Hybrid stars with the Dyson-Schwinger quark model

Abstract

We study the hadron-quark phase transition in the interior of neutron stars.
For the hadronic sector, we use a microscopic equation of state involving
nucleons and hyperons derived within the Brueckner-Hartree-Fock many-body
theory with realistic two-body and three-body forces. For the description of
quark matter, we employ the Dyson-Schwinger approach and compare with the MIT
bag model. We calculate the structure of neutron star interiors comprising both
phases and find that with the Dyson-Schwinger model, the hadron-quark phase
transition takes place only when hyperons are excluded, and that a
two-solar-mass hybrid star is possible only if the nucleonic equation of state
is stiff enough.

Full text

Page 1

arXiv:1107.2497v1 [nucl-th] 13 Jul 2011
Hybrid stars with the Dyson-Schwinger quark model
H. Chen (陈欢), M. Baldo, G. F. Burgio, and H.-J. Schulze
INFN Sezione di Catania, and Dipartimento di Fisica e Astronomia,
Universita´di Catania, Via Santa Sofia 64, 95123 Catania, Italy
We study the hadron-quark phase transition in the interior of neutron stars. For the hadronic sector we
use a microscopic equation of state involving nucleons and hyperons derived within the Brueckner-Hartree-
Fock many-body theory with realistic two-body and three-body forces. For the description of quark matter, we
employ the Dyson-Schwinger approach and compare with the MIT bag model. We calculate the structure of
neutron star interiors comprising both phases and find that with the Dyson-Schwinger model, the hadron-quark
phase transition takes place only when hyperons are excluded, and that a two solar mass hybrid star is possible
only if the nucleonic EOS is stiff enough.
PACS numbers: 26.60.Kp, 12.39.-x, 12.39.Ba
I.INTRODUCTION
The possible appearence of quark matter (QM) in the inte-
rior of massive neutron stars (NS) is one of the main issues
in the physics of these compact objects. Calculations of NS
structure, based on a microscopic nucleonic equation of state
(EOS), indicate that for the heaviest NS, close to the maxi-
mum mass (about two solar masses), the central particle den-
sity reaches values larger than 1/fm3. In this density range
the nucleon cores (dimension ≈ 0.5 fm) start to touch each
other, and it is hard to imagine that only nucleonic degrees of
freedom can play a role. On the contrary, it can be expected
that even before reaching these density values, the nucleons
start to loose their identity, and quark degrees of freedom are
excited at a macroscopic level.
Unfortunately it is not straightforward to predict the rele-
vanceof quarkdegreesof freedomin the interiorof NS forthe
different physical observables, like cooling evolution, glitch
characteristics, neutrino emissivity, and so on. In fact, the
other NS components can mask the effects coming directly
from QM. In some cases the properties of quark and nucle-
onic matter are not very different, and a clear observational
signal of the presence of the deconfined phase inside a NS is
indeed hard to find.
The value of the maximum mass of a NS is probablyone of
the physical quantities that are most sensitive to the presence
of QM. If the QM EOS is sufficiently soft, the quark compo-
nent is expected to appear in NS and to affect appreciably the
maximum mass value. In fact the recent claim of discovery
of a two solar mass NS [1] has stimulated the interest in this
issue. Purely nucleonic EOS are able to accomodate masses
comparable with this large value [2–6]. However, the appear-
ance of hyperons in beta-stable matter is expected to strongly
reduce the maximum mass that can be reached by a baryonic
EOS [6, 7]. In this case the presence of non-baryonic, i.e,
“quark”matter seems to be the only possiblemannerto stiffen
the EOS and reach large NS masses. Heavy NS thus have to
be hybrid quark stars. In this paper we will discuss this issue
in detail.
Unfortunately, while the microscopic theory of the nucle-
onic EOS has reached a highdegree of sophistication, the QM
EOS is poorlyknownatzerotemperatureandat the highbary-
onic density appropriatefor NS. One has, therefore,to rely on
models of QM, which contain a high degree of arbitraryness.
