Properties of Strange Hadronic Matter in Bulk and in Finite Systems

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The hyperon-hyperon potentials due to a recent SU(3) Nijmegen soft-core potential model are incorporated within a relativistic mean field calculation of strange hadronic matter. We find considerably higher binding energy in bulk matter compared to several

a r X i v :n u c l -t h /0005060v 2 19 J u n 2000Properties of Strange Hadronic Matter in Bulk and in Finite

Systems

J¨u rgen Scha?ner-Bielich

RIKEN BNL Research Center,Brookhaven National Laboratory,

Upton,New York 11973

Avraham Gal

Racah Institute of Physics,The Hebrew University,Jerusalem 91904,Israel

The hyperon-hyperon potentials due to a recent SU(3)Nijmegen soft-core potential model are incorporated within a relativistic mean ?eld calculation of strange hadronic matter.We ?nd considerably higher binding energy in bulk matter compared to several recent calculations which constrain the composi-tion of matter.For small strangeness fractions (f S <～1),matter is dominated by N ΛΞcomposition and the calculated binding energy closely follows that calculated by using the hyperon potentials of our previous calculations.For larger strangeness fractions (f S >～1),the calculated binding energy increases substantially beyond any previous calculation due to a phase transition into N ΣΞdominated matter.We also compare bulk matter calculations with ?-nite system calculations,again highlighting the consequences of reducing the Coulomb destabilizing e?ects in ?nite strange systems.I.INTRODUCTION Bodmer and Witten independently highlighted the idea that strange quark matter,with roughly equal composition of u ,d and s quarks leading to a strangeness fraction f S =?S/A ≈1and a charge fraction f Q =Z/A ≈0,might provide the absolutely stable form of matter [1,2].Metastable strange quark matter has been studied by Chin and Kerman [3].Ja?e and collaborators [4,5]subsequently charted the various scenarios possible for the stability of strange quark matter,from absolute stability down to metastability due to weak

decays.Finite strange quark systems,so called strangelets,have also been considered [4,6].For a recent review of theoretical studies and experimental searches for strangelets,see Refs.

[7,8].

Less advertised,perhaps,is the observation made in our previous work [9,10]that metastable strange systems with similar properties,i.e.f S ～1and f Q ～0,might also exist in the hadronic basis at moderate values of density,between twice and three times nu-clear matter density.These strange systems are made out of nucleons (N ),lambda (Λ)and cascade (Ξ)hyperons.The metastability of these strange hadronic systems was established by extending relativistic mean ?eld (RMF)calculations from ordinary nuclei (f S =0)to multi-strange nuclei with f S =0.Although the detailed pattern of metastability,as well as the actual values of the binding energy,depend speci?cally on the partly unknown hyperon potentials assumed in dense matter,the predicted phenomenon of metastability turned out to be robust in these calculations [10,11].

Quite recently,Stoks and Lee[12]have challenged the generality of the above results for strange hadronic systems.These authors constructed G matrices for coupled baryon-baryon

channels,using an SU(3)extension[13]of the Nijmegen soft-core NSC97potentials[14]from the S=0,?1sector(to which data these potentials have been?tted)into the unexplored

S=?2,?3,?4sector.These G matrices were then employed within a Brueckner-Hartree-Fock(BHF)calculation of strange hadronic matter(SHM)in bulk.The results showed that NΛΞsystems are only loosely bound,and that charge-neutral strangeness-rich hadronic

systems are unlikely to exist in nature in metastable form,in stark contrast to our earlier ?ndings[9,10].

This vast di?erence in the predictions for the metastability and binding of SHM between

following a BHF methodology,which uses an SU(3)extrapolated form of the NSC97baryon-baryon potentials,and following a RMF methodology,which is based on mean?elds designed

to mimic the consequences of the Nijmegen hard-core potential model D[15],has prompted us to investigate possible origins of it.In this work we present calculational evidence for the incompleteness of the procedure applied by Stoks and Lee[12].We do so by reproducing

qualitatively their results for the instability and weak binding of NΛΞmatter in bulk, within a constrained RMF calculation in which the mean?elds are now designed to mimic

the consequences of the NSC97model used by Stoks and Lee.The constraints imposed by us,as a check,are identical with those imposed by these authors for the composition of SHM (see Fig.4of Ref.[12]).We argue that this is not the right way to identify minimum-energy

equilibrium con?gurations for SHM.Indeed,doing the unconstrained RMF calculation with the same NSC97-inspired mean?elds,we?nd qualitatively good agreement,for f S<～1 where the bulk matter is NΛΞdominated,between these new results and our old results

in model2[10].For f S>～1,the new unconstrained calculation results in considerably higher binding energies than ever calculated for SHM,due to a phase transition into NΣΞ

dominated matter.

