Thermal evolution of hybrid stars within the framework of a nonlocal Nambu--Jona-Lasinio model

1.1 MB·

by  S. M. de Carvalho

Thermal evolution of hybrid stars within the framework of a non-local NJL model
S. M. de Carvalho1,2, R. Negreiros2, M. Orsaria3,4, G. A. Contrera3,4,5, F. Weber6,7 and W. Spinella6,8
1ICRANet-Rio, Centro Brasileiro de Pesquisas F´ısicas,
Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ, 22290-180, Brazil
2Instituto de F´ısica, Universidade Federal Fluminense, UFF, Niter´oi, 24210-346, RJ, Brazil
3CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina
4Grupo de Gravitacio´n, Astrof´ısica y Cosmolog´ıa,
Facultad de Ciencias Astrono´micas y Geof´ısicas, Universidad Nacional de La Plata,
Paseo del Bosque S/N (1900), La Plata, Argentina
5IFLP, CONICET - Dpto. de F´ısica, UNLP, La Plata, Argentina
6Department of Physics, San Diego State University,
5500 Campanile Drive, San Diego, California 92182
7Center for Astrophysics and Space Sciences,
6 University of California,
1 San Diego, La Jolla, CA 92093, USA and
0 8Computational Science Research Center, San Diego State University,
2
5500 Campanile Drive, San Diego, California 92182

n (Dated: January 13, 2016)
a
J We study the thermal evolution of neutron stars containing deconfined quark matter in their
core. Such objects are generally referred to as quark-hybrid stars. The confined hadronic matter
8
in their core is described in the framework of non-linear relativistic nuclear field theory. For the

quark phase we use a non-local extension of the SU(3) Nambu Jona-Lasinio model with vector
]
h interactions. The Gibbs condition is used to model phase equilibrium between confined hadronic
t matteranddeconfinedquarkmatter. Ourstudyindicatesthathigh-massneutronstarsmaycontain
-
l between 35 and 40% deconfined quark-hybrid matter in their cores. Neutron stars with canonical
c
u masses of around 1.4M⊙ would not contain deconfinedquark matter. The centralproton fractions
of the stars are found to be high, enabling them to cool rapidly. Very good agreement with the
n
temperatureevolution established for theneutron star in Cassiopeia A (Cas A) is obtained for one
[
ofourmodels(basedonthepopularNL3nuclearparametrization),iftheprotonsinthecoreofour

1 stellarmodelsarestronglypaired,therepulsionamongthequarksismildlyrepulsive,andthemass
v of Cas A has a canonical valueof 1.4M⊙.
8
3
9 I. INTRODUCTION The most massive neutron stars observed to date are
2 J1614 2230 (1.97 0.04M⊙) [16] and J0348+ 0432
0 − ±
(2.01 0.04M⊙) [17]. In several recent papers [18–24],
. Exploring the properties of compressed baryonic mat- ±
1 ithasbeenshownthattheymaycontainsignificantfrac-
ter, or, more generally, strongly interacting matter at
0 tionsofquark-hybridmatterin theircenters,despite the
6 high densities and/or temperatures, has become a fore- relatively stiff nuclear equation of state (EoS) that is re-
1 front area of modern physics. Experimentally, the prop-
quired to achieve such high masses. The radii of these
: erties of such matter are being probed with the Rela-
v neutron stars would be between 13 and 14 km, depend-
tivistic HeavyIonColliderRHIC atBrookhavenandthe
i ingonthenuclearEoS[18,19],increasingtorespectively
X Large Hadron Collider LHC at Cern. Great advances in
13.5and14.5kmforlighterneutronstarswithcanonical
r ourunderstandingofsuchmatter arealsoexpectedfrom
a massesofaround1.4M⊙. Suchradiusvaluesliebetween
the next generation of heavy-ion collision experiments
the radius determinations based on X-ray burst oscilla-
at FAIR (Facility for Antiproton and Ion Research at
tions of neutron stars in low-massX-ray binaries [25–28]
GSI) andNICA (Nucloton-basedIon Collider fAcility at
and the estimates of the radius of the isolated neutron
JINR) [1, 2] as well as from the study of neutron stars
star RX J1856-3754[29], emitting purely thermal radia-
(for an overview,see [1, 3–15] and references therein).
tion in the X-ray and in the optical bands.
Neutron stars (NSs) contain nuclear matter com-
pressed to densities which are several times higher than If the dense interior of a neutron star contains decon-
the densities of atomic nuclei. At such extreme condi- fined quark matter, it will most likely be three-flavor
tions, the fundamentalbuilding blocks of matter may no quark matter, since such matter has lower energy than
longerbe justneutronsandprotonsimmersedinagasof two-flavorquarkmatter[30,31]. Furthermore,justasfor
relativisticelectronandmuons,butothernucleardegrees the hyperon content of neutron stars, strangeness is not
offreedomsuchashyperons,deltaparticlesand,mostin- conserved on macroscopic time scales which allows neu-
triguingly, deconfined up, down and strange quarks may tron stars to convert confined hadronic matter to three-
begin to play a role. Neutron stars containing decon- flavor quark matter until equilibrium brings this process
finedquarkmatterintheircentralcorearereferredtoas to a halt. We considered the transition from hadronic
quark-hybrid stars (hybrid stars, for short). to quark matter to be first order. There are two distinct
2
waystoconstructafirstorderphasetransitioninneutron in Sects. V. A discussion of their thermal evolution is
stars. ThefirstoptionisaGibbsconstruction,wherethe presented in VI. Finally, a summary of our results and
electronicandbaryonicchemicalpotentialsaswellasthe important conclusions are provided in Sect. VII.
pressure are continuous during the phase transition; the
second option is a Maxwell construction, where only the II. QUARK MATTER PHASE
baryonic chemical potential and pressure are continuous
and the electronic chemical potential is characterized by
In this section we briefly describe the non-localexten-
a discontinuity at the phase boundary. The surface ten-
sion of the SU(3) Nambu Jona-Lasinio (n3NJL) model.
