Hyperons in neutron stars and supernova cores

0.63 MB·

by  M. Oertel

EPJ manuscript No.
(will be inserted by the editor)
Hyperons in neutron stars and supernova cores
Micaela Oertel1, Francesca Gulminelli2, Constanc¸a Providˆencia3, and Adriana R. Raduta4
1 LUTH,CNRS, Observatoire deParis, Universite Paris Diderot, 5 place Jules Janssen, 92195 Meudon,France
2 ENSICAEN,UMR6534, LPC ,F-14050 Caen c´edex, France
3 CFisUC, Department of Physics, Universityof Coimbra, Portugal
4 IFIN-HH,Bucharest-Magurele, POB-MG6, Romania
6
1 Received: date/ Revised version: date
0
2 Abstract. Thepropertiesofcompactstarsandtheirformationprocessesdependonmanyphysicalingredi-
n ents.Thecomposition andthethermodynamicsof theinvolvedmatterisoneofthem.Wewill investigate
a here uniform strongly interacting matter at densities and temperatures, where potentially other compo-
J nents than free nucleons appear such as hyperons, mesons or even quarks. In this paper we will put the
4 emphasis on two aspects of stellar matter with non-nucleonic degrees of freedom. First, we will study the
phase diagram of baryonic matter with strangeness, showing that the onset of hyperons, as that of quark

] matter, could be related to a very rich phase structure with a large density domain covered by phase
h coexistence. Second, we will investigate thermal effects on theequation of state (EoS), showing that they
t
- favor the appearance of non-nucleonic particles. We will finish by reviewing some recent results on the
l impact ofnon-nucleonicdegreesfreedom incompact starmergersandcore-collapse events,wherethermal
c
u effects cannot be neglected.
n
[ PACS. PACS-key discribing text of that key – PACS-key discribing text of that key

1
v
1 Introduction Forcorecollapsematter,thefulldensity,temperature,
5
3 and Yq-dependence have to be included within the EoS.
4 Thiscomplexityisthemainreasonwhyuntilrecentlyonly
0 Thepropertiesofcompactstars,theirformationprocesses a few hadronic EoSs existed for core collapse simulations.
0 as well as binary mergersdepend on many different phys- These are the one by Hillebrandt and Wolff [1], used by
1. ical ingredients, among them the thermodynamic proper- some groups performing supernova simulations, that by
0 ties of the involved matter entering via the equation of Lattimer and Swesty [2] and finally that by H. Shen et
6 state (EoS). There is an intrinsic connection between the al. [3]. The two latter, publicly available, are most com-
1 properties of matter contained in the EoS for the macro- monly used in core-collapse simulations. They use differ-
v: scopic description of astrophysicalobjects and the under- ent nuclear interactions, but are based on the same lim-
i lying fundamental interactions between particles on the iting assumptions: they take into account non-interacting
X
microscopic level. This makes the study of the aforemen- α-particles, a single heavy nucleus and free nucleons in
r tioned systems very rewarding as they challenge our un- addition to the electron, positron and photon gas.
a
derstanding of nature on both scales. In recent years, several new models have been con-
It is not an obvious task to construct such an EoS. structed,enlargingthevarietyofnuclearinteractionmod-
The main difficulty arises from the fact that very large els. This helps to estimate the uncertainty on astrophys-
rangesof(baryonnumber)densities (10−10 fm−3 < n < ical simulations induced by our limited knowledge about
B
1 fm−3), temperatures (0 <T < 150 MeV) and h∼adron∼ic the interaction in hot and dense matter. Apart from em-
chargefractions(0<Y =n /∼n < 0.7)havetobecov- ploying different models, the full nuclear distribution at
Q Q B
ered. n here denotes the total ha∼dronic charge density, sub-nuclear densities has been included within different
Q
approaches (see e.g. [4,5,6,7,8,9,10]), showing consider-
which in many cases is just given by the proton density.
able differences to that obtained via the single nucleus
Within this range, the characteristics of matter change
approximationin the standard EoS employed in core col-
dramatically, from an ideal gas of different nuclei up to
lapse simulations [2,3].
uniformstronglyinteractingmatter,containinginthesim-
plest case just free nucleons and potentially other com- Up to now, much less effort has been devoted to the
ponents such as hyperons, nuclear resonances or mesons. high density (n > n ) and high temperature (T > 20
B 0
Even a transition to deconfined quark matter cannot be MeV)partoftheE∼OS,includingadditionalparticles,∼such
excluded. as hyperons and mesons or quarks. One reason might be
2 Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores
that the recent observation of two neutron stars with a sible consequences. Second, we will, independently of the
mass of about 2M [11,12] has triggered intensive dis- way they appear, discuss different works on the super-
⊙
cussion on the composition of matter in the central part saturationEoSincluding additionalparticleswith anem-
of neutron stars and its EoS, excluding in the standard phasis on thermal effects. The cold neutron star part will
picture additional degrees of freedom in super-saturation be discussed in other contributions to this issue [33]. We
matter,i.e.atdensitiesaboveabaryonnumber densityof will mention astrophysicalapplications, too,discussing in
n 0.16 fm−3, since they lead to a considerable soften- particularthe impactofadditionalparticlesonblackhole
0
≈
ing of the EoS [13]. This observation, however, does not formation.
exclude them, but only puts stringent constraints on the Thispaperisorganizedasfollows:inSec.2,wereview
respective interaction (e.g. [14,15,16,17,18,19,20,21,22, the main features of the thermodynamic analysis of the
23,24,25,26]). Different solutions with hyperonic and/or phase diagram of a -component system, and the phase
N
quark matter have been proposed without any definite diagramofbaryonicmatterwith strangenessis discussed,
conclusion. in particular, a possible strangeness driven phase transi-
tion and the Coulomb effects on the phase diagram; in
In addition, even if it turns out that finally in cold
Sec. 3 thermal effects on the EoS including non-nucleonic
neutron stars only nucleonic matter is present, in stellar
degrees of freedom are presented, we review relativistic
core-collapse events and neutron star mergers, matter is
mean field models for the EoS, including the hyperonic
strongly heated in addition to being compressed to den-
interaction, and discuss the effect of including hyperons
sities above nuclear matter saturation density. The tem-
andpions in the finite temperature EoS;in Sec. 4 the im-
peratures and densities reached can become so high that
pact of additional particles in astrophysical applications
a traditionaldescription in terms ofelectrons,nuclei, and
is referredand finally in the last sectionsome conclusions
nucleons is no longer adequate. Compared with the cold
are drawn.
neutron star EoS, temperature effects favor the appear-
ance of additional particles such as pions and hyperons
andtheybecomeabundantinthisregime.Atransitionto
quark matter is possible, too [27]. 2 A possible strangeness-driven phase
transition?
