Constraints on the quark matter equation of state from astrophysical observations

0.15 MB·English

by  G. F. Burgio

Constraints on the quark matter equation of state
from astrophysical observations
3
1
0
2

n
a
G. F.Burgio∗
J

7 INFNSezionediCatania,viaS.Sofia64,I-95123Catania,Italy
1 E-mail:[email protected]

] H. Chen
h
t PhysicsDepartment,ChinaUniversityofGeoscience,Wuhan430074,China
-
l E-mail:[email protected]
c
u
H.-J. Schulze
n
[ INFNSezionediCatania,viaS.Sofia64,I-95123Catania,Italy

E-mail:[email protected]
1
v
G. Taranto
0
6 DipartimentodiFisicaeAstronomia,Universita’diCatania,andINFNSezionediCatania,via
0 S.Sofia64,I-95123Catania,Italy
4
E-mail:[email protected]
.
1
0
3 We calculate the structure of neutron star interiors comprisingboth the hadronicand the quark
1
phases.Forthehadronicsectorweemployamicroscopicequationofstateinvolvingnucleonsand
:
v hyperonsderivedwithintheBrueckner-Hartree-Fockmany-bodytheorywithrealistictwo-body
i
X andthree-bodyforces. Forthedescriptionofquarkmatter, weuseseveraldifferentmodels,e.g.
r theMITbag,theNambu–Jona-Lasinio(NJL),theColorDielectric(CDM),theFieldCorrelator
a
method(FCM),andonebasedontheDyson-Schwingermodel(DSM).We findthatatwosolar
masshybridstarispossibleonlyifthenucleonicEOSisstiffenough.
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Constraintsonthequarkmatterequationofstatefromastrophysicalobservations G.F.Burgio
Thepossibleappearance ofquarkmatter(QM)intheinterior ofmassiveneutronstars(NS)is
oneofthemainissuesinthephysicsofthesecompactobjects. Calculations ofNSstructure, based
on amicroscopic nucleonic equation ofstate(EOS),indicate thatforthe heaviest NS,close tothe
maximum mass (about two solar masses), the central particle density reaches values larger than
1/fm3. Inthisdensityrangethenucleonsstarttoloosetheiridentity,andquarkdegreesoffreedom
playarole.
The value of the maximum mass of a NS is probably one of the physical quantities that are
most sensitive to the presence of QM, and the recent claim of discovery of a two solar mass NS
[1] has stimulated the interest in this issue. Unfortunately, while the microscopic theory of the
nucleonic EOS has reached a high degree of sophistication [2], the QM EOS is poorly known at
zero temperature and at the high baryonic density appropriate for NS. One has, therefore, to rely
on models ofQM,whichcontain ahigh degree ofarbitrariness. Atpresent, the best onecan do is
to compare the predictions of different quark models and toestimate theuncertainty ofthe results
fortheNSmatteraswellasfortheNSstructureandmass.
In this paper we will discuss a set of different quark models in combination with a definite
baryonic EOS, which has been developed within the Brueckner-Hartree-Fock (BHF) many-body
approach of nuclear matter, comprising nucleons and also hyperons [3]. In particular, insection 1
wereview the derivation of the baryonic EOSinthe BHFapproach, whereas Section 2isdevoted
totherelevant features ofthehadron quark phasetransition fortheseveralQMEOSdiscussed. In
section 3wepresent the results regarding NSstructure, combining the baryonic and QMEOSfor
beta-stable nuclearmatter,andconclusions aredrawn.
1. EOSofnuclear matter withinBrueckner theory
TheEOSconstructedforthehadronicphaseatT =0isbasedonthenon-relativisticBrueckner-
Bethe-Goldstone (BBG) many-body theory [3], which is a linked cluster expansion of the energy
per nucleon of nuclear matter, well convergent and accurate enough in the density range relevant
for neutron stars. In this approach the essential ingredient is the two-body scattering matrix G,
