ring dia- 1. Note that we include three types of ring diagrams,
grams are summed to all orders within a model space of the proton-proton, neutron-neutron and proton-neutron
decimationscaleΛ. Inthisapproach,weemploythelow- ones. The proton and neutron Fermi momenta are, re-
momentum interaction Vlow−k [21–25]. Briefly speaking, spectively, kFp = (3π2ρp)1/3 and kFn = (3π2ρn)1/3.
this interaction is obtained by solving the following T- With such ring diagrams summed to all orders, we have
3
[32, 33] 10−15%atnormalnuclearmatterdensityandzerotem-
perature. (Pions are not scaled because they are pro-
1 tected by chiral symmetry.) Density-dependent nuclear
∆E(ρ,α)= dλ Ym(ij,λ) interactions obtained by applying the above scaling to
Z0 m ijkl<Λ the light mesons (ω, ρ and σ) which mediate the NN
X X
×Ym∗(kl,λ)hij|Vlow−k|kli, (6) potential have been employed in studying the properties
of nuclear matter [31, 32, 36, 37] and the 14C →14 N
where the transition amplitudes Y are obtaind from a β-decay [38].
pphh RPA equation [31–33]. Note that λ is a strength WeareinterestedintheEOSandE uptodensities
sym
parameter, integrated from 0 to 1. The above ring- as high as ρ≃5ρ , and at such high densities the above
0
diagram method reduces to the usual HF method if linear scaling is clearly not suitable. How to scale the
only the first-order ring diagram is included. In this mesons in such high density region is still by and large
case, the above energy shift becomes ∆E(ρ,α) = uncertain. We shall adopt here the Ericson scaling [41]
HF
21 ninjhij|Vlow−k|iji where nk=(1,0) if k(≤,>)kFp for which is an extension of the BR scaling. In this scaling,
proton and n =(1,0) if k(≤,>)k for neutron. a new relation for the quark condensate hq¯qi based on
k Fn
P
It is well known that the use of the free-space V chiral symmetry breaking is employed, namely
NN
aloneisnotadequatefordescribingnuclearpropertiesat
highdensities. Tosatisfactorilydescribesuchproperties, hq¯q(ρ)i 1
= . (10)
one may need to include the three-nucleon force [40] or hq¯q(0)i 1+ ρΣπN
the in-medium modifications to the nuclear interaction. fπ2m2π
In the present work, we shall employ in our EOS calcu- Note that this relation agrees with the linear scaling re-
lationsnuclearinteractionswhichcontainthein-medium lation of Eq.(8) for small ρ. The above scaling suggests
modifications suggestedby the Brown-Rho(BR) [34,35] a non-linear scaling for meson mass
and Ericson [41] scalings. These scalings are based on
the relation [34, 35, 39] that hadron masses scale with 1/3
∗
m 1
the quark condensate hq¯qi in medium as = (11)
m 1+D ρ
ρ0!
∗ 1/3
m hq¯q(ρ)i
m =(cid:18)hq¯q(0)i(cid:19) (7) with D = ρf0π2Σmπ2πN, and we shall refer to this scaling as
the nonlinear Brown-Rho-Ericson (BRE) scaling. Using
∗
where m is the hadron mass in a medium of density ρ, theempiricalvaluesfor(Σ , ρ , f , m ),wehaveD=
πN 0 π π
and m is that in free space. The quark condenstate hq¯qi 0.35±0.06. Inthepresentwork,weshallemploytheone-
measures the chiral symmetry breaking, and its density boson exchange BonnA potential [30] with its (ρ, ω, σ)
dependence in the low-density limit is related [42, 43] to mesons scaledusing both the linear (Eq.(9)) andnonlin-
the free πN sigma term ΣπN by ear (Eq.(11)) scalings. This potential is chosen because
it has a relatively simple structure which is convenient
hq¯q(ρ)i ρΣ
=1− πN (8) for scaling its meson parameters.
hq¯q(0)i f2m2
π π
wheref =93MeVisthepiondecayconstantandΣ =
π πN III. RESULTS AND DISCUSSIONS
45±7MeV [44]. Applying the above scaling to mesons
inlow-densitynuclearmedium,onehasthelinearscaling
Using boththe unscaledandscaledBonnApotentials,
[35]
we first calculate the ring-diagram EOS for symmetric
m∗ ρ nuclear matter to investigate if they can give saturation
=1−C (9)
m ρ properties in good agreement with the empirical values.
