Deconfinement transition in protoneutron stars: analysis within the Nambu-Jona-Lasinio model

0.24 MB·English

by  G. Lugones

Deconfinement transition in protoneutron stars: analysis within the
Nambu-Jona-Lasinio model
G. Lugones1, T. A. S. do Carmo1, A. G. Grunfeld2,3,4, N. N. Scoccola2,3,5
1 Universidade Federal do ABC, Centro de Ciencias Naturais e Humanas,
Rua Santa Ad´elia, 166, 09210-170, Santo Andr´e, Brazil
2 CONICET, Rivadavia 1917, (1033) Buenos Aires, Argentina.
3 Departmento de F´ısica, Comisio´n Nacional de Energ´ıa Ato´mica, (1429) Buenos Aires, Argentina.
4 Department of Physics, Sultan Qaboos University,
P.O.Box: 36 Al-Khode 123 Muscat, Sultanate of Oman
5 Universidad Favaloro, Sol´ıs 453, (1078) Buenos Aires, Argentina.
Westudytheeffectofcolorsuperconductivityandneutrinotrappingonthedeconfinementtransi-
tionofhadronicmatterintoquarkmatterinaprotoneutronstar. Todescribethestronglyinteracting
0
matter a two-phase picture is adopted. For the hadronic phase we use different parameterizations
1
of a non-linear Walecka model which includes the whole baryon octet. For the quark matter phase
0
weusean SU(3) Nambu-Jona-Lasinio effectivemodel which includescolor superconductivity. We
2 f
imposecolorandflavorconservation duringthetransition insuchawaythatjustdeconfinedquark

n matter is transitorily out of equilibrium with respect to weak interactions. Wefind that deconfine-
a ment is more difficult for small neutrino content and it is easier for lower temperatures although
J
these effects are not too large. In addition they will tend to cancel each other as the protoneutron

1 star cools and deleptonizes, resulting a transition density that is roughly constant along the evo-
1 lution of the protoneutron star. According to these results the deconfinement transition is favored
after substantial cooling and contraction of theprotoneutron star.
]
h
PACSnumbers: 12.39.Fe,25.75.Nq,26.60.Kp
t
-
l
c
u I. INTRODUCTION
n
[
It is currently a matter of speculation the actual occurrence of quark matter during protoneutron star (PNS)

1 evolution. Thestandardscenarioforthebirthofneutronstarsindicatesthattheseobjectsareformedasconsequence
v
of the gravitationalcollapse and supernova explosion of a massive star [1–3]. Initially, PNSs are very hot and lepton-
9
rich objects, where neutrinos are temporarily trapped. During the first tens of seconds of evolution the PNS evolves
0
toformacold(T <1010 K)catalyzedneutronstar[1–3]. As neutrinosareradiated,the lepton-per-baryoncontent
7
1 ofmattergoesdownandtheneutrinochemicalpotentialtendstoessentiallyzeroin 50seconds[3]. Deleptonization
∼
. is fundamental for quark matter formationinside neutron stars, since it has been shownthat the presence of trapped
1
neutrinos in hadronic matter strongly disfavors the deconfinement transition [4, 5]. In fact, neutrino trapping makes
0
0 the density for the deconfinement transition to be higher than in the case of neutrino-free hadronic matter. As
1 a consequence, the transition could be delayed several seconds after the bounce of the stellar core. However, the
: calculationspresentedin[4,5]wereperformedemployingtheMITBagmodelforthedescriptionofquarkmatterand
v
did notinclude the effect ofcolorsuperconductivity. As we shallsee in the presentwork,the use ofthe Nambu-Jona-
i
X Lasinio model and the inclusion of color superconductivity may change qualitatively the effect of neutrino trapping
r in the deconfinement conditions.