The opening of additional degreesof freedom in dense
matter could happen smoothly or could be accompanied
by a phase transition with a considerable effect on the 2.1 Thermodynamic analysis of the phase diagram
thermodynamicsandthehydrodynamicalevolutionofthe
system,seee.g.[27].Inthiscontext,thebestknownexam- In this section we would like to recall the main features
ple is the hadron-quark phase transition, expected to be of the thermodynamic analysis of the phase diagram, see
first order within the density and temperature range rel- e.g. [34]. The phase diagram of a -component system
N
evant for compact stars and core-collapsesupernovae, see is, at constant temperature, a -dimensional volume. If
N
e.g.the contributionby A. Sedrakianto this issue [28]. In thereareM differentcoexistingphases,the boundariesof
practice, anyway, the hadronic and the quark phases are thephasecoexistencedomain(s), nPj ; i=1,..., ; j =
describedwithindifferentmodels,thetransitionnecessar- 1,...,M,aredeterminedbythe( {+i1)}(M 1)conNditions
ilyshowsdiscontinuitiesinthethermodynamicquantities. of thermodynamic equilibrium bNetween the−m,
Itis less knownthat the appearanceofhyperonscouldbe
associated with a “strangeness driven” phase transition, ∂f ∂f
=...= =µ ; i=1,...,
too,similar to the liquid-gastransitionin nuclearmatter. ∂n ∂n i N
Adetailedstudyofthephasediagramofthen,p,Λ-system (cid:18) i(cid:19)P1 (cid:18) i(cid:19)PM
was recently undertaken in Refs. [29,30] within a non- ∂f ∂f
f + n =...= f + n =P,
relativistic mean-field model based on phenomenological − i∂ni! − i∂ni!
functionals. It was shown that under these assumptions Xi P1 Xi PM
first-andsecond-orderphasetransitionexist,andareex- (1)
pected to be explored under the strangeness equilibrium
where f denotes the free energy density and P the pres-
condition characteristic of stellar matter. In Refs. [31,26]
sure. Within the phase coexistence domain(s), a mixture
a phase transition at the onset of hyperons has been dis-
of different phases lowers the free energy of the system
cussed for relativistic mean field models, however in a
as compared with the solutions corresponding to individ-
model with very strong YY attraction.
ualphases.Mathematically,this is equivalentto the pres-
Within this paper, we want to address two aspects of
ence of a convexity anomaly of the thermodynamic po-
the EoS of super-saturation matter. First, we will inves-
tential in the density hyperspace, i.e. the free energy cur-
tigate the influence of a possible phase transition at the vature matrix, C = ∂2f/∂n ∂n , has at least one nega-
ij i j
respective hyperonic thresholds on the phase diagram of
tiveeigenvalue.Thenumberofcoexistingphasesisdeter-
super-saturationbaryonicmatter and discuss possible as- mined by the number of order parameters 1 or, in terms
trophysicalconsequences.Wewillnotdiscussthequestion
whether hyperonic interactions leadto such a phase tran- 1 An order parameter is defined here as the direction in the
sition or not, see e.g. [32] for a thorough study of that observable space corresponding to phase separation, see e.g.
point, but just assume that it exists and explore the pos- Ref. [35] for details
Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores 3
of localproperties,by the number of directions in density stability regions will be characterized by a back-bending
space where spinodalinstabilities develop, i.e. where den- behavior of µ (n ) .
B B |µS,µQ(L)
sity fluctuations are spontaneously amplified finally lead-
ing to phase separation. The latter quantity is related to
the number ofnegativeeigenvaluesofC ,suchthat 2.2 The phase diagram of baryonic matter with
neg ij
M = +N1. strangeness
neg
N
If the direction of phase separation is unique, which
Due to the large incompressibility of electrons present
corresponds to a one-dimensional order parameter, then
to ensure electrical charge neutrality, Coulomb interac-
the problem of phase coexistence in a -component sys-
N tions can have a considerable influence on the phase dia-
tem can by reduced to a problem of phase coexistence
gram, in particular if charged particles are involved. For
ina one-componentsystemby Legendretransformingthe
instance, the well known nuclear liquid-gas phase transi-
thermodynamical potential f with respect to the remain-
tion is strongly quenched by Coulomb effects [36,40,41].
ing ( 1)-chemical potentials [36].
N − On the contrary, in Ref. [30] it has been shown that,
Baryonicmatterwithhyperonscontainseightdifferent
if Λ-hyperons are the only strange baryons, the supra-
particle species. Under the condition of equilibrium with
saturationphasetransitionoccurringinthenpΛ+e-system
respect to the strong interaction, the number of relevant
is only slightly modified by Coulomb interactions com-
degreesof freedomis,however,reducedto three,the den-
paredwith the npΛ-system.The reasonis thatthis phase
sitiesbaryonnumberdensityn ,the totalbaryonic(elec-
B transitionis“strangenessdriven”,i.e.theorderparameter
tric) charge density n and the total strangeness density
Q isdominatedbyn andhasonlyasmallcomponentindi-
S
n . It represents, therefore, a three-component system in
S rectionof the charge.On the other hand, if the hyperonic
the terminology introduced above 2 It is important to re-
couplings are such that charged hyperons are abundant,
markherethattheuseofstrangenessasarelevantdegree
as predicted by many models favoring negatively charged
of freedom does not imply that n is conserved through-
S hyperons, the extension and localization of phase coexis-
out the evolution of the system. In particular, along the
tencedomainscouldbestronglymodifiedbytheCoulomb
strangeness equilibrium trajectory µ = 0 considered in
S interaction.Intheextremecase,itcouldevenmakeitdis-
this study, n obviously varies. Upon adding leptonic de-
S appear for neutron star matter.
grees of freedom in form of electrons and positrons,(elec-
Forthepresentstudywewillemploythenon-relativistic
tron)leptonnumber entersasanadditionalvariable.Due
energydensityfunctionaldevelopedbyBalbergandGal[42],
tothestrictelectricalneutralitycondition,however,charge
which is known to present a rich phase structure [29,30].