At present, the best one can do is to compare the predictions
of different quark models and to estimate the uncertainty of
the results for the NS matter as well as for the NS structure
and mass. Continuing a set of previous investigations using
different quark models [8–12], we employ in this paper the
Dyson-Schwinger model (DSM) for QM [13–16] in combi-
nation with a definite baryonic EOS, which has been devel-
oped within the Brueckner-Hartree-Fock (BHF) many-body
approach of nuclear matter, comprising nucleons and also hy-
perons. Confrontationwith previouscalculations shall also be
discussed.
The paper is organized as follows. In section II we review
the determination of the baryonic EOS in the BHF approach.
SectionIIIconcernstheQMEOSaccordingtotheDSM,com-
paring also with the MIT bag model for reference. In section
IV we present the results regarding NS structure, combining
the baryonicand QM EOS forbeta-stablenuclearmatter. Sec-
tion V contains our conclusions.
II.EOS OF HYPERNUCLEAR MATTER WITHIN
BRUECKNER THEORY
TheBrueckner-Bethe-Goldstonetheoryisbasedonalinked
cluster expansion of the energy per nucleon of nuclear matter
(see Ref. [17], chapter 1 and references therein). The fun-
damental quantity of interest in this many-body approach is
the Brueckner reaction matrix G, which is the solution of the
Bethe-Goldstone equation, written in operatorial form as
Gab[W] =Vab+∑
c∑
p,p′Vac
??pp′?
Qc
W −Ec+iε
?pp′??Gcb[W],
(1)
wheretheindices a,b,cindicatepairs ofbaryonsandthe Pauli
operator Qcand energy Eccharacterize the propagation of in-
termediate baryon pairs. The pair energy in a given channel
c = (B1B2) is
E(B1B2)= TB1(kB1)+TB2(kB2)+UB1(kB1)+UB2(kB2)
(2)

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with TB(k) = mB+k2/2mB, where the various single-particle
potentials are given by
UB(k) =
∑
B′=n,p,Λ,Σ−U(B′)
B
(k)
(3)
and are determined self-consistently from the G matrices,
U(B′)
B
(k) = ∑
k′<k(B′)
F
Re?kk′??G(BB′)(BB′)
?E(BB′)(k,k′)???kk′?.
(4)
The coupled equations (1) to (4) define the BHF scheme with
the continuous choice of the single-particle energies. In con-
trast to the standard purely nucleonic calculation, the addi-
tional coupled-channel structure due to hyperons renders the
calculations quite time-consuming.
Once the different single-particle potentials are known, the
total nonrelativistic baryonic energy density, ε, can be evalu-
ated:
ε =
∑
B=n,p,Λ,Σ−∑
k<k(B)
F
?
TB(k)+1
2UB(k)
?
.
(5)
It has been shown that the nuclear EOS can be calculated with
good accuracy in the Brueckner two hole-line approximation
with the continuous choice for the single-particle potential,
and that the results in this scheme are quite close to the cal-
culations which include also the three hole-line contribution
[18].
The basic input quantities in the Bethe-Goldstone equa-
tion are the nucleon-nucleon (NN), nucleon-hyperon (NY),
and hyperon-hyperon (YY) two-body potentials V. The in-
clusion of nuclear three-body forces (TBF) is crucial in order
to reproducethe correct saturation point of symmetric nuclear
matter. The present theoretical status of microscopically de-
rived TBF is quite rudimentary,and in most approaches semi-
phenomenological TBF are used that involve several free pa-
rameters usually fitted to the relevant data. An important con-
straint is the consistency with a given two-body force, i.e.,
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. Recent results [6, 19] have
been published within this framework,using meson-exchange
TBF that employ the same meson-exchangeparameters as the
underlying nucleon-nucleon potential.
In this paper, we use results obtained in this manner based
on the ArgonneV18(V18) [20], the Bonn B (BOB) [21], and
the Nijmegen93 (N93) [22] potentials, and comparealso with
the widely used phenomenological Urbana-type (UIX) TBF
[23] (in combination with the V18potential). We remind the
reader that in our approach the TBF is reduced to a density-
dependent two-body force by averaging over the position of
the third particle, assuming that the probability of having two
particles at a given distance is given by the two-body correla-
tion function determined self-consistently.