a
As emphasizedinearlierworks[4–11],animportantcharacteristicofthedeconfinementtransitioninneutronstars,
is that just deconfined quark matter is transitorily out of equilibrium with respect to weak interactions. In fact,
depending onthe temperature,the transitionshouldbeginwith the quantumor thermalnucleationof a smallquark-
matterdropnearthecenterofthestar. Ontheotherhand,theflavorcompositionofhadronicmatterinβ-equilibrium
isdifferentfromthatofaβ-stablequark-matterdrop. Roughlyspeaking,thedirectformationofaβ-stablequark-drop
with N quarks will need the almost simultaneous conversion of N/3 up and down quarks into strange quarks, a
∼
process which is strongly suppressed with respect to the formation of a non β-stable drop by a factor G2N/3. For
∼ Fermi
typical values of the critical-size β-stable drop (N 100 1000 [6]) the suppression factor is actually tiny. Thus,
∼ −
quarkflavormustbe conservedduringthe deconfinementtransition[4–10]. When colorsuperconductivityis included
together with flavor conservation, the most likely configuration of the just deconfined phase is 2SC provided the
pairing gap is large enough [9]. The relevance of this 2SC intermediate phase (a kind of activation barrier) has been
analyzed for deleptonized neutron stars [10, 12] but not for hot and lepton-rich objects like PNSs.
In the present paper we shall analyze the deconfinement transition in protoneutron star conditions employing the
Nambu-Jona-Lasinio model in the description of quark matter. For the hadronic phase we shall use a model based
on a relativistic Lagrangian of hadrons interacting via the exchange of σ, ρ, and ω mesons [13]. For simplicity, the
analysis will be made in bulk, i.e. without taking into account the energy cost due to finite size effects in creating a
2
Label composition xσ =xρ xω (gσ/mσ)2 (gω/mω)2 (gρ/mρ)2 b c Mmax
[fm2] [fm2] [fm2]
GM 1 baryon octet + e− 0.6 0.653 11.79 7.149 4.411 0.002947 -0.001070 1.78 M⊙
GM 4 baryon octet + e− 0.9 0.9 11.79 7.149 4.411 0.002947 -0.001070 2.2 M⊙
TABLE I: Parameters of the hadronic equation of state. For each parametrization we give the maximum mass Mmax of a
hadronic star.
drop of deconfined quark matter in the hadronic environment.
The presentarticle is organizedas follows. InSec. II we presentthe main aspects ofthe non–linearWalecka model
describing the hadronic phase. In Sec. III we present the generalities of the model we use for the quark phase. In
Sec. IV we show our numerical results and finally in Sec. V we discuss our results and present the conclusions.
II. THE HADRONIC PHASE
For the hadronic phase we shalluse a non-linear Walecka model (NLWM) [13–15] which includes the whole baryon
octet, electrons and electron neutrinos in equilibrium under weak interactions. The Lagrangianof the model is given
by
= + + , (1)
B M L
L L L L
where the indices B, M and L refer to baryons, mesons and leptons respectively. For the baryons we have
= ψ¯ γµ(i∂ g ω g ~τ ρ~ ) (m g σ) ψ , (2)
B B µ ωB µ ρB µ B σB B
L − − · − −
B (cid:20) (cid:21)
X
with B =n, p, Λ, Σ+, Σ0, Σ−, Ξ−, and Ξ0. The contribution of the mesons σ, ω and ρ is given by
1 b c
= (∂ σ∂µσ m2 σ2) m (g σ)3 (g σ)4
LM 2 µ − σ − 3 N σ − 4 σ
1 1 1 1
ω ωµν + m2 ω ωµ ρ~ ρ~µν + m2 ρ~ ρ~µ, (3)
−4 µν 2 ω µ − 4 µν · 2 ρ µ·
where the coupling constants are g =x g , g =x g and g =x g . The ratios x , x and x are
σB σB σ ωB ωB ω ρB ρB ρ σB ωB ρB
equal to 1 for the nucleons and acquire different values for the other baryons depending on the parametrization (see
TableI).TheleptonicsectorisincludedasafreeFermigasofelectronsandelectronneutrinosinchemicalequilibrium
with the other particles.