The paper is organized as follows.In section II we describe the methodology of?nding equilibrium con?gurations within the RMF formalism,and the input mean?elds entering the new RMF calculations.Section III includes the results of these new calculations for bulk SHM,as well as for?nite multi-strange systems for which BHF calculations have not been done to date.The role of the Coulomb interaction in stabilizing charge-neutral strange systems is highlighted.Our results are summarized and discussed in section IV,where we also comment on the applicability of the SU(3)-extended NSC97potential.

II.METHODOLOGY AND INPUT

We adopt the Relativistic Mean Field(RMF)Model to describe strange hadronic matter (SHM)in bulk and for?nite systems of nucleons and hyperons.The model is an e?ective model where the parameters are adjusted to the known properties of nuclei and hypernuclei. We include in our extended RMF model all the1/2+baryons of the lowest SU(3)?avor octet, as well as hidden-strangeness meson exchange to allow for possibly strong hyperon-hyperon (Y Y)interactions.Here we use model1and model2of Ref.[9].The basic ingredients of

these models are the octet baryons matrix B,the matrices V(8)

µand V(1)

of the vector meson

octet and singlet,respectively,and the two scalar mesonsσandσ?.In addition,a Coulomb term is included in?nite system calculations.The Lagrangian is given as

L=Tr¯B(iγµ?µ?gσBσ?gσ?Bσ??m B)B

?13σ3?c2 ?νσ??νσ??m2σ?σ?2

?g(8)v αTr¯Bγµ V(8)µ,B +(1?α)Tr¯Bγµ V(8)µ,B ?g(1)v Tr¯BγµB·Tr V(1)µ?12Tr m2v V?µVµ+1

3gωN=

gωΣ=gωΞ

gρN=

For theΞnuclear interaction,measurements of the?nal-state interaction ofΞhyperons produced in the(K?,K+)reaction on12C in experiments E224at KEK[24]and E885at the AGS[25]indicate a nonrelativistic potential U(N)

Ξ,nr

of about?16and?14MeV or less,

respectively.Below we will actually vary the value for U(N)

Ξto check its e?ect on the binding

energy of SHM.

The hyperon(Y)potentials U(Y′)

Y in hyperon(Y′)matter,in the absence of direct ex-

perimental data,depend to a large extent on the assumptions made on the underlying Y Y

interactions.In model1,which does not useσ?andφexchanges,the potentials U(Y′)

Y are

rather weak,less than10MeV deep.The exchange of these hidden-strangeness mesons is included in model2,where theσ?coupling to hyperons is adjusted so that the potential of a single hyperon,embedded in a bath ofΞmatter at nuclear saturation densityρ0,becomes

U(Ξ)Ξ(ρ0)=U(Ξ)

(ρ0)=?40MeV,(4)

in accordance with the attractive Y Y interactions of the Nijmegen potential model D[10].

The resulting U(Λ)

Λis about?20MeV,considerably more attractive than in model1.Indeed

the few doubleΛhypernuclear events observed so far in emulsion require a relatively strong ΛΛattractive interaction[26],which lends support to model2over model1,but the actual situation for the other,unknown,Y Y′channels could prove more complex than allowed for by either model.All that may be said at present is that,as far as theΛΛinteraction strength is concerned,model2is a more realistic one than model1.

Since there appears some confusion in the recent literature[12,27]regarding how to calculate self consistently the properties of SHM in bulk,we will ponder on the thermody-namically consistent methodology in more detail.As we will demonstrate in the following, a major property of SHM within the SU(3)-extended NSC97model might have been over-looked in these works.Here we focus on the thermodynamically correct treatment in the RMF approximation.The extension to BHF calculations is then straightforward.Very recently,thermodynamically consistent BHF calculations ofβ-stable strange matter in neu-tron stars have been performed by Baldo et al.[28],using the NSC89model[29]for the Y N interactions,and by Vida?n a et al.[30],using the SU(3)-extended NSC97model[13]for the Y N and Y Y interactions.