sion at the interface between the quark-hadron phase is
The Euclidean effective action for the quark sector, in-
whatdetermineswhetheraGibbsorMaxwellphasetran-
cluding the vector coupling interaction, is given by
sition may be taking place. Severalauthors [34–39] have
attempted to estimate the value of the surface tension,
with mixed results. For what follows, we will assume SE = d4x ψ¯(x)[ i∂/+mˆ]ψ(x)
−
that the surface tension is less than 40 MeV fm−2, such Z
G (cid:8)
thatthe Gibbsconditionisfavoredandamixedphaseof S jS(x) jS(x)+jP(x) jP(x)
− 2 a a a a
quark matter and nuclear matter exists above a certain
density [40]. GP (cid:2)T jS(x)jS(x)jS(x) 3 j(cid:3)S(x)jP(x)jP(x)
In the mixed phase, the presence of quarks allows the − 4 abc a b c − a b c
hadronic component to become more isospin symmet- GV [jµ ((cid:2)x)jµ (x)] , (1(cid:3))
ric, which is accomplished by the transference of electric − 2 Va Va
(cid:27)
charge to the quark phase. Thus, the symmetry energy
can be lowered at only a small cost in rearranging the where ψ stands for the light quark fields, mˆ denotes the
quark Fermi surfaces. The implication of this charge re- current quark mass matrix, and G , G and G are
S P V
arrangementis that the mixed phase regionof a neutron the scalar, pseudo-scalar, and vector coupling constant
star will have positively charged hadronic matter and of the theory, respectively. For simplicity, we consider
negatively chargedquark matter [3, 31, 41]. This should the isospin symmetric limit in which m = m = m¯.
u d
have important implications for the electric and thermal The operator ∂/ = γ ∂ in Euclidean space is defined as
µ µ
properties of NSs. Studies of the transport properties of ~γ ~ +γ4 ∂ ,withγ4 =iγ0. Thescalar(S),pseudo-scalar
quark-hybrid neutron star matter have been reported in (P·∇), and∂vτector (V) current densities jS(x), jP(x), and
a a
[42, 43]. jµ (x), respectively, are given by
As already mentioned above, this study is carried out Va
for neutron stars containing deconfined quark matter in
z z
their centers (so-called quark-hybrid stars). To describe jaS(x)= d4z g(z) ψ¯ x+ 2 λa ψ x− 2 ,
the quark matter phase, we use a non-local extension of Z (cid:16) (cid:17) (cid:16) (cid:17)
z z
tvheectSoUr(in3t)eNraacmtibounsJ.oFnoar-Lthaseinhiaod(rNonJiLc)pmhaosdeewl[e44c–o4n7s]idweirtha jaP(x)= d4z eg(z) ψ¯ x+ 2 i γ5λa ψ x− 2 ,
Z (cid:16) (cid:17) (cid:16) (cid:17)
ntwono-dliinffeearrenretlaptaivraismtiectmrizeaatni-ofinesl,dGmMod1e[l5[34]8–a5n2d]sNoLlv3ed[5f4o]r. jVµa(x)= d4z eg(z) ψ¯ x+ 2z γµλa ψ x− z2 , (2)
The transition from the confined hadronic phase to the Z (cid:16) (cid:17) (cid:16) (cid:17)
deconfined quark phase is treated as a Gibbs transition, where g(z) is a forem factor responsible for the non-local
imposing global electric charge neutrality and baryon character of the interaction, λ with a = 1,...,8 de-
a
number conservation on the field equations. We find notes tehe generators of SU(3), and λ0 = 2/3113×3. Fi-
that the non-local NJL model predicts the existence of nally, the constants T in the t’Hooft term accounting
abc
p
extended regions of mixed quark-hadron (quark-hybrid) for flavor-mixing are defined by
matter in neutron stars with masses up to 2.4M⊙.
The paper is organized as follows. In Sect. II, we de- 1
T = ǫ ǫ (λ ) (λ ) (λ ) . (3)
scribethenon-localextensionoftheSU(3)NJLmodelat abc 3! ijk mnl a im b jn c kl
zero temperature. In Sect. III, the non-linear relativistic
mean-field model, which is used for the description of After the standard bosonization of Eq. (1), the inte-
confined hadronic matter, is briefly discussed. In Sect. gralsoverthequarkfieldscanbeperformedintheframe-
IV, the construction of the quark-hadronmixed phase is workofthe Euclideanfour-momentumformalism. Thus,
discussedforneutronstarmattercharacterizedbyglobal the grand canonical thermodynamical potential of the
charge neutrality. Our results for the global structure model within the mean field approximationat zero tem-
and composition of quark-hybrid stars are discussed perature is given by
3
N ∞ ∞ 1
ΩNL(Mf,µf)=−π3c f=u,d,sZ0 dp0Z0 dp ln ( ωf2+Mf2(ωf2) ωf2+m2f) (4)
X (cid:2) (cid:3)
Nc √µ2f−m2f dp p2 [(µ E )Θ(µ m )b] 1 (σ¯ S¯ + GS S¯2)+ GPS¯ S¯ S¯ ̟f2 ,
−π2 f=u,d,sZ0 f − f f − f − 2f=u,d,s f f 2 f 2 u d s−f=u,d,s4GV
X X X
 
whereNc =3,Ef = p2+m2f,ωf2 =(p0 +iµf)2 +p2, by
and σ¯f denotes the mqean-field values of the quark flavor σ¯ +G S¯ + GP S¯ S¯ = 0,
(f = u,d,s) fields. The vector coupling constant G is u S u d s
V 2
treated as a free parameter and expressed as a fraction
G
of the strong coupling constant GS. σ¯d+GSS¯d+ 2P S¯uS¯s = 0, (10)
The constituent quark masses M are treated as
f G
momentum-dependent quantities. They are given by σ¯s+GSS¯s+ P S¯uS¯d = 0.
2
2 2 The vector mean fields ̟ are obtained by minimizing
M (ω ) = m +σ¯ g(ω ), (5) f
f f f f f Eq. (4), i.e. ∂ΩNL =0.