Therearefiveconstantsinthemodelthataredeterminedbythepropertiesofnuclearmatter,threethatdetermine
thenucleoncouplingstothescalar,vectorandvector-isovectormesonsg /m ,g /m ,g /m ,andtwothatdetermine
σ σ ω ω ρ ρ
the scalar self interactions b and c. It is assumed that all hyperons in the octet have the same coupling than the
Λ. These couplings are expressed as a ratio to the nucleon couplings mentioned above, that we thus simply denote
x , x and x . In the present work we use two parameterizations for the constants. One of them is the standard
σ ω ρ
parameterization GM1 given by Glendenning–Moszkowski [13], as shown in Table I. This parametrization employs
“low” values for x , x and x . The parametrization GM4 employs larger values of these couplings. This makes the
σ ω ρ
EOS stiffer and increases the maximum mass of hadronic stars to 2.2 M , see Table I.
⊙
The derivation of the equations describing the model is given in detail in [16]. The total pressure P and mass -
energy density ρ are given by:
2 −2 2
P = P + 1 gω ρ′2 1 gσ (g σ)2 1bm (g σ)3 1c(g σ)4+ 1 gρ ρ′2, (4)
i 2 m B − 2 m σ − 3 n σ − 4 σ 2 m I3
i=B,L (cid:18) ω(cid:19) (cid:18) σ(cid:19) (cid:18) ρ(cid:19)
X
2 −2 2
ρ= ρ + 1 gω ρ′2+ 1 gσ (g σ)2+ 1bm (g σ)3+ 1c(g σ)4+ 1 gρ ρ′2. (5)
i 2 m B 2 m σ 3 n σ 4 σ 2 m I3
i=B,L (cid:18) ω(cid:19) (cid:18) σ(cid:19) (cid:18) ρ(cid:19)
X
Here P and ρ are the expressions for a Fermi gas of relativistic, non-interacting particles:
i i
1 g p2
P = i d3p (f (T)+f¯(T)), (6)
i 3(2π)3 (p2+m∗2)1/2 i i
Z i
3
g
ρ = i d3p(p2+m∗2)1/2 (f (T)+f¯(T)), (7)
i (2π)3 i i i
Z
where f (T) and f¯(T) are the Fermi - Dirac distribution functions for particles and antiparticles respectively:
i i
f (T)=(exp([(p2+m∗2)1/2 µ∗]/T)+1)−1, (8)
i i − i
f¯(T)=(exp([(p2+m∗2)1/2+µ∗]/T)+1)−1. (9)
i i i
Note that for baryons we use, instead of masses m and chemical potentials µ , “effective” masses m∗ and chemical
i i i
potentials µ∗ given by:
i
m∗ =m +x (g σ), (10)
i i σi σ
2 2
µ∗ =µ x gω ρ′ x I gρ ρ′ , (11)
i i− ωi m B − ρi 3i m I3
(cid:18) ω(cid:19) (cid:18) ρ(cid:19)
where I is the third component of the isospin of each baryon.
3i
′ ′
The weighted isospin density ρ and the weighted baryon density ρ are given by:
I3 B
′
ρ = x I n , (12)
I3 ρi 3i i
i=B
X
′
ρ = x n , (13)
B ωi i
i=B
X
being n the particle number density of each baryon:
i
g
n = i d3p(f (T) f¯(T)). (14)
i (2π)3 i − i
Z
The mean field g σ satisfies the equation:
σ
−2
g
σ (g σ)+bm (g σ)2+c(g σ)3 = x ns, (15)
m σ n σ σ σi i
(cid:18) σ(cid:19) i=B
X
where ns is the scalar density:
i
g m∗
ns = i d3p i (f (T)+f¯(T)). (16)
i (2π)3 (p2+m∗2)1/2 i i
Z i
The hadron phase is assumed to be charge neutral and in chemical equilibrium under weak interactions. Electric
charge neutrality states:
np+nΣ+ −nΣ− −nΞ− −ne =0. (17)
Chemical weak equilibrium in the presence of trapped electron neutrinos implies that the chemical potential µ of
i
each baryon in the hadron phase is given by:
µ =q µ q (µ µ ), (18)
i B n− e e− νe
whereq isitsbaryonchargeandq isitselectriccharge. Forsimplicityweareassumingthatmuonandtauneutrinos
B e
are not present in the system, and their chemical potentials are set to zero.