In the past years, the BHF approachhas beenextendedwith
the inclusion of hyperons [7, 24, 25], which may appear at
sufficiently large baryon density in the inner part of a NS,
and lower the ground state energy of the dense nuclear matter
phase. In our work we use the Nijmegensoft-core NSC89 NY
potential [26] that is well adapted to the available experimen-
tal NY scattering data and also compatible with Λ hypernu-
clear levels [27, 28]. Unfortunately, up to date no YY scat-
tering data and therefore no reliable YY potentials are avail-
able. We therefore neglect these interactions in our calcula-
tions, which is supposedly justified, as long as the hyperonic
partial densities remain limited.
We have previously found rather low hyperon onset densi-
ties of about 2 to 3 times normal nuclearmatter density for the
appearanceof the Σ−and Λ hyperons[7, 24, 25] (other hyper-
ons do not appear in the matter). Moreover, an almost equal
percentage of nucleons and hyperons are present in the stellar
core at high densities. The inclusion of hyperons produces an
EOS which turns out to be much softer than the purely nucle-
onic case, with dramatic consequences for the structure of the
NS (see below). We do not expect substantial changes when
introducing refinements of the theoretical framework, such as
hyperon-hyperonpotentials [25], relativistic corrections, etc..
Three-body forces involving hyperons could produce a sub-
stantial stiffeningof the baryonicEOS. Unfortunatelytheyare
essentially unknown, but can be expected to be weaker than
in the non-strange sector. Another possibility that is able to
produce larger maximum masses, is the appearence of a tran-
sition to QM inside the star. This will be discussed in the next
sections.
III. QUARK PHASE
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
thecomplicatednonlinearandnonperturbativenatureofquan-
tum chromodynamics (QCD). Therefore one can presently
only resort to more or less phenomenological models for de-
scribing QM, and in this paper we illustrate results obtained
by adopting the DSM. A brief comparison with results from
the MIT bag model will also be made.
A.Dyson-Schwinger equations approach
For the deconfinedquark phase, we adopt a model based on
the Dyson-Schwingerequations (DSE) of QCD. DSE provide
a continuum approach to QCD that can simultaneously ad-
dressbothconfinementanddynamicalchiralsymmetrybreak-
ing [13, 14].It has been applied with success to hadron
physics in vacuum [15, 16, 29, 30], and to QCD at nonzero
chemical potential and temperature [31–35]. Recently efforts
have been made to calculate the EOS for cold quark matter
and compact star [36, 37].
Our starting point is QCD’s gap equation for the quark
propagatorS(p;µ) at finite quarkchemical potential µ, which

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reads1
S(p;µ)−1= Z2[iγp+iγ4(p4+iµ)+mq]+Σ(p;µ),
(6)
with the renormalised self-energy expressed as
Σ(p;µ) =
?Λd4q
(7)
Z1
(2π)4g2(µ)Dρσ(p−q;µ)λa
2γρS(q;µ)Γa
σ(q,p;µ),
where
tion of the integral, with Λ the regularisation mass-scale.
Here g(µ) is the coupling strength, Dρσ(k;µ) is the dressed
gluon propagator, and Γa
vertex. Moreover λaare the Gell-Mann matrices, and mqis
the Λ-dependent current-quark bare mass. The quark-gluon
vertex and quark wave function renormalisation constants,
Z1,2(ζ2,Λ2), depend on the renormalisation point ζ, the reg-
ularisation mass-scale Λ, and the gauge parameter.
At finite chemical potential, the quark propagator can as-
sume a general form with rotational covariance
?Λrepresents a translationally invariant regularisa-
σ(q,p;µ) the dressed quark-gluon
S(p;µ)−1= iγp A(p2,p·u,ζ2)+B(p2,p·u,ζ2)
+iγ4(p4+iµ)C(p2,p·u,ζ2),
(8)
where we have written u =(0,iµ). Please note that we ignore
quark Cooper pairing herein. Diquark condensate and color
superconductivity have been considered in the DSM [38–40],
and we will extend our analysis to that case in the future.
The kernel, Eq. (8), depends on the gluon propagator and
the quark-gluon vertex at finite chemical potential. However,
little is known about them except at very high chemical po-
tential, where perturbation theory is applied. We have to ex-
tend to finite µ the DSM that has been successfully applied to
hadron physics at µ = 0.