∂̟f
where g(ω2) is the Fourier transform of the form factor Forquarkmatterinchemicalequilibriumatfiniteden-
f
g(z). The vector mean fields ̟ are associated with the sity,thebasicparticleprocessesinvolvedaregivenbythe
f
vector current densities jµ (x), where a different vector strong process u+d u+s as well as the weak pro-
field for each quark flavorVfa has been considered. cessesd(s) u+e−an↔du+e− d(s). Weareassuming
e → →
thatneutrinos,oncecreatedbyweakreactions,leavethe
Wefollowedthemethoddescribedin[55]toincludethe
system, which is equivalent to supposing that the neu-
vector interactions. The inclusion of vector interactions
trino chemical potential is equal to zero. Therefore, the
shifts the quark chemical potential as follows,
chemical potential for each quark flavor f is given by
2
µf = µf −g(ωf)̟f, (6) µf =µb−Qµe, (11)
2 2 2
ωf = (p0 + iµf) + p . (7) where Q = diag(2/3, 1/3, 1/3) in flavor space and
b − −
µ =1/3 µ is the baryonic chemical potential.
Note that the sbhift in the quarbk chemical potential does bThe contfribfution of free degenerate leptons to the
not affect the Gaussian non-local form factor, P
quark phase is given by
g(ωf2)=exp −ωf2/Λ2 , (8) Ωλ=e−,µ−(µe)=−π12 0pFλ p2 p2+m2λ−µe dp.
avoiding a recursive problem(cid:0)as discus(cid:1)sed in [55–57]. In Z (cid:18)q (cid:19) (12)
Eq.(8),Λplaysthe roleofanultravioletcut-offmomen- Muons appear in the system if the electron chemical po-
tum scale and is taken as a parameter which, together tential µe = µµ is greater than the muon rest mass,
with the quark current masses and coupling constants mµ =105.7 MeV. For electrons we have me =0.5 MeV.
G and G in Eq. (1), can be chosen so as to repro- Thus, the total thermodynamic potential of the quark
S P
ducethephenomenologicalvaluesofpiondecayconstant phaseisgivenbyEq.(4)supplementedwiththeleptonic
fπ, and the meson masses mπ, mη, mη′, as described in contribution of Eq. (12).
[46, 47]. In this work we use the same parameters as in
[18, 19].
III. CONFINED HADRONIC MATTER
Forthestationaryphaseapproximation[45],themean-
field values of the auxiliary fields S¯ in Eq. (4) are given
f
by The hadronic phase is described in the framework
of the non-linear relativistic nuclear field theory [48–
S¯ = 4N ∞ dp0 d3p g(ω2) Mf(ωf2) . 5d2el]t,awshteartees)bairnytoernasct(nveiuatrtohnes,expcrhoatonngse, ohfyspcearloanrs, vaencd-
f − c Z0 2π Z (2π)3 f ω2+Mf2(ωf2) tor and isovector mesons (σ, ω, ρ, respectively). The
(9)
parametrizations used in our study are GM1 [53] and
Duetothechargeneutralityconstraint,wbeconsiderthree NL3 [54]. The lagrangianof this model is given by
scalar fields, σ¯ , σ¯ and σ¯ , which can be obtained by
u d s
solving the coupled system of “gap” equations [45] given = H + ℓ, (13)
L L L
4
with the leptonic lagrangiangiven by The hadronic lagrangianhas the form
= ψ¯ (iγ ∂µ m )ψ . (14)
ℓ λ µ λ λ
L −
λ=Xe−,µ−
1 1 1
LH = ψ¯B γµ(i∂µ−gωωµ−gρρ~µ)−(mN −gσσ) ψB + 2(∂µσ∂µσ−m2σσ2)− 3bσmN(gσσ)3− 4cσ(gσσ)4
B=n,p,Λ,Σ,Ξ
X (cid:2) (cid:3)
1 1 1 1
ω ωµν + m2ω ωµ+ m2ρ~ ρ~µ ρ~ ρ~µν. (15)
−4 µν 2 ω µ 2 ρ µ· − 4 µν
The quantity B sums over all baryonic states which be- and
comepopulatedinneutronstarmatteratagivendensity
[3, 4]. Intriguingly, we have found that, aside from hy- ε=(1 χ)εH +χεq, (18)
−
perons, the ∆− state becomes populated in neutron star
where nH (εH) and nq (εq) denote the baryon number
matter at densities that could be reached in the cores b b
(energy) densities of the hadronic and the quark phase,
of stable neutron stars [19]. Simpler treatments of the
respectively. The quantity χ V /V denotes the vol-
quark-hadron phase transition, based on the MIT bag ≡ q
model [31, 41] do not predict the occurrence of the ∆− ume proportion of quark matter, Vq, in the unknown
volume V. Therefore, by definition χ varies from 0 to
state in stable neutron stars.
1 depending on how much confined hadronic matter has
For the models of this paper, ∆ states become pop-
been converted to quark matter at a given density. The
ulated when the vector repulsion among quarks reaches
condition of global electric charge neutrality is given by
values of G & 0.05G , leading to a substantial stiffen-
V S
the equation
ingofthe EoS.Thisstiffening morethanoffsetsthe soft-
ening of the EoS caused the generation of the ∆ states,
(1 χ) qHnH +χ qqnq =0, (19)
resulting in an EoS which is readily capable to accom- − i i i i
modate even very heavy (2M⊙) neutron stars. Without i=XB,l iX=q,l
this additionalstiffening, it wouldbe difficult to account
where q is the electric charge of particle species i, ex-
i
for 2M⊙ neutron stars with equations of state that are pressed in units of the electron charge. Because of the
characterizedby an earlyappearanceof∆’s, atdensities
globalconservationof electric charge and baryonic num-
between around 2 to 3 times nuclear saturation density
ber, the pressure in the mixed phase increases monoton-
[58].
ically with increasing energy density, as shown in Fig.