All the above equations can be solved numerically by specifying three thermodynamic quantities, e.g. the temper-
ature T, the mass-energy density ρ and the chemical potential of electron neutrinos in the hadronic phase µH.
νe
4
III. THE QUARK MATTER PHASE
InordertostudythejustdeconfinedquarkmatterphaseweuseanSU(3) NJLeffectivemodelwhichalsoincludes
f
color superconducting quark-quark interactions. The corresponding Lagrangianis given by
8
= ψ¯(i/∂ mˆ)ψ+G ψ¯ τa ψ 2+ ψ¯ iγ5τa ψ 2 +2H ψ¯ iγ5τAλA′ ψC ψ¯C iγ5τAλA′ ψ (19)
L −
Xa=0h(cid:0) (cid:1) (cid:0) (cid:1) i A,AX′=2,5,7(cid:2)(cid:0) (cid:1)(cid:0) (cid:1)(cid:3)
where mˆ = diag(m ,m ,m ) is the current mass matrix in flavor space. In what follows we will work in the isospin
u d s
symmetric limit m =m =m. Moreover,τ andλ with i=1,..,8 arethe Gell-Mannmatricescorrespondingto the
u d i i
flavor and color groups respectively, and τ = 2/3 1 . Finally, the charge conjugate spinors are defined as follows:
0 f
ψ =C ψ¯T and ψ¯ =ψTC, where ψ¯=ψ†γ0 is the Dirac conjugate spinor and C =iγ2γ0.
C C p
To be able to determine the relevant thermodynamical quantities we have to obtain the grand canonical thermo-
dynamical potential at finite temperature T and chemical potentials µ . Here, f =(u,d,s) and c=(r,g,b) denotes
fc
flavor and color indices respectively. For this purpose, starting from Eq. (19), we perform the usual bosonization of
thetheory. Thiscanbedonebyintroducingscalarandpseudoscalarmesonfieldsσ andπ respectively,togetherwith
a a
the bosonic diquark field ∆ . In this work we consider the quantities obtained within the mean field approximation
A
(MFA). Thus, we only keep the non-vanishing vacuum expectation values of these fields and drop the corresponding
fluctuations. For the meson fields this implies σˆ =σ τ =diag(σ ,σ ,σ ) and π =0. Concerning the diquark mean
a a u d s a
field, we will assume that in the density region of interest only the 2SC phase might be relevant. Thus, we adopt the
ansatz ∆ =∆ =0, ∆ =∆. Integrating out the quark fields and working in the framework of the Matsubara and
5 7 2
Nambu-Gorkov formalism we obtain the following MFA quark thermodynamical potential (a detailed procedure of
calculation can be found in Refs. [17–19] )
1 Λ 9 1 ∆2
ΩMFA(T,µ ,σ ,σ ,σ , ∆)= dk k2 ω(x ,y )+ (σ2 +σ2+σ2)+ | | , (20)
q fc u d s | | π2 i i 4G u d s 2H
Z0 i=1
X
where Λ is the cut-off of the model and ω(x,y) is defined by
ω(x,y)= x+T ln[1+e−(x−y)/T]+T ln[1+e−(x+y)/T] , (21)
−
h i
with
2 2
(µ µ ) (µ µ )
x =E , x =E , x = E+ ur ± dg +∆2 , x = E+ ug± dr +∆2 ,
1,2 3,4,5 s 6,7 8,9
s 2 s 2
(cid:20) (cid:21) (cid:20) (cid:21)
(µ µ ) µ µ
ur dg ug dr
y =µ , y =µ , y =µ , y =µ , y =µ , y = − , y = − . (22)
1 ub 2 db 3 sr 4 sg 5 sb 6,7 8,9
2 2
Here, E = √k2+M2 and E = k2+M2, where M = m +σ . Note that in the isospin limit we are working
s s f f f
σ =σ =σ and, thus, M =M =M.