In general,we can describe the system by the grand-canonical thermodynamic potential ?,which depends on the temperature T,the volume V,and the independently conserved chemical potentialsµα.At T=0,the pressure is given by:

P(µα)=??(µα,T=0)/V.(5)

For SHM in bulk,since the isospin dependence is usually suppressed,there are just two conserved charges in bulk which are the baryon number B and the strangeness number S. The chemical potentials of the individual baryons can be related to the corresponding baryon chemical potentialµB and strangeness chemical potentialµS by

µi=B i·µB+S i·µS.(6) This ensures that the system is in chemical equilibrium or,in other words,that the strangeness and baryon numbers are conserved in all possible strong-interaction reactions in the medium,such as

Σ+N ?Λ+N Λ+Λ?Ξ+N Λ+Ξ?Σ+Ξ (7)

The Hugenholtz –van-Hove theorem relates the Fermi energy of each baryon to its chemical potential in equilibrated matter

µi =E F,i =

6π2,(9)

where γi is the spin-isospin degeneracy factor.If the solution results in an imaginary Fermi momentum,the particle is not present in the system and the corresponding density is set to zero.In Brueckner theory,one has to solve for an equation of the form

µi =E i (k F,i )=m i +k 2F,i

m 2σσ2?b 4

σ4?12m 2ωω20+12m 2φφ20

+ i =B,l γi 2m 2σ

σ2+b 4σ4+12m 2ωω20+32m 2φφ20+

i =B,l γi k 2+m ?i 2,(11)

respectively.The binding energy per baryon is then obtained by subtracting the properly weighted combination of the rest masses from the energy density of the system

E/A=

ρB =

ρΛ+ρΣ+2ρΞ

UΞ,in models1and2[9,10].For f S=0,there are only nucleons in the system and one

gets the standard equation of state of nuclear matter as function of baryon density.The equilibrium density of nuclear matter is determined by minimizing the binding energy with

respect to the baryon density.The resulting minimum value of binding energy per nucleon is shown then at f S=0in the plots of Fig.1.Next,we increase the strangeness fraction

from zero on,and the system of equations adjusts itself at each?xed value of f S to?nd the

corresponding baryon densities ensuring chemical equilibrium(Eq.(6)).The minimum value of the binding energy per baryon as function of baryon density at each?xed strangeness

fraction is then plotted in Fig.1for the corresponding value of f S.In this way,one gets the

binding energy of SHM as function of the strangeness fraction.It turns out thatΣhyperons do not appear at any value of f S in both models1and2.To display the dependence on the

Ξnuclear potential we chose three di?erent values,UΞ=?10,?18,?28MeV.The variation in the plots of model1is quite pronounced.For UΞ=?28MeV,the minimum is at a

?nite value f S=0.6,with a binding energy per baryon of?17.4MeV.For shallowerΞ

potentials,this minimum disappears and slightly strange matter with f S≈0.1is the most strongly bound con?guration.On the other hand,in model2,varying UΞdoes not lead to

drastic changes.The minimum in the binding energy per baryon for UΞ=?28MeV,at f S=1.3with E/A=?24.6MeV,is shifted to E/A=?21.5MeV for UΞ=?18MeV and

to E/A=?19.6MeV at a slightly higher value f S=1.4for UΞ=?10MeV.The reason is

that in model2the minimum is generated by the Y Y interactions which have been adjusted according to Eq.(4),so that the binding energy curves in model2are not as much a?ected by changing UΞas compared to the e?ect of this change in model1.Note that the constraint (4)ensures that pureΞmatter(f S=2)has the same binding energy,E/A=?8.9MeV,in all three cases.PureΞmatter is always unbound in model1due to the missing attraction in the Y Y channels.