1. In this work we have chosen global rather than lo-
cal electric charge neutrality, since the latter is not fully
IV. QUARK-HADRON MIXED PHASE
consistent with the Einstein-Maxwell equations and the
micro-physicalconditionofchemicalequilibriumandrel-
To determine the mixed phase region of quarks and ativisticquantumstatistics,asshownin[59]. Incontrast
hadrons we start from the Gibbs condition for pressure to local electric charge neutrality, the global neutrality
equilibriumbetween confinedhadronic(PH) matter and condition puts a net positive electric charge on hadronic
deconfined quark (Pq) matter. The Gibbs condition is matter, rendering it more isospin symmetric, and a net
given by negative electric charge on quark matter, allowing neu-
PH(µH,µH, φ )=Pq(µq,µq, ψ ), (16) tron star matter to settle down in a lower energy state
b e { } b e { } that otherwise possible [31, 41].
with µH = µq for the baryon chemical potentials and
b b
µH = µq for the electron chemical potentials in the
e e
hadronic (H) and quark (q) phase. The quantities φ V. STRUCTURE OF NEUTRON STARS
{ }
and ψ stand collectively for the field variables and
{ }
Fermi momenta that characterize the solutions to the
We now turn to the discussionof the structure ofneu-
equations of confined hadronic matter and deconfined
tronstars,whicharecomputedforthe microscopicmod-
quark matter, respectively. By definition, the quark
els of quark-hybrid matter discussed in Sect. II and III.
chemicalpotentialisgivenbyµq =µ /3,whereµ isthe
b n n Weshallexplorethreevaluesforthevectorcouplingcon-
chemical potential of the neutrons. In the mixed phase,
stant G /G , i.e. 0, 0.05, and 0.09. The mass-radius
the baryon number density, n , and the energy density, V S
b relationships of these stars are shown in Fig. 2. One
ε, are given by
sees that increasing values of G lead to higher maxi-
V
n =(1 χ)nH +χnq, (17) mummassesfor bothequationsofstate (GM1 andNL3)
b − b b
5

500 2.4 2.4
GV = 0 (a) (a) MP (b)
400 GV = 0.05 GS 2.0 PSR J0348+0432 2.3 QP MP
-3 m] GV = 0.09 GS PSR J1614-2230 2.2 MP QP
V f 300 M1.6 Nucleons + Hyperons M2.1 QPPSR J0348+0432 MP
Me 200 M/1.2 M/2.0 MP QP
QP
[
P 1.9 PSR J1614-2230
100 Hadron phase Maximum 0.8 GM1 NL3 1.8 QP MP
0 GM1 Mass Star X 0.35 0.4 GGVV==00.05GS Nucleons 1.7 QMPP == PMUIRXEE DQ UPHAARKSE PHASE
500 GV=0.09GS MAXIMUM MASS
GV = 0 (b) 11 12 13 14 15 16 11 12 13 14 15
400 GV = 0.05 GS R [km] R [km]
-3 m] GV = 0.09 GS Fneiguutrroen2s:tars(Cmoaldoer oofnqliunaer)k-(hay)bMridasms-arattdeiru.s(rbe)laEtniolanrsgheipmsenotf
f 300
of the maximum mass region of (a). The labels GM1 and
V
e NL3 label the hadronic model for the EoS, GV indicates the
M 200 strength of thevector coupling constant among quarks.
[

P
100
Hadron phase Maximum Baym-Pethick Sutherland [32] and Baym-Bethe-Pethick
NL3 Mass Star X 0.40 [33] EoS at sub-saturation densities.
0
0 500 1000 1500 2000
-3
[MeV fm ] VI. THERMAL EVOLUTION
Figure 1: (Color online) Pressure, P, as a function of We nowturnourattentiontothe thermalevolutionof
the energy density, ǫ, for the different nuclear parametriza-
neutron stars whose structure and interior composition
tions (GM1, NL3) and vector coupling constant GV/GS aregivenbythemicroscopicmodelsdescribedinthepre-
(0,0.05,0.09) considered in this paper. Panel (a) shows the
vious sections.
hybrid EoSs computed for GM1, panel (b) shows the hybrid
Here we briefly describe the thermal evolution equa-
EoSscomputedforNL3. Thetrianglesinbothpanelsindicate
thecentraldensitiesofthemaximum-massneutronstars(see tions that governthe cooling of neutron stars. The ther-
Fig.2)associated witheachEoS.Thequantityχdenotesthe mal balance and transport equations for a general rela-
fraction of quark matter inside of the most massive neutron tivistic, spherically symmetric object is given by
star for each EoS.
∂(Leν) 4πr2 ∂(Teν/2)
= ǫ eν +c , (20)
ν v
∂r − 1 2m/r ∂t
− (cid:20) (cid:21)
studied in this work. This is expected, since the stiff- Leν p ∂(Teν/2)
ness of the equation of state increases with G . We also = 1 2m/r , (21)
V 4πr2κ − ∂r
notethatallneutronstarscomputedforNL3havelarger
p
radii than those obtained for the GM1 parametrization. where r, m(r), ρ(r), and ν(r) represent the radial dis-
This is so because the NL3 equation of state is stiffer tance, mass, energy density, and gravitational poten-
thanthe GM1 equationofstate,leadingto quarkdecon- tial,respectively. Furthermore,thethermalvariablesare
finement at densities that are lower than for the GM1 given by the interior temperature T(r,t), the luminosity
parametrization, as can be seen in Fig. (1). This fig- L(r,t),neutrinoemissivityǫ (r,T),thermalconductivity
ν
ure also shows that the neutron stars close to the mass κ(r,T), and the specific heat per unit volume c (r,T).
v
peakspossessextendedmixedphaseregionswithapprox- The solution of Eqs. (20)–(21) is obtained with the
imately40%quarkmatterforNL3and35%quarkmatter help of two boundary conditions, one at the core, where
for GM1. In addition, we find that calculations carried the luminosity vanishes, L(r = 0) = 0, since the heat
out for GM1 and a vanishingly small value of G lead flux there is zero. The second boundary condition has
V
to neutron star masses of around 1.8M⊙, which is well to do with the surface luminosity, which is defined by
below the masses observed for neutron stars J1614-2230 therelationshipbetweenthemantletemperatureandthe
(1.97 0.04M⊙) [16] and J0348+0432(2.01 0.04M⊙) surface temperature [60–62]. Furthermore we consider
± ±
[17]. Therefore, this combination of model parameters all neutrino emission processes relevant to the thermal
for the equation of state can be ruled out. The calcula- evolution of compact stars, including the Pair Breaking
tions have been carried out using a combination of the and Formation process (PBF) responsible for a splash
6
of neutrinos on the onset of pair formation. We now
TableI:Propertiesoftheneutronstarswhosethermalevolu-
describe pairing effects and its corresponding effects on
tionsare investigated for thenuclearparametrizations (GM1
thethermalevolutiononthequark-hybridstarsdiscussed
and NL3) of this paper. GV is the vector coupling constant
in this paper.
among quarks, M and R denote the neutron stars’ gravita-
tional masses and radii, respectively.