u d u d
p
The total thermodynamical potential of the quark matter phase (QMP) is obtained by adding to Ω the
MFA
contribution of the leptons. Namely,
Ω (T,µ ,µ ,µ ,σ,σ , ∆)=ΩMFA(T,µ ,σ,σ , ∆)+Ω (T,µ )+Ω (T,µ ) Ω (23)
QMP fc e νe s | | q fc s | | e e νe νe − vac
where Ω and Ω are the thermodynamical potentials of the electrons and neutrinos, respectively. For them we use
e νe
the expression corresponding to a free gas of ultra-relativistic fermions
µ4 µ2T2 7π2T4
Ω (T,µ )= γ l + l + , (24)
l l − l 24π2 12 360
(cid:18) (cid:19)
where l = e,ν and the degeneracy factor is γ = 2 for electrons and γ = 1 for neutrinos. Notice that in Eq.(23)
e e νe
we have subtracted the constant Ω in order to have a vanishing pressure at vanishing temperature and chemical
vac
potentials.
From the grand thermodynamic potential Ω we can readily obtain the pressure P = Ω , the number
QMP QMP
−
densityofquarksofeachflavorandcolorn = ∂Ω /∂µ ,thenumberdensityofelectronsn = ∂Ω /∂µ ,
fc QMP fc e QMP e
− −
5
mu,d [Mev] ms [Mev] Λ [Mev] GΛ2 H/G
set 1 5.5 112.0 602.3 4.638 3/4
set 2 5.5 110.05 631.4 4.370 3/4
TABLE II:The two sets of NJL parameters.
and the number density of electron neutrinos n = ∂Ω /∂µ . The corresponding number densities of each
νe − QMP νe
flavor, n , and of each color, n , in the quark phase are given by n = n and n = n respectively. The
f c f c fc c f fc
baryon number density reads n = 1 n =(n +n +n )/3. Finally, the Gibbs free energy per baryon is
B 3 fc fc u d s P P
P
1
g = µ n +µ n +µ n . (25)
quark n  fc fc e e νe νe
B
fc
X
 
For the NJL model we use two sets of constants shown in Table II. The sets 1 and 2 were taken from [20] and [21]
respectively, but without the ’t Hooft flavor mixing interaction. The procedure, obtained from [22] is to keep Λ and
m fixed, then tune the remaining parameters G and m in order to reproduce M =367.6MeV and M =549.5 MeV
s s
at zero temperature and density. The resulting parameter sets are given in Table II.
In order to derive a quark matter EOS from the above formulae it is necessary to impose a suitable number of
conditions on the variables µ ,µ ,µ ,σ,σ and ∆. Three of these conditions are consequences from the fact that
{ fc} e νe s
the thermodynamically consistent solutions correspond to the stationary points of Ω with respect to σ, σ , and ∆.
s
Thus, we have
∂Ω /∂σ =0 , ∂Ω /∂σ =0 , ∂Ω /∂ ∆ =0. (26)
QMP QMP s QMP
| |
To obtain the remaining conditions one must specify the physical situation in which one is interested in. As in
previous works [4–10], we are dealing here with just deconfined quark matter that is temporarily out of chemical
equilibrium under weak interactions. The appropriate condition in this case is flavor conservation between hadronic
and deconfined quark matter. This can be written as
YH =YQ f =u,d,s,e,ν (27)
f f e
being YH nH/nH and YQ nQ/nQ the abundances of each particle in the hadron and quark phase respectively.