Substantial departures from the universality(Eq.(4))assumed in Refs.[9,10]for the Y Y

interactions occur in the most recent SU(3)-extension of the Nijmegen soft-core potential

model NSC97[13].In particular,theΣΣandΞΞinteractions are predicted to be highly attractive in some channels,leading to bound states.We wish to examine the consequences of this model in our RMF calculation of SHM.The Y Y interactions of Ref.[13]are implemented in our calculation by adjusting the coupling constants of theσ?meson?eld to reproduce qualitatively the hyperon binding energy curves shown in Fig.2of Ref.[12]for set NSC97f. All the other coupling constants are held?xed,so that we still get the hyperon potentials of Eq.(3)in nuclear matter.The resulting binding energy curves,of each baryon species j in its own matter B(j)j,are depicted in Fig.2as function of density.For nucleons,we again use the parameterization TM1so as to get the correct binding energy at the correct saturation densityρ0.Note that NSC97f does not reproduce the correct nuclear matter saturation point,but gives a too shallow minimum at a too high density(see Fig.2of Ref.

[12]).No binding occurs forΛhyperons,and B(Λ)Λreaches+20MeV already at rather low

density,ρ=0.1fm?3.This strong repulsive“potential”is due to the very weak underlying

ΛΛinteraction in the extended NSC97f model which is incompatible with the fairly strong ΛΛattraction necessary to explain the observed doubleΛhypernuclear events(see[26,39] and references therein).On the other hand,Σmatter is deeply bound,by?33MeV per baryon atρ=0.58fm?3which is twice as deep as ordinary nuclear matter,andΞmatter has a binding energy of?23MeV per baryon atρ=0.39fm?3.It is clear from Fig.2that a

0.00.2

0.40.60.8 1.0

Density ρ (fm ?3)

?40?30?20?1001020

B i n d i n g e n e r g y p e r A (M e V )TM1N ΛΣ

FIG.2.Binding energy per nucleon (N )in nucleon matter,compared to the binding energy per hyperon (Λ,Σ,Ξ)in its own hyperonic matter.The hyperonic parameters were chosen to reproduce the binding energy minima of Fig.2in Ref.[12].mixture of Σand Ξmatter must be very deeply bound too,unless there is an overwhelmingly repulsive interaction between Σand Ξhyperons.Actually,the interaction between Σand Ξhyperons is the most attractive one in the extended NSC97f model,giving rise to the deepest bound ΣΞdibaryon state [13].Independently,increasing the number of degrees of freedom will also result in a more deeply bound state.This is the case,for example,when going from unbound neutron matter (γ=2)to bound nuclear matter (γ=4).Therefore,one expects that ΣΞmatter is in fact more deeply bound than Σor Ξmatter alone.In the following,we will denote the parameterization responsible for the curves of Fig.2as model N.

Fig.3shows the binding energy of SHM per baryon in model N as function of strangeness fraction.For comparison,the curves for model 1and 2from Fig.1are also plotted.We performed two di?erent calculations for model N:one where the hyperon fractions χi =ρi /ρB are held equal by hand (χΛ=χΣ=χΞ)as done in Ref.[12],and the self-consistent one where the hyperon fractions are determined so as to ensure chemical equilibrium (denoted as “equil.”in the ?gure).It is evident from Fig.3that the self-consistent treatment gives a substantially lower energy,since it is the unconstrained minimum-energy solution.The disagreement between the results of the two calculations increases with f S .At f S =1the di?erence between the two curves amounts to nearly 10MeV.The curve for model N follows closely the one for model 2up to a strangeness fraction of f S =0.95.At larger values of f S a deep minimum develops due to the highly attractive interaction between the Σand Ξhyperons.Note that an equal mixture of Σ’s and Ξ’s gives f S =1.5and the minimum of the curve is close to that point,i.e.E/A =?79MeV at f S =1.45.For larger values of f S ,the curve denoted by N rises again,ending up at f S =2with the same binding energy per

0.00.5 1.0 1.5 2.0

strangeness fraction f s

?80?60

?40?200

B i n d i n g e n e r g y p e r A (M e V )model 1model 2model N TM1

χΛ=χΣ=χΞparison of the binding energy of SHM per baryon in models 1(dash),2(dash-dot)and N (solid).The upper solid line shows the result for the constrained case of equal hyperon fractions (χΛ=χΣ=χΞ),the lower one shows the curve for the correct,unconstrained equilibrium calculation.

hyperon as that shown in Fig.2for pure Ξmatter.The deep minimum around f S =1.5results from the deep binding of Σplus Ξmatter which is stronger than seen in Fig.2for matter composed of either one of these species separately.This deep minimum structure can only be reached when the hyperon fractions are allowed to adjust self consistently.Fixing the hyperon fractions,as done in Ref.[12],will always give a curve which is higher in energy,risking the loss of some important features of the model.In fact,the curve for the constrained calculation [12]ends up at f S =4/3due to the particular constraint applied.

