Neutrino Emission from Ungapped Quark Matter

0.21 MB·English

by  Thomas Schafer

Neutrino Emission from Ungapped Quark Matter
Thomas Scha¨fer1,2 and Kai Schwenzer1
1Department of Physics, North Carolina State University, Raleigh, NC 27695
2Riken-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973
We study neutrino emission from a normal, ungapped, quark phase in the core of a compact
star. Neutrino emission from non-interacting quark matter leads to an emissivity that scales as
ǫ T7. We show that the emissivity is enhanced by a combination of Fermi liquid and non-Fermi
liq∼uideffects. Fermiliquideffectslead toan emissivity that scales asǫ α T6,as originally shown
s
∼
by Iwamoto. We demonstrate that non-Fermi liquid effects further enhance the rate, leading to
ǫ α3T6log(m/T)2,wheremistheelectricscreeningscaleandm T undertheconditionsfound
∼ s ≫
in compact stars. We show, however, that combined with non-Fermi liquid effects in the specific
heat theenhancement in theemissivity only leads toa modest reduction in thetemperature of the
star at late times. Our results confirm existing bounds on the presence of ungapped quark matter
5
in compact stars. We also discuss neutrino emission from superconducting phases with ungapped
0
fermionic excitations.
0
2

n I. INTRODUCTION ductors that have very small gaps [20, 21], and gapless
a CFL or 2SC phases [22, 23].
J
Compact stars provide a unique opportunity to study In the following we will concentrate on the neutrino
6
cold and dense strongly interacting matter and its rich emissivity of ungapped quark matter. This problem is
2
phase structure. An important source of information important both in connection with the (almost) gapless
2 about the structure of compact stars is their cooling be- phases mentioned above and as a benchmark in order
v havior [1, 2, 3, 4, 5]. For the first 105 years after to exclude the presence of ungapped quark matter in
5 the star is born neutrino emission fro∼m the bulk is the compact stars. The emissivity of a normal quark phase
9 most efficient energy loss mechanism. Since the matter was first studied by Iwamoto [10], see also [24, 25, 26].
3 in the interior of the star is almost degenerate the ex- Iwamoto showedthat the direct Urca process is strongly
0
act nature of the cooling mechanism is very sensitive to suppressed in non-interacting quark matter, but that
1
4 the structure of the low energy excitations. As a result Fermi liquid corrections in interacting matter lead to a
0 the coolingbehavior placesimportant constraintsonthe fast rate ǫ αsT6, where αs is the strong coupling con-
∼
/ phase diagram of dense matter. stant.
h
It is also known, however, that unscreened color mag-
Cooling mechanisms are generally grouped into fast
p
- processes with an emissivity ǫ T6 and slow processes neticinteractionsleadtoabreakdownoftheFermiliquid
o with ǫ T8. Slow processes in∼clude the modified Urca description at temperatures Tnfl mexp( 9π/(4αs)),
r ∼ where m2 = N α µ2/π is the el∼ectric scr−eening scale
t process [6] and neutrino bremsstrahlung. Examples for F s
s [27, 28]. This scale is very small compared to ordinary
fastmechanismsarethe directUrcaprocess[7],neutrino
a
QCDscales,but it is largecomparedto the temperature
: emission from pion or kaon condensates [8, 9], and neu-
v trino emission from an ungapped quark phase [10]. In a ofneutronstarsafter the firstminute or so. For µ=500
i
X fully gapped phase of nuclear or quark matter the emis- MeVwehaveTnfl 500keV.Itwasrecentlyshownthat
∼
r sivity is exponentially small. Near the critical tempera- non-Fermi liquid effects lead to large corrections of the
a tureforsuperconductivitytheemissivityisdominatedby specific heat of degenerate quark matter for T Tnfl
≪