which,alongwiththesingle-particle potentialU,satisfiestheself-consistent equations
G(r ;w ) = V+V (cid:229) |kakbiQhkakb| G(r ;w ), (1.1)
w −e(k )−e(k )
kakb a b
U(k;r ) = (cid:229) hkk′|G(r ;e(k)+e(k′))|kk′i , (1.2)
a
k′≤kF
whereV is the bare nucleon-nucleon (NN) interaction, r is the nucleon number density, w is the
starting energy, |k k iQhk k | is the Pauli operator, e(k)=e(k;r )= h¯2k2+U(k;r ) is the single
a b a b 2m
particle energy, and the subscript “a” indicates antisymmetrization of the matrix element. In the
BHFapproximation theenergypernucleon is
E 3h¯2 k2
(r )= F +D , (1.3)
2
A 5 2m
D = 1 (cid:229) hkk′|G(r ;e(k)+e(k′))|kk′i . (1.4)
2 a
2A
k,k′≤kF
2
Constraintsonthequarkmatterequationofstatefromastrophysicalobservations G.F.Burgio
Theinclusion ofnuclear three-body forces (TBF)iscrucial inordertoreproduce the correct satu-
ration point of symmetric nuclear matter [4, 5]. The present theoretical status of microscopically
derived TBFisquite rudimentary. Recent results [6,7]have shownthat both two-body and three-
body forces should be based on the same theoretical footing and use the same microscopical pa-
rameters in their construction. Results shown here were obtained with the ArgonneV (V18) [8]
18
or the Bonn B (BOB) [9] potentials, and compared also with the widely used phenomenological
Urbana-type (UIX)TBF[10](incombination withtheV18potential). Itshouldbestressed thatin
ourapproach theTBFisreduced toadensity-dependent two-bodyforcebyaveraging overthepo-
sitionofthethirdparticle, assumingthattheprobability ofhavingtwoparticles atagivendistance
isgivenbythetwo-body correlation function determined self-consistently.
Inthepastyears,theBHFapproachhasbeenextendedwiththeinclusionofhyperons[11,12,
13],whichmayappearatbaryon density ofabout 2to3timesnormal nuclearmatterdensity. The
inclusionofhyperonsproducesanEOSwhichturnsouttobemuchsofterthanthepurelynucleonic
case, withdramatic consequences forthestructure oftheNS,i.e.,thevalue ofthemaximummass
issmallerthan thecanonical value1.44 M . Theinclusion offurthertheoretical ingredients, such
⊙
as hyperon-hyperon potentials [12] and/or three-body forces involving hyperons, could alter the
baryonic EOS, but unfortunately they are essentially unknown. Another possibility that is able
to produce larger maximum masses, is the appearance of a transition to QM inside the star. This
scenario willbeillustrated below.
Starting from the EOS for symmetric and pure neutron matter, and assuming stellar matter
composed of neutrons, protons, and leptons [4], the EOS for the beta equilibrated matter can be
obtainedintheusualstandardway[4,14,15]: TheBruecknercalculationyieldstheenergydensity
ofbaryon/leptonmatterasafunctionofthedifferentpartialdensities,e (r n,r p,r e,r m ). Thevarious
chemicalpotentials forthespecies i=n,p,e,m canthenbecomputed straightforwardly,
¶e
m = , (1.5)
i ¶r
i
andtheequations forbeta-equilibrium,
m =bm −qm , (1.6)
i i n i e
(b andq denoting baryonnumberandchargeofspecies i)andcharge neutrality,
i i
(cid:229) r q =0, (1.7)
i i
i
allow onetodetermine theequilibrium composition {r (r )}atgivenbaryon density r andfinally
i
theEOS,
d e ({r (r )}) de
P(r )=r 2 i =r −e =rm −e . (1.8)
dr r dr n
InFig.1wecomparetheEOSobtainedintheBHFframeworkwhenonlynucleonsandleptonsare
present (thicklines), andthecorresponding oneswithhyperons included (thinlines). Calculations
havebeenperformedwithdifferentchoicesoftheNNpotentials,i.e.,theBonnB,theArgonneV18,
andtheNijmegenN93,allsupplemented byacompatiblemicroscopicTBF[6]. Forcompleteness,
3
Constraintsonthequarkmatterequationofstatefromastrophysicalobservations G.F.Burgio