The Ans¨ atze at zero chemical potential are typically imple-
mented by writing
Z1g2Dρσ(p−q)Γa
σ(q,p) =
G((p−q)2)Dfree
ρσ(p−q)λa
2Γσ(q,p),
(9)
wherein Dfree
gauge free gluon propagator, G(k2) is a model effective in-
teraction, and Γσ(q,p) is a vertex Ansatz.
Herein we consider the widely used ’rainbow approxima-
tion’
ρσ(k ≡ p− q) = (δρσ−kρkσ
k2)1
k2is the Landau-
Γσ(q,p) = γσ,
(10)
and a Gaussian-type effective interaction [41]
G(k2)
k2
=4π2
ω6Dk2e−k2/ω2,
(11)
1In our Euclidean metric: {γρ,γσ} = 2δρσ; γ†
∑4
ρ= γρ; γ5= γ4γ1γ2γ3; ab =
i=1aibi; ab = ∑3
i=1aibi; and Pρtimelike ⇒ P2< 0.
involving two parameters D and ω. This is a finite-width
representation of the Munczek-Nemirovsky(MN) model [42]
used in Ref. [37], which expresses the long-range behavior
oftherenormalisation-group-improvedeffectiveinteractionin
Refs. [16, 43, 44]. Equation (11) delivers an ultraviolet-finite
model gap equation. Hence, the regularisation mass-scale Λ
can be removed to infinity and the renormalisation constants
Z1,2set equal to one. In this model, there is no interaction be-
tweendifferentflavorsofquarks. Therefore,thegapequations
for the different flavors are independent of each other. Here
we consider the light flavors u, d, and s, neglecting heavier
ones.
Usually, there exist two solutions of Eq. (6) in the chiral
limit, i.e., when mq= 0. One solution with non-zero quark
mass function M(p) ≡B(p)
represents a phase with dynamical chiral symmetry breaking
and confinement. The other solution with zero mass func-
tion at the chiral limit is called Wigner solution, which repre-
sents a phase with chiral symmetry and deconfinement. The
Nambu phase is realized in vacuum, and provides the base-
ment for describing physics in vacuum. The phase transition
to deconfinement at finite chemical potential, without consid-
ering hadron degrees of freedom, is investigated in Ref. [34].
For strange quarks, only the Nambu solution exists in vacuum
[45]. The appearance of the Wigner phase for strange quarks
at finite chemical potential will be investigated in the follow-
ing.
In Ref. [34] the µq?= 0 Ansatz is specified as the same as in
vacuum. It is reasonable in the Nambu phase, but not in the
Wigner phase. In this article we introduce a further parameter
inordertostudyapossibledensitydependenceoftheeffective
interaction. Consideringasymptoticfreedomat highchemical
potential,we extendthe effectiveinteractionat finite chemical
potential in a simple manner,
A(p)is called Nambu solution, and
G(k2;µ)
k2
=4π2
ω6De−αµ2/ω2k2e−k2/ω2,
(12)
introducing the parameter α, which controls the rate of ap-
proachingasymptoticfreedom. In the followingwe will study
the dependence of our results on this parameter, marked as
’DSα’. Compared to this model, the MIT bag model with
non-interacting quarks corresponds to α = ∞. For simplic-
ity, we assume the same effective interactionEq. (12) for each
flavor.
The parameters D and ω of the model, Eq. (12), and the
quarkmasses canbe determinedbyfitting mesonpropertiesin
vacuum [41]. We choose the set of parameters ω = 0.5 GeV
and D = 1 GeV2. For simplicity, we use current quark masses
mu,d= 0, while ms= 115 MeV is obtained by fitting the K
meson mass in vacuum, which is a little different from the
usually used value ms≈ 150 MeV in the MIT bag model.
The EOS of cold QM is given following Refs. [34, 37]. We
express the quark number density as
nq(µ) = 6
?
1
4π
d3p
(2π)3fq(|p|;µ),
?∞
(13)
fq(|p|;µ) =
−∞dp4trD[−γ4Sq(p;µ)] ,
(14)

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where the trace is over spinorindices only. The quarkthermo-
dynamic pressure at zero temperature can be obtained as
Pq(µq) = Pq(µq,0)+
?µq
µq,0
dµnq(µ).