0
We employ the low-momentum interactions Vlow−k from
where m∗ and m are, respectively, the in-medium and these potentials using a decimationΛ=3.0fm−1, which
free meson mass, and C is a constant of value ∼ 0.15. is chosen because we are to study the EOS up to high
TheabovescalingwillbereferredtoasthelinearBRscal- densities of ∼ 5ρ . As shown in Fig. 2, the EOS (la-
0
ing. Nucleon-nucleoninteractionsaremediatedbymeson belled ’Vlow−k alone’) calculated with the unscaled po-
exchanges, and clearly the in-medium modifications of tential saturates at k ≃ 1.8fm−1, which is too large
F
meson masses can significantly alter the NN interaction. comparedwiththe empiricalvalue,anditalsooverbinds
These modifications could arise from the partial restora- nuclear matter. We then repeat the calculation includ-
tion of chiral symmetry at finite density/temperature ing the medium modifications from the BR scalings. For
or from traditional many-body effects. Particularly im- the linear BR scaling (Eq.(9)), we have used C =0.128,
ω
portant are the vector mesons, for which there is now C =0.113 and C =0.102. These parameters are chosen
ρ σ
evidence from both theory [35, 45, 46] and experiment so as to have satisfactory saturation properties, namely
[47, 48] that the masses may decrease by approximately theygiveE /A≃-15.5MeVandρ ≃0.17fm−3. InFig.
0 0
4
2wealsopresentourresultsobtainedwiththe nonlinear
120
BRE scaling (Eq.(11)) using parameters D =D =0.40
and D =0.30. They were chosen to providωe satisfρactory 100 Vlow-k alone
σ linear BR
results for E0/A and ρ0. It is of interest that for den- 80 nonlinear BRE
sities . ρ0 the EOS given by the linear BR and nonlin- V] 60 Vlow-k with TNF
e
ear BRE scalings are practically eqivalent to each other. M
40
From Eqs.(7-11), we see that the parameters D and C A [
obtained from the density dependence of quark conden- /0 20
sates is 0.29.D .0.41 and C ≃D/3. It is noteworthy E
0
that the C and D parameters we have employed in the
-20
EOS calculations agree well with the above theoretical
values. -40
As also seen from Fig. 2, the above equivalence be- 0.8 1 1.2 1.4 1.6 1.8 2 2.2
-1
gins to disappear for densities larger than ρ . There the k [fm ]
0 F
EOSgiven by the linear scaling is much stiffer than that
given by the nonlinear one; the difference between them
becomes larger and larger as density increases. In ad- FIG.2: Ring-diagramEOSscalculatedwiththeVlow−k inter-
action alone, with the linear BR scaling of Eq.(9), with the
dition to the above two EOS, we have also calculated
nonlinearBRE scaling of Eq.(11), and with theaddition of a
an EOS using the interaction given by the sum of the Skyrme-typethree-nucleon force (TNF).
unscaled Vlow−k and an empiriral Skyrme three-nucleon
force (TNF). The well-known emipirical Skyrme force
[49] is of the form we have considered the nonlinear scaling. As displayed
in Fig. 3, the EOS with the nonlinear BRE scaling is in
V = V(i,j)+ V(i,j,k), (12) much better agreement with the constraint than the lin-
Skyrme
ear BR one. It satisfies the constraint well except being
i<j i<j<k
X X
slightly above the constraint at densiies near ∼4.5ρ . It
0
where V(i,j) is a two-nucleon momentum dependent in- isofinterestthattheEOSusingVlow−k withtheSkyrme-
teraction,andV(i,j,k)isazero-rangethree-nucleoninter- type TNF exhibits even better agreement with the con-
action which has playedan indispensible role for nuclear straint.