1000
BOB
800
V18
]
-3 m N93
600
f UIX
V

e
M N
400
[

P
200
N+Y
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
-3
[fm ]
B
Figure 1: Pressure vs. the baryon numberdensity of hadronic NS matter. Thick curves show results for
purelynucleonicmatter,whereasthincurvesincludehyperons.
wealsoshowresultsobtainedwiththeArgonneV18potential togetherwiththephenomenological
UrbanaIXasTBF.
Wenoticeastrongdependence onboththechosenNNpotential, andontheadoptedTBF,the
microscopiconebeingmorerepulsivethanthephenomenological force. Thepresenceofhyperons
decreases strongly the pressure, and the resulting EOS turns out to be almost independent of the
adopted NN potential, due to the interplay between the stiffness of the nucleonic EOS and the
threshold density of hyperons [6]. The softening of the EOS has serious consequences for the
structure of NS,leading to amaximum massof less than 1.4solar masses [6,13], which isbelow
theobserved pulsarmasses[16].
2. QuarkPhase
The properties of cold nuclear matter at large densities, i.e., its EOS and the location of the
phase transition to deconfined QM, remain poorly known. The difficulty in performing first-
principle calculations in such systems can be traced back to the complicated nonlinear and non-
perturbative nature of quantum chromodynamics (QCD). Therefore one can presently only resort
tomoreorlessphenomenological modelsfordescribing QM.
Oneof the most widely used approach is the MIT bag model [17]. Inthe calculations shown
below, we assumed massless u and d quarks, s quarks with a current mass of m =150 MeV, and
s
eitherafixedbagconstant B=90MeVfm−3,oradensity-dependent bagparameter,
r 2
B(r )=B¥ +(B0−B¥ )exp −b r (2.1)
h (cid:16) 0(cid:17) i
withB¥ =50MeVfm−3,B0=400MeVfm−3,andb =0.17. Thisapproachhasbeenproposedin
[18], and itallows the symmetric nuclear matterto bein thepure hadronic phase atlow densities,
and in the quark phase at large densities, while the transition density is taken as a parameter. We
4
Constraintsonthequarkmatterequationofstatefromastrophysicalobservations G.F.Burgio
2
CDM
NJL
FCM, G=0.012
2
1.5 MIT, B=B(r)
MIT, B=90
o
M
/G 1
M
0.5
0
8 10 12 14 160 0.5 1 1.5 2
R (km) r (fm-3)
c
Figure2: GravitationalNSmassvs.theradius(leftpanel)andthecentralbaryondensity(rightpanel)for
differentEOSofquarkmatter. Seetextfordetails.
findthatthephasetransitiontakesplaceatbaryondensityoftheorderof2–3r ,andthatthemixed
0
phasecontains asmallfractionofhyperons.
The phase transition from the hadronic phase to the pure quark matter phase is usually per-
formedthroughtheMaxwellconstruction, orthemoresophisticated Gibbsconstruction [15]. This
is widely used for the study of neutron star structure, several numerical details can be found in
many references [15], and will not be repeated here. The important point to stress is that the fea-
turesofthephasetransition,suchasitsonsetandthedensityintervaloverwhichitextends,depend
crucially on the models used for describing the hadron and the quark phase. For example, we
found that the phase transition constructed with the CDM [19] model is quite different from the
one obtained using the MIT bag model. In the CDM, the onset of the coexistence region occurs
at very low baryonic density. This implies a large difference in the structure of neutron stars. In
fact,whereasstarsbuiltwiththeCDMhaveatmostamixedphaseatlowdensityandapurequark
coreathigherdensity,theonesobtainedwiththeMITbagmodelcontainahadronphase,followed
by a mixed phase and a pure quark interior. The scenario is again different within the Nambu-
Jona-Lasinio model [20], where at most amixed phase is present, but no pure quark phase. It has
alsobeenfound thatthephase transition fromhadronic toQM occurs athighvaluesofthebaryon
chemicalpotentialwhentheDyson-Schwingermodelisusedtodescribethequarkphase. Insome
extreme cases, for particular choices of the parameters, no phase transition at all is possible. In
fact, the Dyson-Schwinger EOS [21] is generally stiffer than the hadronic one, and the value of
the transition density is high. We also found that with the DSM no phase transition exists if the
hadronic phasecontains hyperons, justlikethephasetransition withNJLmodel.
Itisworthwhiletomentionthatneutronstarsobservations canhelptodeterminefreeparame-
ters, which could bepresent in some quark mattermodels. This isthe case of theQM EOSbased
ontheFieldCorrelatormethod[22],whichdependscruciallyonthevalueofthegluoncondensate
G . It turns out that using value of G ≃0.006−0.007 GeV4, which gives a critical temperature
2 2
T ≃170 MeV, produces maximum masses which are only marginally consistent with the obser-
c
5
Constraintsonthequarkmatterequationofstatefromastrophysicalobservations G.F.Burgio
vational limit, while larger masses are possible if the value of the gluon condensate is increased.
Also in this case, the phase transition only takes place if no hyperons are present in the hadronic
phase.
3. Neutron starstructure
WeassumethataNSisasphericallysymmetricdistributionofmassinhydrostaticequilibrium.
The equilibrium configurations are obtained by solving the Tolman-Oppenheimer-Volkoff (TOV)
equations [14]forthepressure Pandtheenclosed massm,
dP Gme (1+P/e ) 1+4p r3P/m
= − , (3.1)
dr r2 1−(cid:0)2Gm/r (cid:1)
dm
= 4p r2e , (3.2)
dr
beingGthegravitational constant. Startingwithacentralmassdensitye (r=0)≡e ,weintegrate
c
out until the density on the surface equals the one of iron. This gives the stellar radius R and the
gravitational massisthen
R
M ≡m(R)=4p drr2e (r). (3.3)
G Z
0
WehaveusedasinputtheEOSdiscussed above,andtheresultsareplottedinFigs.2and3,where
wedisplay thegravitational mass M (inunits ofthesolarmass M =2×1033g)asafunction of
G ⊙
theradius Randcentral baryondensity r .
c
Calculations displayed in Fig. 2 for neutron star matter are obtained in the BHF theoretical
framework with the V18 nucleon-nucleon potential in combination with i) the CDM (solid black
curve), and ii) the MIT bag model (blue curves) for quark matter. Due to the use of the Maxwell
construction, thecurvesarenotcontinuous [15]: forverysmallcentraldensities (largeradii,small
masses) the stars are purely hadronic. We observe that the values of the maximum mass depend
only slightly on the EOS chosen for describing quark matter, and lie between 1.5 and 1.6 solar
masses. Acleardifference betweenthetwomodelsexistsasfarastheradiusisconcerned. Hybrid
starsbuiltwiththeCDMarecharacterized byalargerradius andasmallercentraldensity,whereas
hybridstarsconstructedwiththeMITbagmodelaremorecompact. Furthercalculationshavebeen
performed using theParispotential asnucleon-nucleon interaction forthehadronic phase, andthe
NJLmodel forthe quark phase (red curve). Inthis case theonset ofpure quark matterleads toan
instability, aswellasintheFCM(greencurve). Inbothcasesthephasetransition takesplaceonly
if nohyperons are present inthe hadronic phase. Unfortunately, forallthe cases discussed above,
the value of the maximum mass lies below the mass observed for the pulsar PSR J1614-2230,
M =1.97±0.04 [1]. Such a high value puts severe constraints on the EOS, and in particular it
⊙
demandsanadditional repulsion intheQMEOS.
InFig.3weshow results obtained intheBHFframework using the BonnBnucleon-nucleon
potential, which produces the stiffest EOS,as shown in Fig. 1. This yields a very high maximum
NS mass, ≈2.50 M , with only nucleons (solid black curve), and 1.37M including hyperons.
⊙ ⊙
Using the DSM for the quark phase, we found that the maximum mass of hybrid stars is only
a little lower than 2.5 M with a = 0.5, and decreases to about 2 M with a = 2, being a a
⊙ ⊙
6
Constraintsonthequarkmatterequationofstatefromastrophysicalobservations G.F.Burgio