(15)
The total density and pressure for the quarkphase are given
by summing contributions from all flavors. For comparison
with the bag model, we write the pressure of the quark phase
as
PQ(µu,µd,µs) = ∑
q=u,d,s
˜Pq(µq)−BDS,
(16)
where
˜Pq(µq) ≡
?µq
µq,0
dµnq(µ),
(17)
BDS ≡ − ∑
q=u,d,s
Pq(µq,0).
(18)
Theoretically, we can choose arbitrary values of µq,0, where
the Wigner phase exists. For massless quarks, the Wigner
phase exists in vacuum and at finite chemical potentials; the
results of nq(µq) are shown in the upper panel of Fig. 1.
Therefore, we choose µu,0= µd,0= 0, and the corresponding
results of˜Pq(µq) are shown in the lower panel of Fig. 1. For
strangequarks,the Wignerphasewith nonzerovalueofρscan
only be obtainedwith chemical potential µsand the parameter
α above some thresholds [45]; see the upper panel of Fig. 2.
Therefore, we set µs,0as the value of the starting point of the
Wigner phase with each α. The single-quark number density
and pressure˜Psfor strange quarks are shown in Fig. 2.
Now only BDSneeds to be fixed in this model. For u and d
quarks, we can use the ’steepest-descent’ approximation[46],
P[S] = TrLn?S−1?−1
2Tr[ΣS] ,
(19)
which is consistent with the gapequationwithin the ’rainbow’
approximation. In vacuum we obtain the pressure difference
between the Nambu phase and the Wigner phase for massless
quarks [34]
B ≡ P[SN]−P[SW] = 45 MeVfm−3.
(20)
InterpretingtheNambuphaseastherealvacuumwithP[SN]=
0, we then obtain the pressure of the Wigner phase for light
quarks in vacuum Pu,d(µ0= 0) = −45 MeVfm−3and the ef-
fective bag constant Bnf=2
DS
the introductionof the parameter α, the ’steepest-descent’ ap-
proximation is not consistent with the gap equation and we
cannot use it to calculate Ps(µs,0). Therefore,in our model we
simply set BDS= 90MeVfm−3as a parameter, neglecting the
ambiguity from Ps(µs,0).
In summary, with respect to the MIT bag model with free
quarks, the pressure of the DSM at a fixed chemical potential
is lower, which corresponds to an increasing bag constant as
chemical potential (density) increases. In the next section, we
will show a density-dependenteffective bag constant for beta-
stable QM in the interior of NS.
= 90 MeVfm−3. However, due to
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
FIG. 1: (Color online) Quark number density (upper panel) and pres-
sure˜Pq, Eq. (17), (lower panel) for massless quarks at finite chemi-
cal potentials. Different curves correspond to different values of the
parameter α = 0,0.5,1,2,4,∞. The curve denoted ’DS-MN’ corre-
sponds to the results in Ref. [37].
B.MIT bag model
We adopt the MIT bag model assuming massless u and d
quarks, s quarks with a current mass of ms= 150 MeV, and
either a fixed bag constant B = 90 MeVfm−3, or a density-
dependent bag parameter,
B(ρ) = B∞+(B0−B∞)exp
?
−β
?ρ
ρ0
?2?
(21)
with B∞= 50MeVfm−3, B0= 400MeVfm−3, and β = 0.17.
This approach has been proposed in [8], and it allows the
symmetric nuclear matter to be in the pure hadronic phase at
low densities, and in the quark phase at large densities, while
the transition density is taken as a parameter. Several possi-
ble choices of the parameters have been explored in [8], and
all give a NS maximum mass in a relatively narrow interval,
1.4 M⊙? Mmax? 1.7 M⊙.
Ithasalsobeenfound[8,47]thatwithintheMITbagmodel
(without color superconductivity) with a density-independent
bag constant B, the maximum mass of a NS cannot exceed a
value of about 1.6 solar masses. Indeed, the maximum mass

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?
?
?
?
?
?
?
?
?
?
?
?
?
FIG. 2: (Color online) Same as Fig. 1, but for strange quarks with
ms= 115 MeV.
increases as the value of B decreases, but too small values of
B are incompatible with a transition density ρ ? (2,...,3)ρ0
in symmetric nuclear matter, as demandedby heavy-ion colli-
sion phenomenology. For a more extensive discussion of the
MIT bag model, the reader is referred to [8].