isnolongeranindependentdegreeoffreedomonceleptons
The total baryonic energy density is then given by the
areincludedandthesystemremainsthree-dimensional[38,
sum of mass, kinetic and potential energy density,
30], see also Ref. [39], in terms of the number densities
nB,nS and nL. In the absence of neutrinos, nL, the elec- e = n m c2+ ¯h2 τ +e ( n ) . (3)
tronlepton density, denotes here the net electrondensity, B i i 2m i pot { i}
nL =ne =ne− ne+ andchargeneutralitygivesnL =nQ i=Xn,p,Y (cid:18) i (cid:19)
−
Theequilibriumconditionsthusreducethedimension- The single-particle densities and kinetic energy densities
ality of the phase space from 8 (9 with electrons), corre- are thereby given by the Fermi integrals
sponding to the number of different particle species, to
three, described by baryon, lepton and strangeness num- 4π 2mi 32 8π3 2mi 25
n = F (βµ˜ ); τ = F (βµ˜ ),
ber densities. To further reduce the dimensionality for i h3 β 21 i i h5 β 23 i
studying phase coexistence, one may then perform the (cid:18) (cid:19) (cid:18) (cid:19) (4)
Legendretransformationwithrespecttoanyset(µB,µS), with F (η) = ∞dx xν being the Fermi-Dirac in-
(µ ,µ ) and (µ ,µ ). µ thereby stands either ν 0 1+exp(x−η)
S Q(L) B Q(L) Q(L) tegral, β = T−1 is the inverse temperature, m denotes
for µQ in a purely baryonic systems or for µL if electrons R i
the i-particle massand µ˜ the effective chemicalpotential
are included. In practice, we found that the order param- i
of particle species i-species.
eter is always one dimensional. This means that a single
ThepotentialenergydensityproposedbyBalbergand
Legendre transformation is enough to spot the thermo-
Gal [42] has the following form,
dynamics provided that the order parameter is not or-
thogonal to the controlled density. The most convenient e(BG)( n )= e(BG)(n ,n );
framework to easily access the physical trajectories is the pot { i} ij i j
one controlling the nB-density: Xi,j
δ
e(BG)(n ,n )= 1 ij (a n n +b n n
f¯(n ,µ ,µ )=f(n ,n ,n ) µ n µ n . ij i j − 2 ij i j ij i3 j3
B S Q(L) B S Q(L) − S S− Q(L) Q(L) (cid:18) (cid:19)
(2) nγij+1n +nγij+1n
Coexisting phases, if any, will then be characterized by + c i j j i), (5)
ij
equal values of µ = ∂f¯/∂n and P and the phase in- ni+nj
B B
Forsimplicity the same functionalformis employedinall
2 SeeRef.[37]foradiscussionofthephasediagramifstrong channelsanda unique value is usedfor the exponentgov-
equilibrium is not assumed. eringtheshort-rangerepulsion.Thevaluesofthecoupling
4 Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores
constants, listed in Table 1, have been chosen to satisfy
the experimental constraints available at that time, sym- 1300
metry arguments or, in the case of the NN-channel, to
agree with popular models. Concerning the latter, the re- 1250
sulting values are in good agreement with experimental
contraintsforthesaturationdensity,n =0.155fm−3,and 1200
0 V)
energypernucleonofsymmetricnuclearmatteratsatura- e
M 1150
tion, -15.9 MeV. The incompressibility modulus with 375 (B
MeV is, instead, largely overestimated. The well depths
m 1100
ofvarioushyperonicspeciesinuniformsymmetricnuclear
matter at saturation density are: UΛ(N)(n0)=UΣ(N)(n0)= 1050
26.6 MeV, U(N)(n ) = 22.8 MeV and the value of Λ-
− Ξ 0 − 1000
potential in Λ-matter U(Λ)(n /5)=-12.8 MeV. Both, the 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Λ 0 n (fm-3)
ΞN-andΛΛ-potentialsaretooattractiveinviewofactual B
experimental data. AGS-E885 data indicate for U(N)(n )
Ξ 0
thevalueof-14MeV[43].ThevalueofU(Λ)(n /5)canbe 0.25
Λ 0
related to the bond energy of double-Λ hypernuclei [44].
Experimental data for 6 He [45] suggest that a much 0.2
ΛΛ
larger value of U(Λ)(n /5) = -0.67 MeV is more realis-
Λ 0
tic. The situation of the ΣN-potential is ambigous but, 3) 0.15
-m
vcoeruyplipnrgobcaobnlsyt,anittsiswrheipchulosivveer.coAmlteertnhaetidvreawvablaucekssooff tthhee n (fi 0.1
originalparameterizationhavebeenproposedinRef.[20].
Thermodynamic consistency allows to infer the rela-
0.05
tionbetweenthe chemicalpotentialsµ ∂e /∂n
i ≡ B i|nj,j6=i
and the effective ones µ˜i as µi = µ˜i + mic2 +Ui, with 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ui =∂epot/∂ni.Allremainingthermodynamicquantities, n (fm-3)
such as pressure or entropy can then be derived from the B
energy density via standard thermodynamic relations.
Fig. 1. (Color online) Baryonic chemical potential (top) and
particle abundances(bottom) as a function of baryon number
2.2.1 The n,p,Y-system without electrons density for µS = 0 and µQ = 0 at T=1 MeV, employing the
BGI parameterization [42].
InordertohighlighttheeffectoftheCoulombinteraction,
we will start within this section by analyzing the phase
diagram of pure baryonic matter including the complete phase coexistence regions exist, induced by the onset of
baryon octet, but without electrons. The upper panel of each hyperonic family.
Fig. 1 illustrates the evolution of the baryonic chemical
Differentthermodynamicalconditions,i.e.differentval-
potential as a function of baryon number density at con-
ues of (µ ,µ ) and T, will obviously change the par-
stant values of µ = 0, µ = 0 and T=1 MeV 3. Three S Q
S Q ticle thresholds, their abundances, and the location of
back-bendingregionsexist,whichareallcorrelatedwitha
phase coexistence regions. By varying µ and µ , the
S Q
particlethreshold,seethebottompanelofFig.1wherethe
whole 3-dimensional phase diagram for a given temper-
corresponding particle abundances are displayed. Since
ature could be explored. Since we are mostly interested
within the model of Balberg and Gal [42] the hyperon-
here in astrophysical systems which can be considered in
hyperon(YY)-interactiondependsonlyweaklyonthepar-
strangeness changing weak equilibrium, µ = 0, we will
S
ticular channel, the order of the particle thresholds is
limit the analysis to this case. The corresponding projec-
mainly given by their respective rest masses and chem-
tions of the phase diagram on the n -n (panel (a)) and
B Q
ical potentials. In the present case, µ = µ = 0, this
S Q the n -n -plane (panel (b)) are represented in Fig. 2.
S Q
means that the Λ is the first one to appear, followed by
The arrows indicate the directions of phase separation.
thealmostdegenerateΣ-hyperonsandthentheCascades.
In general, increasing µ leads to higher values of n .