A. Pairing Models Parametrization GV/GS M [M⊙] R [km] ρc [MeV/fm3]
GM1 0 1.4 13.84 375.27
In addition to the microscopic model described in the 1.87 12.63 961.10
previous sections, we now devote some time to the dis- GM1 0.05 1.4 13.85 375.27
cussion of pairing of nucleons, which will be used in our 2.0 12.48 819.58
cooling simulations below. Pairing among nucleons has GM1 0.09 1.4 13.91 355.08
received enormous interest recently due to the unusual
2.0 12.74 443.28
thermaldata observedforthe neutronstarinCassiopeia
NL3 0 1.4 14.32 343.30
A(CasA)(seee.g.[63–65]forarecentstudyoftheeffects
2.0 13.75 541.10
of pairing in the thermal evolution of compact stars).
A full-blown microscopic description of pairing among NL3 0.05 1.4 14.47 333.76
neutrons and protons in beta stable matter at high den- 2.2 13.62 675.07
sitiesisachallengingtask,andsofartherearestillmany NL3 0.09 1.4 14.68 311.33
uncertainties,particularlywithrespecttoproton-pairing 2.0 14.12 693.88
at high-densities [66]. In this work we use a phenomeno-
logical description, as described in [67]. We assume that
1
neutrons form singlet S0 pairs in the crust and triplet
3P2 pairs in the core. Fig.3weshowthecriticaltemperatureforthethreemod-
As for the protons, there is still much discussion as to els used for proton1S0 singlet pairing in the cores of the
how high densities protons may form pairs in the cores stars studied in this paper.
ofneutronstars. Asdiscussedin[68],thepresenceofthe
direct Urca process in the core of neutron stars depends
strongly on the symmetry energy and on its possible de- B. Neutron Star Cooling Curves
pendence on a so-called “quartic term” (a term that is
offourthorderinthe deviationfromsymmetricmatter).
We now show the thermal evolution obtained by nu-
Aspointedoutin[68],itisverydifficulttointerpretneu-
merically integrating the energy balance and transport
tronstarcoolingwithoutmoreinformationregardingthe
equations (20)–(21). We have chosen to perform sim-
symmetry energy and its “quartic term” dependence at
ulations on two different neutron stars; the first has a
high densities. For that reason we have chosen to study
gravitationalmass of1.4M⊙, andthe secondhas a mass
three different models for proton pairing, which are re-
that is closer to the maximum-mass value of each stel-
lar sequence. The thermal evolution of neutron stars
8´1011
with masses between these two limiting cases will then
lie within the bounds of these twocooling curves. In Ta-
Shallow
6´1011 ble I we show the properties of the neutron stars whose
Medium thermal evolution is being studied.
We show in Fig. 4 the surface temperature T as a
HLK 4´1011 Deep function of time t (in years) for the GM1 paramsetriza-
Tc
tion. The results for the NL3 parametrizationareshown
in Fig. 5. Figure 4 shows that the shallow and medium
2´1011
proton pairing cases obtained for the GM1 parametriza-
tion exhibit very little difference. For both cases pairing
0 is not strong enoughto completely suppress all fast neu-
0.1 0.2 0.3 0.4 0.5 0.6 0.7
trino processes so that these stars exhibit fast cooling.
n Hfm-3L
b Stars with a lower mass show slightly slowercooling due
to their smaller core densities (and thus smaller proton
Figure 3: (Color online) Critical temperature, TC, for the fractions). The situation is different for the deep pair-
onset of proton (1S0) singlet pairing in neutron star matter ing model where only for the lighter stars (GV = 0 and
as a function of baryon numberdensity,nb. G = 0.05G ) the neutrino process is completely sup-
V S
pressed. Furthermore, for the deep pairing model, all
ferred to as shallow, medium, and deep. As the labeling starsobtainedfor G =0.09G havetheir fastneutrino
V S
indicates,thesemodelscorrespondtoprotonpairingthat processes suppressed.
ends at low, medium and high densities, respectively. In As for the NL3 parametrization,Fig. 5 shows that the
7
shallow and medium cases have similar behavior in that
2´107
thereisnocompletesuppressionofthefastneutrinopro- HaLShallow
cipcseeassrsssaoeemmss.aeeTtrwrehihzteaaotttdiaeoolenlpyp.pscouaOspistnepee,rehtssooesweewsdevhitenahrta,btoewxttehhhielfbiogifuthasntsdtaannbfdoeeruhhtaetrvhaiinveooyrGntphMerauo1t-- @DK 251´´´111000667 GGGGGGvvvvvv======000000......000000995500GGGGGGSSSSSS,,,,,,MMMMMM======212111......04248400007MMMMMMŸŸŸŸŸŸ
tron stars when GV =0 and GV =0.05GS. In contrast Ts1´106
tothis,whenGV =0.09GS weseethatthefastneutrino 5´105
processes are only suppressed in lighter neutron stars.