IV.RESULTS AND DISCUSSION
A. Beta-stable hadronic matter
In order to study the structure of NS, we have to calculate
the composition and the EOS of cold, neutrino-free,catalyzed
matter. We require that the NS contains charge-neutral mat-
ter consisting of baryons (n, p, Λ, Σ−) and leptons (e−, µ−)
in beta equilibrium, and compute the EOS in the following
standard way [2, 3, 48]: The Brueckner calculation yields the
energy density of baryon/lepton matter as a function of the
different partial densities, ε(ρn,ρp,ρΛ,ρΣ,ρe,ρµ), by adding
the contribution of noninteracting leptons to Eq. (5). The var-
ious chemical potentials (of the species i = n,p,Λ,Σ−,e,µ)
?
?
?
?
?
?
?
?
?
FIG. 3: (Color online) Pressure vs. the baryon number density of
hadronic NS matter. Thick curves show results for purely nucleonic
matter, whereas thin curves include hyperons.
can then be computed straightforwardly,
µi=∂ε
∂ρi,
(22)
and the equations for beta-equilibrium,
µi= biµn−qiµe,
(23)
(biand qidenoting baryon number and charge of species i)
and charge neutrality,
∑
i
allow one to determine the equilibrium composition {ρi(ρ)}
at given baryon density ρ and finally the EOS,
P(ρ) = ρ2d
dρρ
In Fig. 3 we compare the EOS obtained in the BHF frame-
work when only nucleons and leptons are present (thick
lines), and the corresponding ones with hyperons included
(thin lines). Calculations have been performed with different
choices of the NN potentials, i.e., the Bonn B, the Argonne
V18, and the Nijmegen N93, all supplemented by a compat-
ible microscopic TBF [6]. For completeness, we also show
results obtained with the ArgonneV18 potential together with
the phenomenologicalUrbana IX as TBF.
We notice a strong dependence on both the chosen NN po-
tential, and on the adopted TBF, the microscopic one being
more repulsive than the phenomenological force. The pres-
ence of hyperons decreases strongly the pressure, and the re-
sulting EOS turns out to be almost independentof the adopted
NN potential, due to the interplay between the stiffness of the
nucleonicEOS and the thresholddensity of hyperons[6]. The
softening of the EOS has serious consequences for the struc-
ture of NS, leading to a maximum mass of less than 1.4 solar
masses [6, 7], which is below observed pulsar masses [49].
ρiqi= 0,
(24)
ε({ρi(ρ)})
= ρdε
dρ−ε = ρµn−ε .
(25)

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?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
FIG. 4: (Color online) Baryon density (upper panel) and pressure
(lower panel) vs. the baryon chemical potential of NS matter for dif-
ferent models. The vertical dotted lines indicate the positions of the
phase transitions under the Maxwell construction.
B.Quark matter in beta-equilibrium
In compact stars, we need to consider matter in beta-
equilibrium,
d ↔ u+e+νe↔ u+µ +νµ↔ s,
(26)
and charge neutrality, which gives the constraints in pure QM
µd= µu+µe= µu+µµ= µs,
2nu−nd−ns
3
(27)
−ρe−ρµ= 0.
(28)
The numerical results of baryon number density ρB=
(nu+nd+ns)/3 and pressure versus baryon chemical poten-
tial µB= µu+2µdare shown in Fig. 4 for the several cases
discussed above.
For the hadroniccase, we display results only for the Bonn-
B NN potential, which gives the stiffest EOS without hy-
perons, and thus is the most favored for reaching large NS
masses. In this case the thick solid (dashed) curve indicates
results obtainedwithout (with) hyperons. For QM, we plot re-
sults with the MIT bag model, with mu,d= 0, ms=150 MeV,
and a bag constant B = 90 MeVfm−3, or a density-dependent
B(ρ), Eq. (21) (thin curves). The remaining curves are re-
sults obtained with the DSM and several choices of the model
parameter α.