2.5
2.0
M
1.5
M/
1.0 BOB(N)
BOB(N+Y)
DS0.5
0.5 DS2
DS4
MIT90
MIT-B
0.0
0.0 0.5 1.0 1.5 2.0 9 12 15
-3
[fm ] R [km]
c
Figure3: GravitationalNSmassvs.theradius(rightpanel)andthecentralbaryondensity(leftpanel)for
differentEOSemployingtheBOBhadronicmodel.
model parameter which controls the rate of approaching asymptotic freedom. With increasing a
wecanobtainasmooth change fromthepurehadronic NStotheresultswiththeMITbagmodel.
Moreover, nophasetransition canoccurandnohybridstarcanexistifhyperonsareintroduced. If
hyperons areexcluded, thephase transition from nucleon mattertoQMtakes place atratherlarge
baryon density, andtheonsetofthephasetransition isdetermined inthiscasebytheparameter a .
Thepossible effects ofthe hadron-quark phase transition are very different withthe MITbag
modelandtheDSM:InthecaseoftheMITmodel, thephase transition beginsatverylowbaryon
densityandthuseffectivelyimpedestheappearance ofhyperons. Consequently theresultingmax-
imummassoftheMIThybridstaris1.5M ,lowerthanthevalueofthenucleonic star,buthigher
⊙
than thatofthehyperon stargivenbefore. Acleardifference between thetwomodels existsasfar
as the radius is concerned. Hybrid stars built with the DSM are characterized by a larger radius
and a smaller central density, whereas hybrid stars constructed with the MIT bag model are more
compact.
In conclusion, a hybrid star with 2 M is only allowed if the nucleonic EOS is stiff enough,
⊙
andthehadron-quark phasetransition takesplacewithouthyperons inthehadronic phase.
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