2´105
1´105
0.1 10 1000 105 107
2´107
HaLShallow t@yD
@DTKs51251´´´´´111110000056667 GGGGGGvvvvvv======000000......000000995500GGGGGGSSSSss,,,,,,MMMMMM======112121......840404700000MMMMMMŸŸŸŸŸŸ HLK 2512´´´´111100006677 GGGGGGvvvvvv======000000......b000000L995500MGGGGGGSSSSSSe,,,,,,dMMMMMMiu======m212111......042484000000MMMMMMŸŸŸŸŸŸ
2´105 Ts1´106
5´105
1´105
0.1 10 1000 105 107
2´105
t@yD
1´105
0.1 10 1000 105 107
2´107
HbLMedium tHyL
HLTKs51251´´´´´111110000056667 GGGGGGvvvvvv======000000......000000995500GGGGGGSSSSss,,,,,,MMMMMM======112121......840404700000MMMMMMŸŸŸŸŸŸ HLK 2512´´´´111100006677 GGGGGGvvvvvv======000000......000000995500cGGGGGGLDSSSSSS,,,,,,eMMMMMMep======212111......042484000000MMMMMMŸŸŸŸŸŸ
2´105 Ts1´106
5´105
1´105
0.1 10 1000 105 107
2´105
tHyL
1´105
0.1 10 1000 105 107
2´107
cLDeep tHyL
HLK 251´´´111000667 GGGGGGvvvvvv======000000......000000995500GGGGGGSSSSSS,,,,,,MMMMMM======212111......040484000070MMMMMMŸŸŸŸŸŸ Fpaigruamreet5r:iza(Ctioonlo.r online) Same as Fig. 4, but for the NL3
Ts1´106
5´105 C. Comparison with Observed Data
2´105
We complete our study of the thermal evolution by
1´105
0.1 10 1000 105 107 comparing our results with the thermal behavior ob-
served for compact stars, and, in particular, that of the
tHyL
neutronstarsinCassiopeiaA(CasA).Thisobjectisthe
youngest known neutron star from which the thermal
Figure 4: (Color online) Thermal evolution of neutron stars
emission has been observed continuously for a decade.
for the different proton pairing scenarios (shallow, medium,
Heinke&HofoundthatthesurfacetemperatureofCasA
anddeep)considered inthispaper. Thecalculations arecar-
has droppedby 4%between2000and2009,from2.12to
ried out for the GM1 parametrization and vector coupling
2.04 106 K [69]. The rapid coolinghas been attributed
constants ranging from zero to 0.09GS. ×
to the onset of neutron superfluidity in the stellar core.
TheobserveddatafortheneutronstarinCasAhasbeen
8
5×106 1,75×106 5×106 GCaVs= A0.05GS, M=1.4Msun ba-- 1REX 1J200872.24--45224079
GV=0.05GS, M=2.2Msun dc-- RPXSR J 00080323+-465246
GV=0.00, M=1.8Msun
1,7×106 GV=0.00, M=1.4Msun
Spin­down
Kinematic
2×106 300 325 350 2×106
K) K] a
( [
T s T s b
8
106 106 c 7
1- PSR 0633+1746
Gv = 0.05Gs 2- PSR 0656+14
Cas A 3- PSR B2334+61 6
4- PSR 1706-44 d
Gv = 0 65-- PPSSRR 10085353--5425 5 3 1
4
7- PSR 0538+2817
5×105 5×105 8- PSR J1119-6127 2
100 1000 104 10 100 1000 104 105 106
t (yrs) t [y]
Figure6: (Coloronline)CoolingcurvesofaM =1.4M⊙neu- Figure 7: (Color online) Theoretical cooling curves of neu-
tron star computed for NL3 and vector coupling constants tron stars, computed for NL3 and vector coupling constants
GV = 0 and GV = 0.05GS. The inset shows the data ob- GV = 0 and GV = 0.05GS, compared with observed data.
served for Cas A over a time period of one decade. (Data Pink (green) diamonds denotespin-down (kinematic) age es-
taken from [70].) timates.
revisitedrecentlybyHoetal. (2015)[70],wheretwonew be considered very carefuly, perhaps serving only as an
upper limit on the true age of a given neutron star.
Chandra ACIS-S graded observations are presented. We
In Fig. 7, we compare the cooling tracks of neutron
note, however, that the statistical significance of Cas A
starscomputedfortheNL3parametrizationwiththeob-
observeddatahasbeencalledintoquestion,asdiscussed
served data. We note that our model agrees fairly well
in Ref. [71].
withsomeobserveddata,butfails toreproducethe data
For the models of this paper, only the 1.4M⊙ neutron of several other neutron stars. This is not an inherent
starcomputedfortheNL3parametrization(G =0and
V feature of our model but, rather, appears symptomatic
G = 0.05G ) and with deep pairing for the protons
V S for mostthermal models that agreewith the Cas A data
agreeswith the data observedfor Cas A. The reasonbe-
(seeforinstance[64]). Itmayindicatethatthese objects
ing that this model stronglysuppressesthe fast neutrino
are subjected to a heating mechanism which keeps them
coolingprocesses,whichis necessaryto explainthe ther-
warm during their evolution.
mal behavior of Cas A as discussed in [63, 64]. We show
this result in Fig. 6. We note that agreement with the
observed Cas A data is obtained for neutron stars with
massesof1.4M⊙,beforetheonsetofquarkmatterinthe VII. SUMMARY AND CONCLUSIONS
stellarcore. Thisresultisinagreementwithestimatesof
the mass of Cas A [70], which indicates that the mass of In this work, we have used an extension of the non-
this object is probably too low to contain quark matter. local3-flavorNambu-JonaLasiniomodel to study quark
WenowconfrontthetwoNL3modelsthatareinagree- deconfinement in the cores of neutron stars. Confined
ment with Cas A (i.e., G = 0 and G = 0.05G with hadronic matter is described by non-linear relativis-
V V S
deepprotonpairing)withobservedthermaldataoncom- tic nuclear field theory, adopting two popular hadronic
pact stars. We use two sets of observed data (see [62] parametrizationslabeledGM1andNL3. Thephasetran-
and references therein), one for age estimates based on sitionfromconfinedhadronicmattertodeconfinedquark
the stars’ spin-down properties, and the other based on matter is modeled via the Gibbs condition, imposing
the so called kinematic age. We note that the kinematic globalelectric chargeneutrality onthe particle composi-
agesconstituemorerealisticageestimatesastheyareas- tion of neutron star matter. Repulsive forces among the
sociatedwith kinematicpropertiesofsupernovaebelived quarks are described in terms of a vector coupling con-
to be the progenitors of the neutron stars in question. stant, GV, whose value ranges from zero (no repulsion)
For the few cases where both kinematic and spin-down to 0.9GS, where GS denotes the scalar strong coupling
ages have been estimated, large discrepancies have been constant of the theory.