The deep minimum seen in Fig.3emerges due to a second minimum in the corresponding equation of state at high strangeness fraction,connected with a ?rst order phase transition from matter consisting of N ΛΞbaryons to N ΣΞbaryonic matter.This transition is visu-alized in Fig.4where the binding energy is drawn versus the baryon density for several representative ?xed values of f S .For f S =0.8,there is a global minimum at a baryon density of ρB =0.27fm ?3.A shallow local minimum is seen at larger baryon density at ρB =0.72fm ?3.Increasing the strangeness fraction to f S =0.9lowers substantially the local minimum by about 20MeV,whereas the global minimum barely changes.At f S =1.0this trend is ampli?ed and the relationship between the two minima is reversed,as the min-imum at higher baryon density becomes energetically lower than the one at lower baryon density.The system will then undergo a transition from the low density state to the high density state.Due to the barrier between the two minima,it is a ?rst-order phase transition from one minimum to the other.

Fig.5demonstrates explicitly that the phase transition involves transformation from N ΛΞdominated matter to N ΣΞdominated matter,by showing the calculated composition

0.0

0.20.40.60.8 1.0Density ρ (fm ?3

)

?30?20?1001020

B i n d i n g e n e r g y p e r A (M e V )TM1N ΛΞN ΣΞ

f s =0.80.91.0FIG.4.Transition from N ΛΞmatter to N ΣΞmatter in model N.A second minimum appears at higher density for a strangeness fraction of f S =0.8,becomin

g more stable for higher strangeness fraction (f S =1).0.0

0.5 1.0 1.5 2.0strangeness fraction f s

0.00.20.4

0.60.81.0

P a r t i c l e f r a c t i o n N Λ

ΣΞmodel N

position of SHM in model N versus the strangeness fraction.Σhyperons appear around f S =1.

of SHM for model N as function of the strangeness fraction f S .The particle fractions χi for each baryon species change as function of f S .At f S =0,one has pure nuclear matter,whereas at f S =2one has pure Ξmatter.In between,matter is composed of baryons as dictated by chemical equilibrium.A change in the particle fraction may occur quite drastically when new particles appear,or

existing ones disappear in the medium.A sudden change in the composition is seen in Fig.5for f S =0.2when Ξ’s emerge in the medium,or at f S =1.45when nucleons disappear.The situation at f S =0.95is a special one,as Σ’s appear in the medium,marking the ?rst-order phase transition observed in the previous ?gure.The baryon composition alters completely at that point,from N Ξbaryons plus a rapidly vanishing fraction of Λ’s into ΣΞhyperons plus a decreasing fraction of nucleons.At the minimum of the binding energy curve in Fig.3,the matter is composed mainly of Σ’s and Ξ’s with a very small admixture of nucleons.

0.00.5 1.0 1.5 2.0

strangeness fraction f s

?22?20

?18?16?14?12?10?8?6

B i n d i n g e n e r g y p e r A (M e V )FIG.6.Binding energy of multi-strange ?nite systems built on a 100Sn nuclear core in models 1(circles)and 2(squares,stars),with and without Coulomb e?ects.The curves for bulk SHM (solid lines)are also shown.

Last but not least,we discuss ?nite systems of SHM for which the Coulomb interaction plays a signi?cant role.We also performed calculations switching o?the Coulomb interaction in order to study separately the e?ects due to the possibly strong Y Y interactions.Yet,we will not pursue for ?nite systems the implications of the deep minimum found in model N,but rather stick in the following to the more conservative models 1and 2.Our choice for the Ξpotential is U Ξ=?18MeV,in accordance with the recent observations [24,25].Following the procedure outlined in [9,10]we start from a “normal”nucleus (here 100Sn,for example)and add Λhyperons to the system.As soon as the strong decay reactions p Ξ?→ΛΛor n Ξ0→ΛΛare Pauli-blocked,we start adding Ξhyperons.For each given system of nucleons and hyperons,we check that the two reactions are blocked also in reverse,so that the whole multi-strange nucleus is metastable,decaying only via weak interactions.