saturation. For nucleons in a nuclear medium of density
ρ, this three-nucleon force becomes a density-dependent
two-nucleon force commonly written as
t 100
3
V (i,j)= ρδ(~r −~r ). (13)
ρ 6 i j 3]
m
InFig. 2theEOSlabelled’Vlow−k withTNF’isobtained V/f
e
using the combinedinteraction of Vlow−k (unscaled) and M 10
Vρ. The parameter t3 is adjusted so that the resulting P[
EOSgivessatisfactorysaturationpropertiesforsymmet- linear BR
nonlinear BRE
ric nuclear matter. The EOS shown has t =2000 MeV-
3 V with TNF
fm6. low-k
1
It is of interest that the above three EOS (linear BR 1 1.5 2 2.5 3 3.5 4 4.5 5
and nonlinear BRE, TNF) are nearly identical for densi- ρ/ρ
0
ties .ρ , butthey deviate fromeachotherwith increas-
0
ingdensities. Withoutexperimentalguidelinesaboutthe
nuclearmatterEOSaboveρ ,itwouldbe difficulttode- FIG. 3: Comparison of the calculated equations of state for
0
termine which of these three EOS has the correct high symmetricnuclearmatterwiththeconstraint(solid-linebox)
of Danielewicz [9]. See text for more explanations.
density behavior. Fortunately, heavy-ioncollision exper-
iments conducted during the last severalyears have pro-
vided us with constraints of the EOS at high densities. So far we have studied the effects of the BR scalings
Danielewicz et al. [9] have obtained a constraint on the andtheTNFthree-nucleonforceontheEOSforsymmet-
EOS for symmetric nuclear matter of densities between ric nuclear matter. The neutron matter EOS is also an
2ρ and 4.5ρ , as shown by the red solid-line box in Fig. interestingandimportanttopic[50,51]. Itplaysacrucial
0 0
3. Comparing our three EOSs with their constraint, the roleindeterminingthenuclearsymmetryenergiesaswell
linear BR EOS is clearly not consistent with the con- asthepropertiesofneutronstars. Itshouldbeofinterest
straint and should be ruled out. This linear scaling is to study also the effects of the above BR/BRE scalings
suitable for low densities, but definitely needs modifica- and the TNF force on the EOS of neutron matter. Us-
tionathighdensities. Itisprimarilyforthispurposethat ing the same Vlow−k ring-diagram framework employed
5
for symmetric nuclear matter and the same C, D and t3 80
parameters, we have caculated the neutron matter EOS ρ=0.090 fm-3 ρ=0.090 fm-3
upto4.5ρ0. Ourcalculatedneutron-matterEOSaredis- 70 ρρ==00..232382 ffmm--33 ρρ==00..232382 ffmm--33
played in Fig. 4. Danielewicz et al. [9] have given two V] 60
e
different constraints for the neutron matter EOS: a stiff M
one(upperblacksolid-linebox)andasoftone(lowerred )[ 50
0
solid-linebox)whicharebothdisplayedinFig. 4. As we ρ, 40
(
cansee, the linear BREOS is againproducing too much E
pressure. The nonlinear BRE EOS agrees well with the α)- 30
stiff constraint (upper box) while the TNF EOS is fully ρ(, 20
E
within the soft constraint box. To further test these two
10
EOS(nonlinearBREandTNF),itwouldbeveryhelpful
tohavenarrowerexperimentalconstraintsontheneutron 0
matter EOS. 0 0.2 0.4 0.6 0.8 α2 0.2 0.4 0.6 0.8 1
FIG. 5: Ring-diagarm equations of state of asymmetric nu-
clear matter. See text for more explanations.