Aninvestigationoff¯(n )confirmsthatanyback-bending Q Q
B The µ = 0 trajectory considered in Fig. 1 is here rep-
Q
canbecuredbyaMaxwellconstructionandthatthemix-
resented with dashed lines. For the sake of completeness,
tureofstablephaseslowersthefreeenergycomparedwith
also the phase coexistence region of the (n,p,Λ) systems
the individual phases and corresponds thus to the ener-
at supra-saturation densities is illustrated as a hatched
getically favored solution. This means that three distinct
yellow area [29,30].
3 Thistemperaturevaluehasbeenchosenforcomputational Most of the time, the phase coexistence domains re-
convenience. The presented results are very close to the zero latedto the onsetofΛ-andΣ-hyperonsmergeto asingle
temperature limit. domainextending overa verylarge range of baryonnum-
Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores 5
Table 1. Coupling constants corresponding to the stiffest interaction proposed in Ref. [42], called BGI within this paper, and
those corresponding to the nuclear interaction used by Lattimer and Swesty [2] supplemented with the Λ-hyperon [30,46]. In
case of BGI, the only hyperon pair with non-vanishingisospin-component is Σ-Σ.
Parameter aNN bNN cNN aΛΛ bΛΛ cΛΛ aΛN bΛN cΛN γNN γΛN γΛΛ
set MeVfm3 MeVfm3 MeVfm3γNN MeVfm3 MeVfm3 MeVfm3γΛΛ MeVfm3 MeVfm3 MeVfm3γΛN
BGI -784.4 214.2 1936.0 -486.2 0 1553.6 -340.0 0 1087.5 2 2 2
LS220Λ -1636.2 214.2 1869.2 -486.2 0 1553.6 -340.0 0 1087.5 1.26 2 2
220g3 -1636.2 214.2 1869.2 -90.0 0 1000.0 -270.0 0 4000.0 1.26 3 3
Parameter aΣN bΣN cΣN aΞN bΞN cΞN aYY bYY cYY γΣN γΞN γYY
set MeVfm3 MeVfm3 MeVfm3γΣN MeVfm3 MeVfm3 MeVfm3γΞN MeVfm3 MeVfm3 MeVfm3γYY
BGI -340.0 214.2 1087.5 -291.5 0 932.5 aΛΛ 0/428.4 cΛΛ 2 2 2
220g3 450.0 214.2 250.0 -170.0 0.0 2900.0 aΛΛ 0/430.0 cΛΛ 3 3 3
increasingabsolute value of µ , closer to the threshold of
Q
0.4 neutral Λ-hyperons.At some criticalvalue,the two phase
0.35 m S=0 coexistence regions existing at µQ = 0 merge into one
single domain.
0.3 Asexpected,the componentofthe orderparameterin
3) 0.25 nQ-direction is more important if the number of charged
-m particlesparticipatinginthephasecoexistenceislarge,i.e.
(fQ 0.2 for large values of µQ. In fact, although the total charge
n 0.15 n israthersmallforlargeµ ,theabundancesofcharged
Q Q
baryons itself are large. For instance, the transition in-
0.1
duced by Λ- and Σ-hyperons shows an almost vanishing
0.05 (a) component in n -direction close to µ = 0, where three
Q Q
0 phase coexistence regions exist and the Σ-onset is almost
0.3 0.4 0.5 0.6 0.7 0.8 0.9
n (fm-3) degenerate,see Fig. 2, whereas it becomes larger with in-
B creasing µ , i.e. in the regions where only two phase
Q
| |
coexistence domains exist. The Ξ-induced phase transi-
0.4
tion has an order parameter with important contribution
(b)
0.35 along nQ whenever both Ξ0 and Ξ− are created as their
total charge cannot vanish. With increasing µ the Ξ−
0.3 Q
production threshold is shifted to higher densities, finally
3) 0.25 leaving the considered density domain. Consequently, the
-m charge dependence of the order parameter becomes very
(fQ 0.2 weak for large nQ.
n 0.15
0.1
2.2.2 Coulomb effects on the phase diagram
0.05 m =0
S
We now turn to investigate the influence of Coulomb ef-
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
fects on the phase diagram. For simplicity, we will con-
n (fm-3)
S sider only electrons and neglect other charged leptons or
mesons. The total free energy can be written as the sum
Fig. 2. (Color online) Phase diagram of the (n,p,Y)-system of a baryonic, leptonic and photonic contribution, f =
forµS =0atT=1MeVasprovidedbytheBGIparameteriza- fB+fL+fγ,where leptonsandbaryonsarecoupledonly
tion [42] in the nB-nQ (a) and nS-nQ (b) planes. The arrows via the strict electrical neutrality condition, nQ = nL.
indicatethedirectionsofphaseseparation.TheµQ =0trajec- Leptons and photons are well described by, respectively,
toryisillustratedwitha(red)dashedline.Thehatchedyellow fermionicandbosonicidealgases[2].Noneofthemaffects
area corresponds to the strangeness driven phase transition
the phase structure.
domain of the simpler (n,p,Λ)-system studied in Ref. [30].
In the case of the n,p,Λ-system, where the order pa-
rameter has only a small component in charge direction,
the effect of electrons on the phase diagram is small [30].
ber density, whereas the Cascade thresholds remainsepa- As discussed in the previous section, upon including the
ratedleadingto a seconddistinct phase transitionregion. full octet with all charged hyperons, the dependence on
This can be understood as follows: nonzero values of µ , the charged component becomes larger and we thus ex-
Q
positiveornegative,liftthedegeneracyoftheΣ-hyperons, pect more important Coulomb effects on the phase dia-
favoringthe appearance of chargedones, negative or pos- gram. This is confirmed by the results, see Fig. 3, where
itive depending on the sign of µ . The corresponding thephasediagramisdisplayedintheplanen -n .Asbe-
Q B L
threshold for Σ+ or Σ− is shifted to lower densities with fore, the arrows mark the directions of the order param-
6 Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores
eter. For the sake of completeness the phase coexistence
domain ofthe simpler (n,p,Λ,e)-systemis representedas m =0
0.25 S
a hatched yellow area. The qualitative structure of the
phasediagramissimilartothecasewithoutelectronsdis-
0.2
cussed before, see Fig. 2. It does not come as a surprise,
however,that the phase coexistence region extents over a 3)
-m 0.15
muchsmaller rangein n ,i.e. the phase transitionregion
is quenchedby CoulombBeffects, inparticular,for low nL- n (fL
values, where the the number of charged baryons partici- 0.1
patinginthephasetransitionislarge.Comparedwiththe
case without electrons, the direction of phase separation 0.05
isrotatedinordertoreducethedifferenceinn =n be-
L Q
tweenthe twophases.The reasonis the largeincompress- 0
0.3 0.4 0.5 0.6 0.7 0.8 0.9
ibilityofelectrons,effectivelysuppressingelectrondensity n (fm-3)
fluctuations.Formatterinβ-equilibrium,relevantforcold B
neutron stars, a large density domain is actually covered
by phase coexistence, roughly between 0.3 < n < 0.43 Fig. 3. (Coloronline)Phasediagramofthe(n,p,Y,e)-system
B
fm−3, see the dashed line in Fig. 3. ∼ ∼ atT=1MeV forµS =0as providedbyBGI parameterization
[42] in the nB-nL-plane. The (red) dashed curve marks the
As easy to anticipate, for small values of µ , corre-
Q pathcorrespondingtoβ-equilibrium.Theyellowhatchedband
sponding to the largest n -values shown in the figure,
L marksthephasecoexistenceregionobtainedwithinthesimpler
where the Λ and Σ production thresholds are sufficiently (n,p,Λ,e)-system [30].