These calculations also include the e?ects of theρmeson?eld,in order to properly account

for symmetry-energy contributions to the binding energy.

Our results are summarized in Fig.6.The solid lines are the binding energy curves for

SHM in bulk for model1(upper curve)and model2(lower curve),taken from Fig.1for the value UΞ=?18MeV.The?lled symbols denote systems where the Coulomb interaction

is included,the open symbols stand for the case where the Coulomb interaction has been

switched o?.For model1,shown in circles,SHM is slightly more bound than the core nucleus 100Sn.For the highest strangeness fraction we found,f S=0.375,the system is bound by E/A=?9.5MeV,compared to E/A=?8.3MeV for100Sn.When the Coulomb interaction

is turned o?,the curve shifts down by several MeV per baryon.The chargeless core nucleus 100Sn is then bound by E/A=?12.2MeV,whereas the strangeness richest object has now a slightly lower binding energy per baryon of E/A=?11.9MeV.Hence,the main e?ect for the increased binding energy of SHM in model1actually comes from Coulomb e?ects.

The negatively chargedΞ?hyperons neutralize the positively charged protons,making SHM

more bound than ordinary nuclei.Still,the binding energies found are well above the curve for bulk matter due to?nite size e?ects,such as surface tension.

The situation is di?erent in model2,for which results are shown by squares in Fig.

6.Due to the attractive Y Y interactions in this model,the binding energy per baryon

increases substantially to a value of E/A=?19MeV at f S=1.3,which is even deeper

than the binding energy of nuclear matter in bulk.This high value of binding is obtained irrespective of whether or not the Coulomb interaction is included,since for such high values

of strangeness fraction the total charge fraction of the system is close to zero.Obviously,

in model2the tremendously increased binding energy of SHM originates mostly from the attractive Y Y interactions,and only to a minor extent from the reduced Coulomb repulsion.

Note that the binding energy for the deepest lying systems in model2is close to the values for SHM in bulk matter which are shown by the lower solid line.These systems are quite

heavy,with mass numbers of about A=400and higher,so that?nite size e?ects become

quite small.

In addition,we also plotted the binding energies of purely hyperonic systems which

consist ofΛΞ0Ξ?hyperons solely and,therefore,do not need to be Pauli blocked in order

to keep them metastable[9,10].Since these systems can decay only via weak interaction, arbitrary numbers for the three di?erent hyperon species are allowed.We?nd that purely

hyperonic objects are bound up to E/A=?12MeV per hyperon.The binding energies, denoted by stars in the?gure,follow the trend of the bulk calculation curve(solid line).

IV.SUMMARY AND CONCLUSIONS

In the present work we have calculated the minimum-energy equilibrium composition of bulk SHM made out of the SU(3)octet baryons N,Λ,ΣandΞ,over the entire range

of strangeness fraction0≤f S≤2,for meson?elds which generate,within the RMF model,qualitatively similar baryon potentials to those generated from the SU(3)extension

of the NSC97potential model[13]using the BHF approximation[12].Our main results

are displayed in Fig.3which shows that SHM is comfortably metastable in this model N for any allowed value of f S>0.The NΛΞcomposition and the binding energy calculated for equilibrium con?gurations with f S<～1resemble those of model2in our earlier work

[9,10].The use of models2and N[10,13]implies sizable Y Y attractive interactions which di?er,however,in detail between the two models.The model of Ref.[13]yields particularly attractiveΞΞ,ΣΣandΣΞinteractions,but vanishingly weakΛΛand NΞinteractions. We remark that this extent of weakness is in fact ruled out by the little information one has fromΛΛhypernuclei[26]and fromΞ-nucleus interactions[24,25].On the other hand, model2of Ref.[10]accounts more realistically for the attractiveΛΛand NΞinteractions, but ignores altogetherΣhyperons which require exceptionally strong binding in order to overcome the strong-interactionΣB→ΛB conversion which in free space releases about 75MeV.Yet,all these di?erences between the two models regarding the relative size of interactions within NΛΞdominated matter hardly matter when it comes to establishing the stability and binding pattern of this multi-strange matter.In this sense,SHM is a robust phenomenon.The metastability of SHM has also been recently con?rmed within the modi?ed Quark-Meson Coupling Model[40].