100
3m] obtained with the ’Vlow−k with TNF’ interaction while
V/f for the right panel the ’nonlinear BRE’ interaction is
e used. As seen, E(ρ,α) varies with α2 almost perfectly
M 10 linearly, for a wide range of ρ. (Note that in Fig. 5
P[
we plot the energy difference E (ρ,α)−E (ρ,0).)
linear BR sym sym
nonlinear BRE Thisisadesirableandremarkableresult,indicatingthat
Vlow-k with TNF ourring-diagramsymmetryenergycanbeaccuratelyob-
1 tained from the simple relation given by Eq.(2), namely
1 1.5 2 2.5 3 3.5 4 4.5 5
theenergydifferencebetweenneutronandsymmetricnu-
ρ/ρ
0 clear matter.
In Fig. 6, the ’shaded area’ represents the empirical
FIG. 4: Comparison of the calculated equations of state for constraint, Eq.(14), of Li et al. [10]. As seen, there are
neutron-matter with the constraints of Danielewicz [9]. See large uncertainties in the high-density region. The em-
text for more explanations. pirical relation Eq.(15) of Tsang et al. [7] is given by
the ’second curve from bottom’ in the figure. As seen,
the density dependence of this relation is slightly below
The symmetry energyE is a topic ofmuchcurrent
sym
the softest limit (lower boundary of the shaded area) of
interest, and extensive studies have been carried out to
Eq.(14). Our ’nonlinear BRE’ results are in the middle
extract its density dependence from heavy-ion collision
oftheshadedarea,ingoodagreementwiththeempirical
experiments[7,10]. Basedonsuchexperiments,Li et al.
constraint of [10]. Our results with the TNF force are
[10] suggested an empirical relation
below the empirical ones of both [10] and [7], giving a
E (ρ)≈31.6(ρ/ρ )γ; γ =0.69−1.1, (14) softer density dependence than both. It may be noticed
sym 0
that for densities (ρ . ρ ), the calculated and empirical
0
for constraining the density dependence of the symme- results are all in good agreement with each other. The
try energy. Also based on such experiments, Tsang et symmetry energies given by them at ρ0 are all close to
al. [7] recently proposeda new empiricalrelation for the ∼ 30MeV, which is also the only well determined em-
symmetry energy, namely pirical value. Furthermore, our calculated symmetry en-
ergies all increase monotonically with density. We have
C ρ 2/3 C ρ γi required our nuclear matter EOS to satisfy certain em-
s,k s,p
Esym(ρ)= + (15) pirical constraints, and with such requirements it may
2 ρ 2 ρ
(cid:18) 0(cid:19) (cid:18) 0(cid:19) be difficult for our present calculations to have a soft
where Cs,k = 25MeV, Cs,p = 35.2MeV and γi ≈ 0.7. It Esym(ρ) as soft as the supersoft one of [8] which satu-
shouldbeusefulandofinteresttocheckifourcalculated rates at density near ∼1.5ρ0.
E (ρ) is consistent with the above relations. Wehavefoundthatoursymmetryenergiescanbewell
sym
Using the ring-diagram framework described earlier, fitted by expressions of the same forms as Eqs.(14) and
we have calculated the ground-state energy E(ρ,α) for (15),withthe exponentsγ andγ treatedasparameters.
i
asymmetric nuclear matter. (Recall that the asymmetry In Table I, we compare the exponents determined from
parameter is α = (ρ −ρ )/ρ.) Some representative re- ourresults withthe empiricalonesof[10] and[7]. The γ
n p
sultsareshowninFig. 5: theresultsintheleftpanelare exponent given by the nonlinear BRE scaling is in good
6
agreement with the empirical values of [10]. The empir- saturationpropertiescansatisfactorilybereproducedby
ical γ of [7] is, however, about half-way between the γ including the medium corrections from the well-known
i i
obtained with ’nonlinear BRE’ and that with ’TNF’. linearBrown-Rhoscalingforthein-mediummesons,this
scaling produces an EOS which is too stiff compared
200 with the Danielewicz constraints. We have found that
theEOSobtainedwiththenonlinearBrown-Rho-Ericson
Tsang et al.