different for the corresponding phase coexistence regions
to be separated,the coexistence domainassociatedto the
onsetofΛs inthe complete systemsits exactlyonthe top we do not expect any qualitative change upon including
ofthatcorrespondingtothe(n,p,Λ,e)-system.Atsmaller the full octet, with quantitative values of the same or-
µ values,i.e.forsmallvaluesofn ,thetwosystemshave der. These values are typically reached within the cool-
L L
differentphasecoexistenceregionswiththeoneofthefull ing proto-neutron star, meaning that effects of critical-
baryonicoctetwiderthanthatofthesimplermixturedue ity should be experienced if there is a strangeness-driven
to the participation of Σ-hyperons. In particular, in the phase transition.
leftbottompartofFig.3thelowdensityboundaryofthe In particular, the neutrino mean free path, λ, due to
phase coexistence region of the (n,p,Y,e)-system is dis- scatteringoffbaryonsshouldbestronglyinfluenced.Inor-
tinct from that corresponding to the (n,p,Λ,e)-mixture. der to explore this point, we show calculations of the lat-
There are two reasons for that. For the lowest considered ter including the long-rangecorrelations,essential for the
n -values, the low density phase of the restricted system study of criticality. The linear response approximation is
L
is purely nucleonic while, upon considering the full bary- employed.Inaddition,sincewefocushereontheimpactof
onic octet, the low density phase contains Σ−, too, the densityfluctuationsclosetothecriticalpoint,wherespin-
first hyperons to appear in that case. At slightly higher densityfluctuationsareexpectedtobe small[47,48],only
but still low n -values, the discrepancy arises from the the vector channel will be considered. Then, in the non-
L
composition of the high density phase, which contains Σ- relativistic limit for the baryonic components the mean
hyperons in addition to nucleons and Λ-hyperons. free path at temperature T of a neutrino with initial en-
ergy E is given by [49,50],
It is important to stress that the phase diagram will ν
obviouslydependontheassumedcouplingconstantsinall 1 1 G2
channels.Forinstance,ifarepulsivepotentialisemployed λ = λV(E ,T) = 16πF2 (1+cosθ)SV(q,T)(1−fν(k3))dk3.
forΣ-hyperons,assuggestedbytheanalysesof(π−,K+)- ν Z
(6)
spectra, their threshold will be shifted to higher densities
G denotes here the Fermi constant, θ is the angle be-
and the associated phase transition will consequently be F
tween the initial and final neutrino momentum (=k3), q
shifted.
isthetransferredenergy-momentum,q =(ω,q),andf is
ν
For the simpler n,p,Λ+e-system, it has been shown theFermi-Diracdistributionoftheoutgoingneutrino.SV
that this phase transition persists in this model at finite represents the dynamical response function in the vector
temperature,witha phaseseparationdirectionalmostin- channel. It is defined as
dependentofT [30].Increasingthetemperature,thewidth
2 1
ofthecoexistenceregion,shownforT =0inFig.3,atlow V(q,T)=
S −π1 exp( ω/T) ×
values of n shrinks, leading to the appearence of a criti-
L − −
calpointabovesomefinitevalueofT oftheorder15MeV cn
V
which survives up to very high temperature. It moves to cn cp cΛ ΠV(q,T) cp , (7)
whihgehreertnhLe-cvraitluiceaslwteimthpienrcarteuarseinagndtetmhepceorarrteusrpeo,nsedeinFgiegl.e4c-, (cid:0) V V V (cid:1) cVΛV 
 
tron fraction Y = n /n are shown. Since the charac- where ΠV(q,T) is the vector-polarization matrix for the
e L B
ter of the phase transition remains “strangeness-driven”, three species n,p, and Λ. In mean field approximation it
Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores 7
3 Thermal effects on the EoS including
30
Y non-nucleonic degrees of freedom
e 0.6
T
Crit
In heavy-ion collisions (HICs), core-collapse supernovae
V) 25 andneutronstarorblack-holeneutronstarmergers,mat-
e 0.4
M e ter is strongly heated and thermal effects on the EoS be-
T (Crit 20 Y cfeormenetiminpHorItCasntc.oOmfpacoreudrsew,itthhethceonadsittrioonpshyasriecavleervyendtisf-.
0.2
In particular, matter in HICs cannot be considered in
strangeness changing weak equilibrium, but rather the
15 zeronetstrangenessofthetwocollidingnucleiisconserved
0
0.3 0.35 0.4 duringtheentirecollisionduetotheveryshorttimescales.
nB (fm-3) However,thermaleffectsleadtoaconsiderableproduction
ofnon-nucleonicdegreesoffreedom,hyperons,nuclearres-
Fig. 4. (Color online) Electron fraction, Ye, and nB at the onances,strangeandnon-strangemesons,andintheearly
corresponding critical temperature for µS = 0 within the phase probably a quark-gluonplasma. This clearly shows
(n,p,Λ,e)-system. Figure taken from Ref. [30].
the importance of thermal effects on the composition of
matter. Therefore in compact star astrophysics, where in
additiontobeingheated,matteriscompressedtodensities
abovenuclearmattersaturationdensity,n ,non-nucleonic
10 10 0
Mean-field (a) (b) En = m n degrees of freedom are expected to occur, too.