found. This indicates that the spin-down age needs to Each one of our models for the EoS of (quark-hybrid)
9
neutron star matter accommodates high-mass neutron take into account pairing among the quarks will be pre-
stars with masses up to 2.4 solar masses as long as the sented in a future work. We have also compared the
valueofG issufficientlylarge. Allhigh-massstarscon- thermal behavior predicted for Cas A with that of other
V
tain extended quark-hybrid matter cores in their cen- compact stars. In contrast to Cas A, however, these ob-
ters, but a pure quark matter is never reached for any servationsonlytemperature”snapshots”existforthelat-
of our model parametrizations. The maximum neutron ter, with age estimates based on either their kinematic
star masses drop if the strength of the vector repulsion or spin-down properties. The results show that the NJL
among quarks is reduced, falling below the 2M⊙ limit models that are in agreement with Cas A (i.e., GV = 0
set by pulsars J1614 2230 (1.97 0.04M⊙) [16] and andGV =0.05GS withdeepprotonpairing)leadtogood
− ±
J0348+0432 (2.01 0.04M⊙) [17] for some parameter agreement with the observed data of several other com-
±
combinations. Examples of this are neutron stars com- pact objets, while failing to reproduce sevealother data.
puted for the GM1 parametrizationwith G =0, which This, however, is not an inherent feature of our model
V
yields a maximum-mass neutron star of 1.8M⊙, in but,rather,appearssymptomaticformostthermalmod-
∼
conflict with the recent mass determinations mentioned els that agree with the Cas A data [64].
just above.
The quark-hybridstars of our study possess relatively
high proton fractions in their cores so that fast neutrino
processes,mostnotablythedirectUrcaprocess,isactive,
Acknowledgments
leading to very rapid stellar cooling. An agreementwith
the thermal evolution data of the neutron star in Cas
A can be obtained, however, if one assumes that strong S.M.C. acknowledge the support by the International
proton-pairing is occurring in the core of this neutron Cooperation Program CAPES-ICRANet financed by
star, which is known to strongly suppress fast cooling. CAPES – Brazilian Federal Agency for Support and
Under this condition, a 1.4M⊙ neutron star computed EvaluationofGraduateEducationwithintheMinistryof
forthe NL3modelandvaluesofthevectorcouplingcon- Education of Brazil. R. N. acknowledges financial sup-
stants between G =0 and G =0.05G lead to excel- portbyCAPESandCNPq. M.O.andG.C.acknowledge
V V S
lent agreement with the observed data. It is important financialsupportbyCONICET,Argentina. F.W.issup-
to note that the proton pairing model has not been fine portedby the NationalScienceFoundation(USA) under
tuned to the Cas A data. Further studies where we also Grant PHY-1411708.
[1] StronglyInteractingMatter-TheCBMPhysicsBook,B. Rev. Mod. Phys. 80, 1455 (2008).
Friman, C. H¨ohne, J. Knoll, S. Leupold, R. Randrup, J. [14] F. Weber, G. A. Contrera, M. G. Orsaria, W. Spinella,
Rapp, P. Senger, (Eds.), Lecture Notes in Physics 814, and O.Zubairi, Mod. Phys. Lett. A 29, 1430022 (2014)
960 pages, (Springer, 2011). [15] H. Grigorian, D. Blaschke, and D. N. Voskresensky,
[2] SeeNICAWhitePaperathttp://nica.jinr.ru/files/ arXiv:1502.03080 [astro-ph.HE].
WhitePaper.pdf. [16] P. B. Demorest, T. Pennucci, S. M. Ranson, M. S. E.
[3] N. K. Glendenning, Compact Stars, Nuclear Physics, Roberts,andJ.W.T.Hessels,Nature467,1081(2010).
Particle Physics, and General Relativity, 2nd ed. [17] Lynch et al. Astrophys. J 763, 81 (2013); J. Antoniadis
(Springer-Verlag, NewYork,2000). et al.,Science 340, no. 6131 (2013).
[4] F. Weber, Pulsars as Astrophysical Laboratories for Nu- [18] M.Orsaria,H.Rodrigues,F.Weber,andG.A.Contrera,
clear and Particle Physics, High Energy Physics, Cos- Phys. Rev.D 87, 023001 (2013).
mology andGravitation Series (IOPPublishing, Bristol, [19] M.Orsaria,H.Rodrigues,F.Weber,andG.A.Contrera,
Great Britain, 1999). Phys. Rev.C 89, 015806 (2014).
[5] Physics of Neutron Star Interiors, ed. by D. Blaschke, [20] H. Chen, G. F. Burgio, H.-J. Schulze, and N. Yasutake,
N. K. Glendenning, and A. Sedrakian, Lecture Notes in Astron. Astrophys.551, A13 (2013).
Physics 578 (Spring-Verlag, Berlin, 2001). [21] H. Chen, J.-B. Wei, M. Baldo, G. F. Burgio, and H.-
[6] J. M. Lattimer and M. Prakash, Astrophys. J. 550, 426 J. Schulze,Phys. Rev.D 91, no. 10, 105002 (2015).
(2001). [22] D.E.Alvarez-Castillo andD.Blaschke,arXiv:1412.8463
[7] F. Weber,Prog. Part. Nucl.Phys. 54, 193 (2005). [astro-ph.HE].
[8] D. Page and S. Reddy, Ann. Rev. Nucl. Part. Sci. 56, [23] R. Lastowiecki, D. Blaschke, T. Fischer, and T. Klahn,
327 (2006). arXiv:1503.04832 [nucl-th].
[9] P. Haensel, A. Y. Potekhin, and D. G. Yakovlev, Neu- [24] V.Dexheimer,R.Negreiros,andS.Schramm,Phys.Rev.
tron Stars 1, Astrophysics and Space Science Library, C 91, no. 5, 055808 (2015).