The crossing points of the baryon and quark pressure
curves(markedwithasquare)representthetransitionbetween
baryon and QM phases under the Maxwell construction. The
projection of these points (dotted lines) on the baryon and
quark density curves in the upper panel indicate the corre-
sponding transition densities from low-density baryonic mat-
ter to high-density QM. Some qualitative considerations can
be done. In particular,we notice that the phase transition from
hadronic to QM occurs at low values of the baryon chemi-
cal potential when the MIT bag model is used to describe the
quark phase, whereas much higher values are required with
the DSM. In some extreme cases, such as DS0, no phase tran-
sition at all is possible. In fact, the DSM EOS is generally
stiffer than the hadronic one, and the value of the transition
density is high. We also notice that with the DSM no phase
transition exists if the hadronic phase contains hyperons.
After these indications, we study in the following the phase
transition with the more sophisticated Gibbs construction.
C. Phase transition in beta-stable matter
A realistic model of the phase transition between bary-
onic and quark phase inside the star is the Gibbs construc-
tion [2, 11, 50], which determines a range of baryon densi-
ties where both phases coexist, yielding an EOS containing a
pure hadronic phase, a mixed phase, and a pure QM region.
The crucial point of the Gibbs construction, as suggested by
Glendenning[50], is that both the hadron and the quark phase
are separately charged, while preserving the total charge neu-
trality. This implies that NS matter can be treated as a two-
component system, and therefore can be parametrized by two
chemical potentials. Usually one chooses the pair (µe,µn),
i.e., electron and baryon chemical potential. The pressure is
the same in the two phases to ensure mechanical stability,
while the chemical potentials of the different species are re-
lated to each other, satisfying chemical and beta stability. As
a consequence, the pressure turns out to be a monotonically
increasing function of the density. We note that our Gibbs
treatment is the zero surface tension limit of the calculations
including finite-size effects [11, 51].
The Gibbs conditions for chemical and mechanical equilib-
rium between both phases read
µu+µe = µd= µs,
µp+µe = µn= µΛ= µu+2µd,
µΣ− +µp = 2µn,
pH(µe,µn) = pQ(µe,µn) = pM(µn).
(29)
(30)
(31)
(32)

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?
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?
?
?
FIG. 5: (Color online) Pressure vs. baryon density of NS matter with
the Gibbs phase transition construction for different models.
From these equations one can calculate the equilibriumchem-
ical potentials of the mixed phase corresponding to the inter-
section of the two surfaces representing the hadron and the
quark phase, which allows one to calculate the charge densi-
ties ρH
by QM in the mixed phase, i.e.,
cand ρQ
c and therefore the volume fraction χ occupied
χρQ
c+(1−χ)ρH
c= 0.
(33)
From this, the energy density εMand the baryon density ρM
of the mixed phase can be determined as
εM = χεQ+(1−χ)εH,
ρM = χρQ+(1−χ)ρH.
(34)
(35)
In Fig. 5 we display results for the EOS including the
hadron-quark phase transition, using the same conventions as
in Fig. 4. We notice that the phase transition constructed with
the DSM turns out to be quite different from the one obtained
using the MIT bag model. In the former case, if the coexis-
tence regiondoes exist, it is shifted to higherbaryonicdensity.
In order to clarify the fundamental differencebetween MIT
modelononesideandDSMontheotherside, weplotinFig.6
the effective density-dependentbag constant
B(ρ) ≡ ε(ρ)−εfree(ρ)
(36)
obtained with the DSM in NS matter. One observes that B(ρ)
is a monotonically increasing function in contrast to the em-
pirical density dependence introduced with the MIT model.
D.Neutron star structure
We assume that a NS is a spherically symmetric distribu-
tion of mass in hydrostatic equilibrium. The equilibrium con-
figurations are obtained by solving the Tolman-Oppenheimer-
?
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?
?
?
?
?
?
?
?
?
FIG. 6: (Color online) Effective bag constant obtained with the MIT
model and the DSM.
Volkoff (TOV) equations [48] for the pressure P and the en-
closed mass m,
dP
dr
dm
dr
= −Gmε
r2
(1+P/ε)?1+4πr3P/m?
1−2Gm/r
,
(37)
= 4πr2ε ,
(38)
being G the gravitational constant. Starting with a central
mass density ε(r = 0) ≡ εc, we integrate out until the density
on the surface equals the one of iron. This gives the stellar
radius R and the gravitational mass is then
MG≡ m(R) = 4π
?R
0
dr r2ε(r).