+RPA E n=m n+2T This hasbeen recognizedshortlyafterthe firstdiscus-
vMean-field En=m n-2T sainodnsEaobSosuhtavheypbeereonndseavneldopqeudartkosstiundcyotldheniemuptraocnt osftaards-,
vl (m) 1 Mfieeladn- vl / Mean-field+RPA 1 daevsitoimlounetsaioolnnpaocorftnpidcrleoentsos,a-sntueecuahtnraodsnqhsutyaaprrkesr,omsneasettaee.nrgd.fo[m5rm4e]saoftoniros,nanaosnewatrhellyel
l review. Inhomogeneous matter in the outer layers and
0.1
0.1 the formation of the neutron star crust do not influence
0.30 0.32 0.34 0.36 0.38 0.40 0.30 0.32 0.34 0.36 0.38 0.40 strongly the proto-neutron star evolution, such that in a
nB (fm-3) nB (fm-3) first approximation it can be neglected. Therefore most
Fig.5.(Coloronline)(a)Neutrinomeanfreepathforthescat- of the works investigating thermal effects on the EoS,
tering off n, p, and Λ at T = 20 MeV along a constant-Ye = treat homogeneous matter at some given temperature or
0.2981 trajectory in the phase diagram for Eν = µν, µν ±T entropy per baryon and hadronic charge fraction, YQ =
as a function of the baryonic density, nB. The result of the nQ/nB, with values relevant for proto-neutron star evo-
mean-fieldapproximationiscomparedwithamean-field+RPA lution, see e.g. [55,56,57,58,59,60]. Only recently, some
calculation. (b) The ratio of themean free path within mean- models have been proposed, including non-nucleonic de-
field+RPA over mean-field approximation is shown. Figure grees of freedom and treating the low-density and low-
taken from Ref. [30]. temperatureinhomogeneousnuclearmatter,too,seee.g.[20,
27,61,62].
Quarkmatterformationisveryinterestinginthiscon-
text,inparticularifthetransitionoccursduringtheearly
post-bouncephase,sincethentherelatedphasetransition
reduces to the Lindhard functions [51] and in mean-field
is visible in the supernova neutrino signal [27]. Such an
+ RPA approximation it is the solution of the Bethe-
early phase transition requires, however, a low transition
Salpeterequations[48,49,50].Thecouplingconstants,ci ,
V density and therefore a relatively soft EoS which seems
are 1(n),0.08(p), 1(Λ) [52]. The residual p-h interac-
− − difficult to be reconciled with a maximum cold neutron
tion entering the calculation of the polarization matrix is
starmassabove 2M inagreementwithrecentobserva-
closely related to the curvature matrix without electrons ∼ ⊙
tions[11,12],seeRef.[63].Inthecontractingandpartially
[53]andreflectsthereforethecriticalityofthephasetran-
deleptonizedproto-neutronstar,higherdensitiesandtem-
sition.
peratures are reached, and the conditions are, therefore,
In Fig. 5 we show the calculation within the n,p,Λ,e- more favorable for quark matter to appear. Such a tran-
system4.Thedepletionofthemean-freepathinthevicin- sition could explain some gamma-ray-bursts or, via the
ity of the critical point is clearly visible. The possible im- scenario of so-called “quark-novae”, some unusual super-
pact of a phase transition on the hydrodynamics of core nova light-curves, see the discussion in Ref. [64] and ref-
collapse will be discussed in Sec. 4. erencestherein.Inaddition,quarkmatteratnottoo high
temperatures has a very rich phase structure related to
different color superconducting phases with consequences
for compact star phenomenology, for instance on neutron
4 Electrons are not relevant for this calculation. star cooling, see e.g. [65] for a review.
8 Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores
1
domH,eir.ee.,hwyepweriollncsoanncdenmtreastoenso,nahnaddprroensiecndtesgormeeessoeflefcrteeed- +2(∂µσ∗∂µσ∗−m2σ∗σ∗2)
EoS in order to illustrate thermal effects on the appear- +1(∂ δ∂µδ m2δ2)
anceoftheseparticles.Duetothecomputationalcomplex- 2 µ − δ
ity, only some microscopic calculations exist, see e.g. the 1 1 1
W† Wµν P† Pµν R† Rµν
BHF calculations of[66,67,68]. Most models employ phe- −4 µν − 4 µν − 4 µν ·
nomenological interactions, non-relativistic or relativistic 1 1
+ m2ω ωµ+ c (ω ωµ)2
mean field (RMF) ones. In the following section we will, 2 ω µ 4 3 µ
therefore, briefly introduce the latter, the former has al- 1 1
ready been discussed in Sec. 2.2. +2m2φφµφµ+ 2m2ρρµ·ρµ , (8)
Pions,andtolessextentkaons,havealreadybeencon-
sideredtwentyyearsagoaspossiblecandidatesforthehot whereψj denotesthefieldofbaryonj,andWµν,Pµν,Rµν
and dense matter in supernova cores and proto-neutron are the vector meson field tensors of the form
stars, see e.g. [55,56,69]. The authors of Ref. [69] argue Vµν =∂µVν ∂νVµ . (9)
that the temperature of the supernova core could be in- −
creased by the presence of pions. As a consequence, the σ,σ∗arescalar-isoscalarmesonfields,couplingtoallbaryons
number of electron neutrinos would increase resulting in (σ) and to strange baryons(σ∗), respectively. δ induces a
a higher neutrino luminosity favoring a successful explo- scalar-isovectorcoupling.
sion. However, the employed pion-nucleon interaction is In mean field approximation, the meson fields are re-
probably too attractive, and a more realistic interaction placed by their respective mean-field expectation values,
decreasesthenumberofpionseventuallypresentinsuper- which are given in uniform matter as
nova coresand thus the effect onthe neutrino luminosity.