f ≡ f B i ≡ f B
In other words,the just deconfined quark phase must havethe same “flavor”composition than the β-stable hadronic
phase from which it has been originated. Notice that, since the hadronic phase is assumed to be electrically neutral,
flavor conservation ensures automatically the charge neutrality of the just deconfined quark phase. The conditions
given in Eq. (27) can be combined to obtain
n =ξ n , n =η n , n =κ n , 3n =2n n n , (28)
d u s u νe u e u− d− s
wheren isthe particlenumberdensityofthei-speciesinthequarkphase. Thequantitiesξ YH/YH, η YH/YH
andκ iYH/YH arefunctionsofthepressureandtemperature,andtheycharacterizethecom≡podsitionuofth≡ehasdronuic
≡ νe u
phase. These expressions are valid for any hadronic EOS. For hadronic matter containing n, p, Λ, Σ+, Σ0, Σ−, Ξ−
and Ξ0, we have
ξ = np+2nn+nΛ+nΣ0 +2nΣ− +nΞ−, (29)
2n +n +n +2n +n +n
p n Λ Σ+ Σ0 Ξ0
η = nΛ+nΣ+ +nΣ0 +nΣ− +2nΞ0 +2nΞ−, (30)
2n +n +n +2n +n +n
p n Λ Σ+ Σ0 Ξ0
nH
κ = νe . (31)
2n +n +n +2n +n +n
p n Λ Σ+ Σ0 Ξ0
Additionally, the deconfined phase must be locally colorless; thus it must be composed by an equal number of red,
green and blue quarks
n =n =n . (32)
r g b
6
Also, ur, ug, dr, and dg pairing will happen provided that ∆ is nonzero, leading to
| |
n =n , n =n . (33)
ur dg ug dr
In order to have all Fermi levels at the same value, we consider [9]
n =n , n =n . (34)
ug ur sb sr
These two equations, together with Eqs. (32) and (33) imply that n =n =n =n and n =n =n [9].
ur ug dr dg sr sg sb
Finally, including the conditions Eqs.(26) we have 13 equations involving the 14 unknowns (σ, σ , ∆, µ , µ and
s | | e νe
µ ). Forgivenvalueofoneofthe chemicalpotentials (e.g. µ ), the setofequationscanbe solvedoncethe values
fc ur
{ }
of the parameters ξ, η, κ and the temperature T are given. Instead of µ , we can provide a value of the Gibbs free
ur
energy per baryon g or the pressure P and solve simultaneously Eqs. (28)-(34) together with Eq. (26) in order
quark
to obtain σ, σ , ∆, µ , µ and µ .
s | | e νe { fc}
IV. DECONFINEMENT TRANSITION IN PROTO-NEUTRON STAR MATTER
Inorderto determinethe transitionconditions,weapplytheGibbs criteria,i.e. weassumethatdeconfinementwill
occur when the pressure and Gibbs energy per baryon are the same for both hadronic matter and quark matter at a
given common temperature. Thus, we have
gH =gQ , PH =PQ , TH =TQ , (35)
wherethe indexH refersto hadronmatterandtheindex Qto quarkmatter. Accordingtothese conditions(together
with the equations of Sections 2 and 3), for a given temperature TH and neutrino chemical potential of the trapped
neutrinos in the hadronic phase µH, there is anunique pressureP at which the deconfinement is possible. Instead of
νe
P, we may characterize the transition point by giving the Gibbs free energy per baryong, or alternatively, the mass-
energy density of the hadronic phase ρ (see Figs. 1-3). We emphasize that, according to the present description, P
H
and g are the same in both the hadronic phase and the just deconfined phase. However, the mass-energy density ρ
H
and ρ at the transitionpoint are different in general. Similarly, while the abundance Y of neutrinos is the same in
Q νe
both the hadronic and just deconfined quark phases, the chemical potentials µQ and µH are different.
νe νe
According to numerical simulations [1–3], during the first tens of seconds of evolution the protoneutron star cools
from T 40 MeV to temperatures below 2-4 MeV. In the same period, the chemical potential µH of the trapped
∼ νe
neutrinos evolves from 200 MeV to essentially zero. Thus, in order to consider typical PNS conditions we have
solvedEqs. (35)togethe∼rwith the equationsofSections2 and3fortemperatures inthe range0 60MeV andµH in
− νe
the range 0 200 MeV. The results are displayed in Figs. 1-3 for all the parameterizations of the equations of state
−
given in previous sections.
InFig. 1wedisplaytheresultsfortheGM1parametrizationofthehadronicEOS.IntheleftpanelofFig. 1weshow
theGibbsfreeenergydensityperbaryong atwhichdeconfinementoccursversusthetemperatureT forthreedifferent
values of the neutrino chemical potential in the hadronic phase (µH =0,100,200 MeV). In the right panel the same
νe
resultsareshownbutasafunctionofthemass-energydensityofthehadronicphase(inunitsofthenuclearsaturation
density ρ = 2.7 1014 g cm3). In both figures, if the thermodynamic state of hadronic matter (characterized e.g.