The di?erence between models2and N clearly shows up for f S>～1,whereΣ’s replace Λ’s in model N due to their exceptionally strong attraction toΣandΞhyperons.Figs.3,4, 5of the present work give evidence for a phase transition,from NΛΞdominated matter for f S<～1to NΣΞdominated matter for f S>～1,with binding energies per baryon reaching as much as80MeV.This e?ect has gone unnoticed in previous works which by constraining the composition of matter in bulk did not allow for the most general minimum-energy equilibrium con?gurations.In contrast,our model2produces a much smoother pattern of binding over the entire range of f S,with a gain of only approximately5MeV per baryon (at f S≈1.3)for the bulk matter calculation.However,for?nite multi-strange systems the gain can be considerably bigger,due to getting rid of most of the Coulomb repulsion for such approximately charge-neutral systems,amounting to almost11MeV per baryon for the examples of Fig.6.

We checked also for the critical strength of theΞ-nuclear potential below which?nite systems of SHM would consist only of nucleons andΛhyperons in model1.Of course,the critical value for UΞdepends on the size of the system.For a nuclear core of16O with8Λ’s ?lling up the s-and p-shells,Ξ’s cannot be added to the system for a potential shallower than U cΞ=?13MeV.In the case of56Ni,this critical value shifts to U cΞ=?10MeV.For the100Sn nuclear core used to demonstrate?nite systems of SHM in our present calculation (see Fig.6),we?nd a critical strength U cΞ=?7MeV for which theΞN→ΛΛstrong-interaction conversion is barely Pauli-blocked.However,UΞneeds to become more attractive than the above critical values demonstrate,in order that the corresponding multi-strange ?nite systems also remain particle stable.We remind the reader that the value UΞ=?18 MeV used in the?gure was designed to agree with the present phenomenological estimates [24,25].In bulk matter,Ξ-nucleus potentials as repulsive as UΞ=+40MeV still admit boundΞ’s just before SHM gets unbound at f S=0.7.The reason for this behavior is that the constraint f S=0.7introducesΞ’s in matter even though their interaction is repulsive. The Fermi momentum of theΛ’s become su?ciently high so that it pays to create some seemingly unfavorableΞ’s in order to lower theΛFermi momentum.Note that SHM in model2always containsΞ’s,irrespective of the nature of UΞ,by virtue of the attractive underlying Y Y interactions.

While it is true that the RMF model2is a schematic model and is linked only indirectly to the underlying baryon-baryon interactions,it is nevertheless constrained byΛandΞnuclear

phenomenology,and by the fewΛΛhypernuclear species reported to date.The extrapolation to Y Y channels which underlie the hyperon potentials in hyperon matter is more conservative

in this model than in model N inspired by the SU(3)extension of the NSC97potential model [13].We emphasize that although the NSC97model[14]has been tuned up to reproduce

certain characteristics ofΛhypernuclei,particularly its version NSC97f,the predictions elsewhere of these models appear invariably ruled out by whatever experimental hints one has to date.In addition to the exceedingly weakΛΛand NΞinteractions already mentioned

above,the NSC97model overbindsΛhyperons in nuclear matter(UΛ～?38MeV)and gives rise to quite attractiveΣnuclear potential(UΣ～?20MeV)in BHF calculations [27],whereas the phenomenology ofΣ?atoms[18]and‘hypernuclei’[22]indicates a much

weaker,if not a repulsive,Σnuclear potential.Furthermore,the NSC97model gives rise to a sizableΣnuclear symmetry energy which is opposite in sign[27]to that of the earlier NSC89 model[29]and,more importantly,to that established phenomenologically[18,22,23].We therefore suggest that the consequences of the NSC97model in dense SHM,as exempli?ed here,should be taken with a grain of salt.More dedicated work is required to amend the pitfalls of this model(for a recent discussion in this direction see Ref.[41]).

ACKNOWLEDGMENTS

JSB thanks RIKEN,Brookhaven National Laboratory and the U.S.Department of En-ergy for providing the facilities essential for the completion of this work.The work of AG is supported in part by a DFG trilateral grant GR243/51-2.AG also wishes to thank Larry McLerran,Sidney Kahana,and John Millener for their hospitality during a supported visit to the Nuclear Theory Group at the Brookhaven National Laboratory in February2000.

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