(39)
We have used as input the EOS discussed above and shown in
Fig. 5. For the description of the NS crust, we have joined the
hadronic EOS with the ones by Negele and Vautherin [52]
in the medium-density regime, and the ones by Feynman-
Metropolis-Teller [53] and Baym-Pethick-Sutherland[54] for
the outer crust.
The results are plotted in Figs. 7 and 8, where we dis-
play the gravitational mass MG(in units of the solar mass
M⊙= 2×1033g) as a function of the radius R and central
baryon density ρc. We present results obtained with two ex-
treme choices of the hadronicEOS yielding rather low or very
highmaximumNSmasses, namelytheUIXandBOB models,
respectively. Themaximummassesareinthesecases 1.84M⊙
and2.50M⊙with onlynucleonsand1.30M⊙and1.37M⊙in-
cluding hyperons.
The possible effects of the hadron-quark phase transition
are very different with the MIT model and the DSM: In the
case of the MIT model, the phase transition beginsat very low
baryondensity and thus effectivelyimpedesthe appearanceof
hyperons [11]. Consequently the resulting maximum mass of
the MIT hybrid star is 1.5 M⊙, lower than the value of the

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stars. In fact, whereasstars built with the MITbagmodelhave
a purehadronphase at low density,followedbya mixedphase
and a pure quark core at higher density, the ones obtained
with the DSM containonlya hadronphase anda mixedphase,
and probably no pure quark interior. The scenario resembles
theoneobtainedwithintheNambu-Jona-Lasinio(NJL)model
[9], where at most a mixed phase is present, without a pure
quark phase.
A clear difference between the two models exists as far 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.
?
?
FIG. 7: (Color online) Gravitational NS mass vs. the radius (right
panel) and the central baryon density (left panel) for different EOS
employing the BOB hadronic model.
?
?
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?
?
?
?
?
?
FIG. 8: (Color online) Same as Fig. 7, but with the UIX model.
nucleonic star, but higher than that of the hyperon star given
before.
On the contrary, with the DSM no phase transition can oc-
cur and no hybrid star can exist if hyperons are introduced.
If hyperons are excluded, the phase transition from nucleon
matter to QM takes place at rather large baryon density. The
maximum mass of the hybrid star has a slightly smaller value
than that with only nucleons, and decreases as the density of
the phase transition decreases. For example, when the nu-
cleonic Bonn-B NN potential is used, the maximum mass of
hybrid stars is only a little lower than 2.5 M⊙with α = 0.5,
and decreases to about 2 M⊙with α = 2. The same happens
if the nucleonic UIX interaction is adopted, in which case the
maximummass of the hybridstar cannotexceed1.84 M⊙. We
remindthat with the valueα =0 correspondingto the unmod-
ified DSM without density-dependent interaction strength, no
phase transition at all is possible! With α increasing to ∞ we
can obtain a smooth change from the pure hadronic NS to the
results with MIT bag model.
We also notice a large difference in the structure of hybrid
V.CONCLUSIONS
We have investigated the capability of the DSM for QM to
provide hybrid NS configurations in combination with a mi-
croscopic hadronic EOS obtained within the BHF formalism
including also the appearance of hyperons. We found that the
unmodified DSM does not allow a hadron-quark phase tran-
sition, thus requiring the introduction of an empirical density
dependence of the quark interaction strength. Even in this
case, however, the early appearance of hyperons in hadronic
matter inhibits the phase transition. Only in the fictitious case
ofrestrictingtopurenucleonmatter,a phasetransitionat large
baryondensity is possible, and a hybridstar with 2 M⊙is only
allowed if the nucleonic EOS is stiff enough to produce a NS
with 2 M⊙.
These features of the DSM are very different from MIT-
type quark models, and we have attributed it to the attractive
interaction between quarks in the DSM, which can be repre-
sented as a density-dependent bag constant. We have found a
different density dependence of the effective bag constant in
the DSM and the MIT quark model.
Acknowledgments
ThisworkwaspartiallysupportedbyCompStar,aResearch
Networking Programmeof the EuropeanScience Foundation,
and by the MIUR-PRIN Project 2008KRBZTR.
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