0
by TH, ρ and µ×H) lies to the left of the curve corresponding to the same µH, then the deconfinement transition is
H νe νe
notpossible. In the rightside regionofa givencurvethe preferredphase is deconfined quarkmatter. Notice that the
transition’s Gibbs free energy is an increasing function of µH. However, the transition density of the hadronic phase
νe
slightlydecreasesasµH increases. InFig. 2wedisplaytheresultsforthe GM4parametrizationofthehadronicEOS.
νe
The results are qualitatively the same but the transition densities are smaller than those for GM1 by 30%.
∼
InFig. 3weshowthebehaviorofthetransition’sdensityasafunctionofthechemicalpotentialoftrappedneutrinos
µH for two specific temperatures (T = 2 and 30 MeV). It is clearly seen that for a fixed temperature the effect of
νe
deleptonizationis toinhibit the transition. This effectisnotverylarge;atfixedtemperaturethere isaslightincrease
by less than a 10% when µH falls from 200 MeV to 0 MeV. On the other hand, the effect of cooling works in the
νe
opposite direction because pairing tends to help the transition and the gap increases as the temperature goes down.
The effect of cooling is also small; at fixed µH there is a slight decrease of the transition density ρ by less than a
νe H
10%when the temperature falls from30 MeV to 2 MeV. Both effects tend to cancel eachother as the PNS cools and
deleptonizes, resulting a transition density that is roughly constant along the evolution of the protoneutron star.
7
GM 1 GM 1
60 60
e = 0 MeV e = 0 MeV
50 e = 100 MeV 50 e = 100 MeV
e = 200 MeV e = 200 MeV
40 V] 40
T [MeV] 30 T [Me 30
20 20
10 10
set 2 set 1 set 2 set 1
0 0
1250 1300 1350 1400 1450 1500 1550 1600 4 5 6 7 8 9 10 11
g [MeV] h/
FIG. 1: Left panel: the Gibbs free energy density per baryon g at which deconfinement occurs versus the temperature T for
threedifferentvaluesoftheneutrinochemicalpotentialinthehadronicphase(µH =0MeV indashedline,µH =100MeVin
νe νe
dottedlineandµH =200MeVinfullline). Rightpanel: themass-energydensityofthehadronicphaseatwhichdeconfinement
νe
occurs versus the temperature T, for the same values of µH given in the left panel (density is given in units of the nuclear
νe
saturation density ρ0 ). The hadronic phase is described by the GM1 parametrization of the EOS. For the quark phase we
adopt the two parameterizations of the NJL model given in Table II. In both figures, if the thermodynamic state of hadronic
matter (characterized by { TH, gH, µHνe} or by {TH, ρH, µHνe } ) lies to the left of the curve corresponding to the same µHνe,
then the deconfinement transition is not possible. In the right side region of a given curve the preferred phase is deconfined
quark matter.
GM 4 GM 4
60 60
e = 0 MeV e = 0 MeV
50 e = 100 MeV 50 e = 100 MeV
e = 200 MeV e = 200 MeV
40 40
V] V]
e e
T [M 30 T [M 30
20 20
10 10
set 2 set 1 set 2 set 1
0 0
1250 1300 1350 1400 1450 1500 1550 1600 4 5 6 7 8 9 10 11
g [M eV] h/
FIG. 2: Same as Fig. 1 but employing theGM4 parametrization of thehadronic equation of state.
V. CONCLUSIONS
Inthispaperwehaveinvestigatedtheroleofcolorsuperconductivityinthedeconfinementtransitionfromhadronic
matter to quark matter at finite temperature and in the presence of a trapped neutrino gas. The study presented
here is relevant for the first tens of seconds of evolution of newly born protoneutron stars.