(Springer-Verlag, NewYork,2006). [25] F. O¨zel , D. Psaltis , T. Gu¨ver, G. Baym , C. Heinke,
[10] T. Kla¨hn et al.,Phys.Rev. C 74, 035802 (2006). and S.Guillot, arXiv:1505.05155.
[11] A.Sedrakian, Prog. Part. Nucl. Phys. 58, 168 (2007). [26] S. Guillot and R. E. Rutledge, Astrophys. J. Lett. 796,
[12] T. Kla¨hn et al.,Phys.Lett. B 654, 170 (2007). L3 (2014).
[13] M.G.Alford,A.Schmitt,K.Rajagopal, andT.Sch¨afer,
10
[27] A. W. Steiner, J. M. Lattimer, and E. F. Brown, Astro- (1986).
phys.J. Lett. 765, L5 (2013). [53] N. K. Glendenning and S. A. Moszkowski, Phys. Rev.
[28] V. Suleimanov, J. Poutanen, M. Revnivtsev, and Klaus Lett. 67, 2414 (1991).
Werner,Astrophys.J. 742, 122 (2011). [54] G.A.Lalazissis,J.Konig,andP.Ring,Phys.Rev.C55,
[29] J. Tru¨emper, Prog. Part. Nucl. Phys.66, 674 (2011). 540 (1997).
[30] J. Ellis, J. I. Kapusta, and K. A. Olive, Nucl. Phys. [55] D. B. Blaschke, D. Gomez Dumm, A. G. Grunfeld,
B348, 345 (1991). T. Klahn, and N. N. Scoccola, Phys. Rev. C 75, 065804
[31] N.K. Glendenning, Phys.Rev.D 46, 1274 (1992). (2007).
[32] G. Baym, C. J. Pethick, and P. Sutherland, Astrophys. [56] K. Kashiwa, T. Hell, and W. Weise, Phys. Rev. D 84,
J. 170, 299 (1971). 056010 (2011).
[33] G.Baym,H.A.Bethe,andC.J.Pethick,Nucl.Phys.A [57] G. A. Contrera, A. G. Grunfeld, and D. B. Blaschke,
175, 225 (1971). Phys. Part. Nucl. Lett. 11, 4, 342 (2014).
[34] K.Iida and K. Sato, Phys. Rev.C 58, 2538 (2008). [58] A.Drago,A.Lavagno,G.Pagliara,andD.Pigato,Phys.
[35] L.F.PalharesandE.S.Fraga,Phys.Rev.D82,125018 Rev. D 90, 065809 (2014).
(2010). [59] M. Rotondo, Jorge A. Rueda, R. Ruffini, and S.-S.Xue,
[36] T. Endo, Phys. Rev.C 83, 068801 (2011). Phys. Lett. B 701 667 (2011).
[37] M.B.Pinto,V.Koch,andJ.Randrup,Phys.Rev.C86, [60] D.Page,J.M.Lattimer,M.Prakash,andA.W.Steiner,
025203 (2012). Astrophys. J. Suppl.155 623 (2004).
[38] G.Lugones,A.G.Grunfeld,andM.A.Ajmi,Phys.Rev. [61] D.YakovlevandC.Pethick,Annu.Rev.Astron. Astro-
C 88, 045803 (2013). phys.42 169 (2004).
[39] W.Y.KeandY.X.Liu,Phys.Rev.C89,074041(2015). [62] D.Page,J.M.Lattimer,M.Prakash,andA.W.Steiner,
[40] M. Alford, K. Rajagopal, S. Reddy, and F. Wilczek, Astrophys. J. 707 1131 (2009).
Phys.Rev.D 64, 074017 (2001). [63] D.Page,M.Prakash,J.M.Lattimer,andA.W.Steiner,
[41] N.K. Glendenning, Phys.Rep. 342, 393 (2001). Phys. Rev.Lett. 106, 081101 (2011).
[42] S. Reddy, G. Bertsch, and M. Prakash, Phys. Lett. B [64] P. S. Shternin, D. G. Yakovlev, C. O. Heinke, W. C. G.
475, 1 (2000). Ho, and D. J. Patnaude, Monthly Notices of the Royal
[43] X.Na,R.Xu,F.Weber,andR.Negreiros,Phys.Rev.D Astronomical Society: Letters 412, L108-L112 (2011).
86, 123016 (2012). [65] D. Blaschke, H. Grigorian, D. N. Voskresensky, and F.
[44] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 Weber, Phys. Rev.C 85, 022802(R) (2012).
(1961); Phys.Rev. 124, 246 (1961). [66] J.M.C.Chen,J.W.Clark,R.D.Dav´e,andW.Khodel,
[45] A. Scarpettini, D. Gomez Dumm, and N. N. Scoccola, Nucl. Phys. A555, 59 (1993).
Phys.Rev.D 69, 114018 (2004). [67] D. G. Yakovlev, A. D. Kaminker, U. Y. Gnedin, and P.
[46] G. A. Contrera, D. Gomez Dumm, and N. N. Scoccola, Haensel, Phys. Reports354, 1 (2001).
Phys.Lett. B 661 113 (2008). [68] A. W. Steiner, Phys.Rev.C 74, 045808 (2006).
[47] G. A. Contrera, D. Gomez Dumm, and N. N. Scoccola, [69] C.O. HeinkeandW.C.G.Ho,Astrophys.J.719,L167
Phys.Rev.D 81, 054005 (2010). (2010).
[48] J. D. Walecka, Ann.Phys. 83, 497 (1974). [70] W. C. G. Ho, K. G. Elshamouty, C. O. Heinke, and
[49] J. Boguta and A.R. Bodmer, Nucl. Phys. A292, 413 A. Y.Potekhin, Phys. Rev.C 91, 015806 (2015).
(1977). [71] B.Posselt,G.G.Pavlov,V.SuleimanovandO.Kargalt-
[50] J. Boguta and J. Rafelski, Phys. Lett. 71B, 22 (1977). sev, Astrophys.J. 779, 186 (2013).
[51] J.BogutaandH.St¨ocker,Phys.Lett.120B,289(1983).
[52] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1