For cold neutron stars, it is now commonly assumed that m2σσ¯+g2σ¯2+g3σ¯3 = gσinsi (10)
there is an s-wave πN repulsion, preventing pions from i∈B
X
foonrmthinegEaoSBfoosreh-Eoitnastnedindecnosnedemnasattteer. nMeogsletctrefcoernstimwpolrikcs- m2σ∗σ¯∗ = gσ∗insi (11)
i∈B
ity any interaction and use a free pion gas [20,46,61,62]. X
This will be the case for the results shown below, too. At m2δδ¯= gδit3insi (12)
least for the high temperatures and low densities thermal i∈B
X
effects shoulddominateandthe interactionshouldbe less m2ω¯+c ω¯3 = g n (13)
ω 3 ωi i
important. Let us mention that in Ref. [61] it is shown
i∈B
that a parameterized s-wave πN repulsive interaction al- X
lows to avoidpion condensationat low temperatures but, m2φφ¯= gφini (14)
sincetheeffectsoftheinteractionontheEoSareonlyim- i∈B
X
portantatlow temperatures,andpionic effects aregener- m2ρ¯= g t n , (15)
ρ ρi 3i i
ally not the dominantcontributionto the EoS,the model
i∈B
dependence of the results becomes very weak. X
where δ¯= δ , ρ¯= ρ0 , ω¯ = ω0 , φ¯= φ0 , and t rep-
h 3i h 3i h i h i 3i
resentsthethirdcomponentofisospinofbaryoniwiththe
3.1 Relativistic mean field models for the EoS convention that t3p =1/2. The scalar density of baryon i
is given by
The literature on phenomenological RMF models is large 1 M∗
ns = ψ¯ψ = k2 i f[ǫ (k)]+f¯[ǫ (k)] dk,
andmanydifferentversionsexist(seee.g.[70]).The basic i h i ii π2 √k2+M∗2{ i i }
idea is that the interaction between baryons is mediated Z (16)
bymesonfields.Thesearenotrealmesons,butintroduced and the number density by
onaphenomenologicalbasiswiththeirquantumnumbers
1
indifferentinteractionchannels.Earliermodels introduce n = ψ¯γ0ψ = k2(f(ǫ (k)) f¯(ǫ (k)))dk . (17)
non-linear self-couplings of the meson fields in order to i h i ii π2 i − i
Z
reproduce correctly nuclear matter saturation and prop-
f and f¯ represent here the occupation numbers of the
erties of nuclei, whereas more recently density-dependent
respective particle and antiparticle states with the single-
couplingsbetweenbaryonsandthemesonfieldshavebeen
particle energies, ǫ (k) = √k2+M∗2, which reduce to a
widely used. The Lagrangianof the model canbe written i
step function at zero temperature. The effective baryon
in the following form
mass M∗ depends on the scalar mean fields as
i
L= ψ¯j(iγµ∂µ−mj +gσjσ+gσ∗jσ∗ Mi∗ =Mi−gσiσ¯−gσ∗iσ¯∗−gδit3iδ¯, (18)
Xj∈B andtheeffectivechemicalpotentials,(µ∗)2 =(M∗)2+k2 ,
+g δ I g γ ωµ g γ φµ g γ ρµ I )ψ i i Fi
δj j ωj µ φj µ ρj µ j j are related to the chemical potentials via
· − − − ·
1 1 1
+2(∂µσ∂µσ−m2σσ2)− 3g2σ3− 4g3σ4 µ∗i =µi−gωiω¯−gρit3iρ¯−gφiφ¯−Σ0R . (19)
Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores 9
The rearrangementterm matter at saturation density is given, too, for which re-
cently a range
∂g ∂g ∂g
Σ0R = ∂nωjω¯nj +t3j ∂nρjρ¯nj + ∂nφjφ¯nj 14.1< E/A(n0)< 21.4MeV. (24)
j∈B(cid:18) j j j ∼ ∼
X
−∂∂gnσjσ¯nsj − ∂∂gnσ∗jσ¯∗nsj −t3j∂∂gnδjδ¯nsj .(20) hraalsnbuecelneadrefroivrecdesfr[7o8m].mTihcirsosqcuoapnictitcyalicsuplaatritoincsulwairtlhyininctehri--
j j j (cid:19) esting for the EoSof compactstars,completing the infor-
mation about symmetric matter, since very asymmetric
is present in density-dependent models to ensure thermo-
matter close to pure neutron matter is encountered. Ex-
dynamic consistency.
cept for the DDHδ-model, for which the value is too low,
In the present paper we consider a set of models fre-
the employed models lie within the indicated range. The
quently usedin the literaturethatsucceedin describinga
symmetry energy E and its slope, L,containing infor-
2M neutron star. In particular, we will show results for sym
⊙
mationabout the isospindependence of the EoS,too, are
three non-linear models, GM1 [71], TM1 [72] and TM1-
as well important in this respect. For instance, it is well
2[73],andtwodensity-dependentones,DDHδ [74,75]and
known that the radius of compact stars is very sensitive
DD2 [76]. For the GM1 parameterization, c =0, and the
3 to L [79,80,81]. The models considered here span a wide
δ-field is absent in GM1, TM1, TM1-2 and DD2. The
rangeofvaluesofL,withsomeofthembeingattheupper
TM1-2 parametrization has properties similar to TM1 at
end of possible values [82].
saturation but is stiffer at large densities. TM1-2 has a
Hyperonic data are scarce and it is therefore very dif-
quite large symmetry energy slope L at saturation (L =
ficulttoobtaininformationonthe interactionparameters
110MeV),thereforewehaveconsiredamodificationwith
in the hyperonic sector. Many recent works, see e.g. [18,
a smaller slope, L = 55 MeV, by including a ωρ mixing
23,84], use a procedure inspired by the symmetries of the
term which allows us to discuss the influence of the den-
baryon octet to reduce the number of free parameters.
sitydependenceofthesymmetryenergyontheresults.In
The individual isoscalar vector meson-baryon couplings
Table 2 we show two parametrizations for TM1-2 corre- can then be expressed in terms of g and a few addi-
ωN
sponding to the two values of L. The density-dependent tional parameters, α,θ,z =g /g , see e.g. [85] for details,
1 8
models assume g2 = g3 = c3 = 0 (no non-linear terms) as follows
and the couplings become density dependent,
gωΛ = 1− √2z3(1−α)tanθ , gφΛ =− tanθ+ √2z3(1−α) ,
gi(nB)=gi(n0)hi(x) , x=nB/n0 . (21) gωN 1− √z3(1−4α)tanθ gωN 1− √z3(1−4α)tanθ
There exist different parameterization employing mostly gωΞ = 1− √z3(1+2α)tanθ , gφΞ =− tanθ+ √z3(1+2α) ,
thesamefunctionalforms.WithintheDDHδandtheDD2 gωN 1− √z3(1−4α)tanθ gωN 1− √z3(1−4α)tanθ
pisaorsacmaleatrecroizuaptliionng,st,he followingformsareassumedforthe ggωωNΣ = 11−+√√z23z3((11−−4αα))ttaannθθ , ggωφΣN = 1−−ta√nz3θ(+1−√2z43α(1)−tanα)θ ,
1+b (x+d )2 g tanθ+ z (1−4α)
h (x)=a i i (22) φN = − √3 . (25)
i i1+ci(x+di)2 gωN 1− √z3(1−4α)tanθ
The following values are commonly assumed: tanθ =
and
1/√2, corresponding to ideal ω-φ-mixing, α = 1, and
h (x)=a exp[ b (x 1)] c (x d ) . (23)
i i i i i
− − − − z =1/√6. The latter value reflects an underlying SU(6)-
for the isovector ones. In the following we will show re- symmetry,andonly recentstudies in view of the observa-
sults for extensions of the STOS [3] and the statistical tionofhighmassneutronstarshaverelaxedthis assump-
model EoS by Hempel and Schaffner-Bielich [6] (BHB tion, for example [18,22,61,84], or even varied freely the
model),thathaveasunderlyingparametrizations,respec- hyperonic isoscalar vector couplings, see e.g. [26].