160 nonlinear BRE scaling are in good agreementwith the Danielewicz con-
Vlow-k with TNF straints. We have considered another method to ren-
V] der the effective interaction density dependent, namely
e 120
M adding a Skyrme-type three-nucleon force (TNF) to the
[m unscaled Vlow−k interaction. The EOS so obtained are
80
y alsoingoodagreementwiththeDanielewiczconstraints,
s
E
but the resulting neutron matter EOS is significantly
40 softer than that with the nonlinear scaling. The three
methods (linear and nonlinear scalings, and TNF) all
0 havereproducedwellthe empiricalsaturationproperties
0 1 2 3 4 5 ofnuclearmatter(ρ ≈0.17fm−3andE /A≈−15MeV),
0 0
ρ/ρ0 but their results at high densities are different. We have
determinedthe scaling parametersC (linear BRscaling)
and D (nonlinear BRE scaling) by fitting the above sat-
FIG. 6: Comparison of the density dependence of our cal-
uration properties. It is encouraging that the results,
culated symmetry energies with the empirical results of [7]
(dot-dash line) and [10] (shaded area). (0.102.C .0.128) and (0.30.D .0.40), so obtained
areactuallyingoodagreementwiththetheoreticalresult
D ≃0.35±0.06≃3C given by Eqs.(7-11).
Including the above medium modifications, we pro-
TABLE I: Comparison of the density exponents for the nu-
ceed to calculate the nuclear symmetry energies. We
clearsymmetryenergy Esym(ρ). Theexponentsγ andγi are
have found that the E (ρ,α) given by our asymmet-
defined respectively in Eqs. (14) and (15). sym
ric ring-diagram calculations depends on α2 almost per-
γ γ
i
fectly linearly. This is a rather surprising and useful re-
Li et al. [10] 0.69-1.1
sult, suggesting that the symmetry energy can be reli-
Tsang et al. [7] 0.7
ably obtained from the simple energy difference between
non-linear BRE 0.82 1.04
TNF 0.53 0.43 symmetricnuclearmatterandneutronmatter. Oursym-
metry energies obtained with the nonlinear BRE scaling
agree well with the empirical constraints of [10], and are
slightly above the empirical values of [7]. Our results
with the TNF force is slightly below the empirical re-
IV. SUMMARY AND CONCLUSION
sults of both [10] and [7]. The non-linear Ericsonscaling
has given satisfactory results for the equations of states
EmployingtheVlow−k low-momentuminteractions,we of nuclear matter and nuclear symmetry energies up to
havecalculatedthenuclearsymmetryenergyE (ρ)up
sym a density of ∼ 5ρ . We believe this scaling provides a
0
to a density of ∼ 5ρ using a ring-diagram framework
0 suitableextensionofthelinearBRscalingtomoderately
where pphh ring diagrams are summed to all orders. We
high densities of . 5ρ . Our calculated E (ρ) all in-
0 sym
firstcalculatetheEOSforsymmetricnuclearmatterand
crease monotonically with ρ up to ∼ 5ρ . It may be of
0
neutron matter and compare our results with the corre-
interest to carry out further studies about the possibil-
sponding empirical constraints of Danielewicz et al. [9].
ity of obtaining a supersoftsymmetry energywhich may
To have satisfactory agreements with such constraints,
saturate at some low density of ∼1.5ρ [8].
0
wehavefounditnecessarytoincludecertainmediumcor-
rectionstothefree-spaceNNinterations. Inotherwords, Acknowledgement We thank Professor Danielwicz
the effective NN interactions in medium are different for sending us the experimental data, and G.E. Brown
fromthoseinfreespace,andwhenusingtheminnuclear and E. Shuryak for many helpful discussions. This work
many-body problems it may be necessary to include the is supported in part by the U.S. Department of Energy
renormalizationeffectsduetothepresenceofothernucle- under Grant Nos. DE-FG02-88ER40388 and DE-FG02-
ons. We have considered severalmethods to incorporate 03ER41270(R.M.), andthe U.S. NationalScience Foun-
such medium corrections. Although the nuclear matter dation under Grant No. PHY-0099444.
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