In our analysiswe used a two phase description where we employedthe Nambu-Jona-Lasiniomodel in the descrip-
tion of quark matter (Sec. III) and a non-linear Walecka model which includes the whole baryon octet, electrons
and electron neutrinos in equilibrium under weak interactions in the descriptionof hadronic matter (Sec. II). Decon-
finement is assumed to be a first order phase transition and the just deconfined quark phase is assumed to have the
same“flavorcomposition”thantheβ-stablehadronicphasefromwhichithasbeenoriginated(see[12]andreferences
therein). Whencolorsuperconductivityisincludedtogetherwithflavorconservation[9],themostlikelyconfiguration
of the just deconfined phase is 2SC provided the pairing gap is large enough. This just deconfined phase is out of
chemical equilibrium under weak interactions and thus it is very short lived but it is a a kind of “activation barrier”
8
G M1 G M4
11 11
set 1 T = 2 MeV
10 10 T = 30 MeV
9 9
set 2
0
/h0 8 /h 8
7 7
set 1
6 6
T = 2 MeV
5 T = 30 MeV 5 set 2
4 4
0 50 100 150 200 0 50 100 150 200
e ( MeV) e (MeV)
FIG. 3: The mass-energy density of thehadronic phase at which deconfinementoccurs as a function of the chemical potential
of trappedneutrinosµH. Resultsare given for twotemperatures: T =2MeV in full lineand T =30 MeV in dashed line. We
νe
employed the GM1 (left panel) and the GM4 (right panel) parametrization of the hadronic EOS. Notice that there is a small
decrease of thetransition density ρ for large µH.
h νe
that determines the onset of the deconfinement transition.
The main result of the present paper is that, within the NJL model, deconfinement is more difficult for small
neutrino content and it is easier for lower temperatures. This effect is not very large, at least for the here-used
parameterizations of the NJL model. At fixed temperature there is a slight increase by less than a 10% when µH
νe
falls from 200MeV to 0 MeV (see Fig. 3). The effect of cooling is also small; at fixed µH there is a slight decreaseof
νe
the transition density ρ by less than a 10% when the temperature falls from 30 MeV to 2 MeV (see Fig. 3). This
H
is due to the fact that the pairing gap becomes larger as the temperature decreases and therefore the increase of the
condensationtermfavorsthe transitionat low temperatures. Botheffects tend to canceleachother as the PNS cools
and deleptonizes, resulting a transition density that is roughly constant along the evolutionof the protoneutronstar.
Thehere-foundbehaviorisqualitativelyoppositetowhatitwasfoundwithintheMITbagmodel. Infact,previous
analysiswithoutincludingtheeffectofcolorsuperconductivity[4,5]showthatthepresencetrappedneutrinospushes
up the transition density to values much larger than for neutrino free matter. It was also found in [4, 5] that the
transition is easier for larger temperatures. More recent results including the effect of color superconductivity within
the MIT Bag model [23] show that the transition density increases with neutrino trapping but (in coincidence with
the here found results) the pairing gap favors the transition as the temperature decreases.
In spite of some differences between the results within the NJL and the MIT bag model description of quark
matter some general conclusions may be obtained about the effect of color superconductivity in the deconfinement
transition. First, when color superconductivity is present the deconfinement density is not so strongly affected by
neutrino trapping as it is in the unpaired case. Second, color superconductivity makes the transition easier at lower
temperatures and the dependence of the deconfinement density with T is much smaller than in the unpaired case.
Duringcoolinganddeleptonizationofthe protoneutronstarthetemperatureandthechemicalpotentialoftrapped
neutrinos fall abruptly in a few seconds and there is also some contractionof the whole neutronstar. It is interesting
to note that although the density increase is not too large, it may be comparativelyimportant for the deconfinement
transitionbecausetheeffectsoftemperatureandneutrinotrappingaresmoothedbycolorsuperconductivity. Accord-
ingtoourresultsthedeconfinementtransitionisfavoredaftersubstantialcoolingandcontractionoftheprotoneutron
starbutfullnumericalsimulationsofprotoneutronstarevolutionareneededinordertodeterminewhetherandwhen
the deconfinement conditions are attained.
VI. ACKNOWLEDGEMENTS
This workwas supported in partby CONICET (Argentina)grant# PIP6084and by ANPCyT (Argentina)grant
# PICT07 03-00818. T. A. S. do Carmo acknowledges the financial support received from UFABC (Brazil). G.
9
Lugones acknowledges the financial support received from FAPESP (Brazil).
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