tively, TM1 and DD2, and for the non-relativistic EOS, In the isovector sector, not the same procedure is ap-
LS220[77].Asextradegreesoffreedompions,Λ-hyperons, plied, since this would lead to contradictions with the
orallhyperonsofthebaryonicoctethavebeenconsidered, observed nuclear symmetry energy. g is therefore left
ρN
and accordingly, we will add the termination π, Λ or Y as a free parameter, adjusted to the desired value of the
to the name othe EoS. The original references are given symmetryenergy,andtheremainingisovectorvectorcou-
in Table 3, together with the hyperonic interaction em- plings are fixed by isospin symmetry.
ployed. Forthe scalarsector,differentmethods areapplied.In
The wealth of nuclear data allows to constrain rea- Ref. [21] a symmetry inspired procedure is discussed to-
sonably the parameter values of the interaction between getherwiththeconstraintsimposedbyhypernucleardata.
nucleons. The corresponding parameter values of the dif- In [83], the value of the σΛ-coupling is takenfrom a fit of
ferentmodelscanbefoundintheabovereferencesandthe the binding energies of single Λ-hypernuclei resulting in
resulting nuclear matter properties are listed in Table 2. R = 0.621. In many other works, see e.g. Refs. [18,23,
σΛ
In addition to standard properties of isospin symmetric 26,61], the informationfromhypernucleardata on hyper-
nuclear matter, the energy per baryon of pure neutron onic single-particle mean field potentials is used to con-
10 Micaela Oertel et al.: Hyperonsin neutron stars and supernovacores
K Esym n0 B L E/A(n0)
[MeV] [MeV] [MeV] [MeV] [ MeV] [ MeV]
GM1 300 32.5 0.153 16.3 94 18.6
TM1 281 36.9 0.145 16.3 111 21.1
TM1-2 282 37.2 0.146 16.4 111/55 21.8/17.1
DD2 243 31.7 0.149 16.0 55 18.2
DDHδ 240 25.1 0.153 16.3 44 10.6
LS220 220 28.6 0.155 16.0 74 14.4
Table 2. Nuclear matter properties of themodels considered in thisstudy for symmetric nuclearmatter at saturation, except
for the last column where theenergy perbaryon of neutron matter at n is given with theneutron mass subtracted.
0
Model Nuclear Rσ∗Λ Rσ∗Ξ Rσ∗Σ RωY RφY UΛ(Λ)(n0/5) UΞ(Ξ)(n0/5) UΣ(Σ)(n0/5) Reference(s)
interaction [MeV] [MeV] [MeV]
BHBΛ DD2 0 - - 1 0 -5 - - [23]
BHBΛΦ DD2 0 - - 1 1 7 - - [23]
STOSΛ TM1 0 - - 1 0 6 - - [83]
LS220Λ LS220 - - - - - -5 - - [30,46]
STOSY TM1 0.67 1.23 0.67 1 1 -11 -8 5 [61]
GM1 Y6 GM1 0 0.55 0 2 2 -7 -10 13 [26]
TM1-2 Y1 TM1-2(111) 0 0 0 1 1 1.7 21.1 16.2 [26]
TM1-2 Y2 TM1-2(55) 0 0 0 1 1 1.7 21.1 16.2 [26]
TM1-2 Λ4 TM1-2(111) 1.68 1.68 1.68 1.5 2 -41.1 36.0 -21 [26]
TM1-2 Λ6 TM1-2(55) 1.58 1.58 1.58 1.5 2 -33.7 44.4 -12.8 [26]
DDHδ Y4 DDHδ 1.03 0 0 1.5 0.85 -5 79 62 [26]
220g3 LS220 - - - - - -2.73 -2.73 -2.73 [20]
Table 3. Summaryofcouplingparametersusedwithinthedifferentmodels.Thenuclearinteractionisindicatedinthesecond
column and those of hyperons in columns 3-7. The couplings to the isoscalar vector mesons are defined with respect to the
respective SU(6) values and the couplings to σ∗ are defined with respect to gσN. The coupling parameters of the two non-
relativistic models, LS220Λ and 220g3 are given in Table 1. In addition, the values of the hyperon single particle potentials in
hyperonmatteratn /5aregivenforinformation.Thelastcolumnindicatesthereference(s)totheoriginalwork.Themodelsin
0
theupperparttherebyconsideronlyΛ-hyperonswhereasthoseinthlowerpartallowallhyperonstohavenonzeroabundances.
strain the coupling constants. Let us emphasize that al- Among the relativistic models, the parameter sets TM1-
most no information is available on the hyperon-hyperon 2Λ4/TM1-2Λ6arechosenatthelimitofpresentingather-
(YY)-interaction apart from a few light double-Λ-hyper- modynamic instability at the onset of Λ-hyperons and
nuclei,thatconstrainonlythelowdensitybehavior.There- stable with respect to all other hyperons [26]. The other
fore, the corresponding couplings, within the RMF mod- models do not show any instability related to the onset
els, in particular σ∗ and φ are very poorly constrained. of hyperons. To summarize, we believe that the large set
In most recent models φ-mesons are added in order to of employed models and coupling constants gives a repre-
be compatible with the 2 M neutron star, whereas of- sentative estimate of the present theoretical uncertainties
⊙
ten σ∗ is neglected (see e.g. [16,18,23]). This leads, how- in the high density equation of state modelling at finite
ever, to a very repulsive YY-interaction already at very temperature.
lowdensities[26,61,86]whichdoesnotappearveryrealis-
tic in view of the double-Λ-hypernucleardata. In Table 3
we summarize the coupling parameters used within the
3.2 Hyperons and pions in the finite temperature EoS
different models employed here. We list, in addition, the
values of the corresponding hyperonic single particle po-
tentials in hyperonic matter. Data on the bond energy of Let us start with some comments on the EoS of cold β-
double-Λ-hypernucleicanbe reinterpretedinterms ofthe equilibratedneutronstarsincludingnon-nucleonicdegrees
Λ potentialinΛ matteratthe averagedensityofΛ inside offreedom,formoredetailssee[33].Bysimplearguments
those nuclei [44,87]. Mean-field calculations have shown based on the Pauli principle, for a system composed of
that in light nuclei (from He to C) the average Λ density fermionic particles, additional degrees of freedom tend to
is close to one fifth of the saturation density [44,87]. We softentheEoS.Inturn,thisreducestheneutronstarmax-
have thereforechosenthis density for the reference values imummass,eventuallybeingincontradictionwiththere-
of the hyperonic potentials. cently observed masses [11,12]. The way out is of course
that the interactionmust be much more repulsive at high
ThemodelLS220Λpresentsastrangeness-drivenphase density than presently assumed.This is true for hyperons
transition at the onset of Λ-hyperons [29,30], see Sec. 2. and quarks. On the quark side, this leads to the problem