pair breakingandrecombinationwitha temperature be- [29]. In the present work we compute non-Fermi liquid
corrections to the emissivity and study the cooling be-
havior that is intermediate between fast and slow mech-
anisms [11, 12]. havior of degenerate ungapped quark matter.
In this paper we wish to focus on the emissiv-
ity of a possible quark matter phase. At asymptoti-
cally high baryon density quark matter is in the color- II. QUARK DISPERSION RELATION
superconducting CFL phase [13]. In this phase all
quarkexcitationsare gappedandthe emissivity is domi- In a Fermi gas of free quarks the direct Urca process
natedbyexponentiallysmallprocessesinvolvingmassive is strongly suppressed. As a consequence, the rate is
pseudo-Goldstone modes [14, 15, 16]. At densities that very sensitive to modifications of the quark dispersion
are relevantto compactstars distortions of the CFL due relation. Iwamoto noticed that Fermi liquid corrections
to the non-zero strange quark mass cannot be neglected to the relation between the Fermi energy and the Fermi
andthephasestructureismuchmorecomplicated. Most momentumleadtoasignificantenhancementoftheneu-
of the phases that have been proposed in the literature trino emissivity [10]. Non Fermi liquid effects due to
involvequarkswithvanishingorverysmallgaps. Exam- unscreened transverse gauge boson interactions dramat-
ples are the 2SC phase [17, 18, 19] in which one of the ically alter the dispersion relation in the vicinity of the
colorsremainsungapped,singleflavorspin-onesupercon- Fermi surface [28, 29, 30, 31, 32, 33, 34] and thereby
2
v’ v’ v’ v’ v’ v’
v v v v v v
FIG. 2: Leading order contributions to the fermion self en-
v’ v’ v’ v’
ergy intheeffectivetheory. Thetriangle denotesagluon self
energy insertion and the square is the four fermion operator
definedin Fig. 1.
v v v v
is reproduced by the collinear interaction while the ex-
change term has to be matched against a contact term
FIG.1: Feynmandiagrams thatappearinthematchingpro- [33,36]. Thespin-color-flavorsymmetricpartisgivenby
cedurefortheforwardscatteringamplitudetoleadingorderin
thecouplingconstant. Theupperpanelshowsthedirect and
exchangetermsinQCD.Thelower panelshowsthecollinear C g2
and contact terms in theeffective theory. L=f0s(ψv†ψv)(ψv†′ψv′), f0s = 4NFN p2 , (3)
c f F
with C = (N2 1)/(2N ) and all other fs = 0. The
could have a similar effect. In this section we wish to F c − c i
parameters v and δµ are determined by computing
study the interplay of Fermi liquid and non-Fermi liq- F
the contribution to the fermion dispersion relation from
uid effects in dense quark matter and derive the quark
statesfarawayfromtheFermisurface. Thiscanbedone
dispersion relation.
mosteasilyusingtheharddenseloopapproximation. We
We are interested in the propagation of quarks in
find
the vicinity of the Fermi surface. Since the momentum
p vµ is large, typical soft scatterings cannot change m2 C α
the∼momentumbyverymuchandthe velocityisapprox- p0 =|p~|+ p~f, m2f = 2Fπs µ2, (4)
imately conserved. An effective field theory for particles | |
and holes moving with velocity v = (1,~vF) is given by which gives
[33, 35]
C α C α
F s F s
1 v =1 , δµ= µ. (5)
= ψ†(iv D+δµ)ψ Ga Ga + F − 2π π
L v · v− 4 µν µν LHDL
+ flΓRl(v·v′)(ψv†Γψv)(ψv†′Γψv′)+.... (1) TorhyesoefrFeelramtioinlisqcuaidnsa.lsLoabnedaduersihveodweudsinthgattheGLalainlediaiunvtahrei--
Xv′,Γ
ance implies a relation between the effective interaction
Here, Dµ is the covariant derivative and δµ is a coun- onthe Fermisurface,encodedin the Landauparameters
terterm for the chemical potential. The effective theory fs, and the parameters v and δµ. These arguments
l F
containstwotypesoffourfermionoperators,correspond- were generalized to the relativistic case by Baym and
ing to forward scattering v+v′ v+v′ and the BCS Chin [37]. They show that
processv+( v) v′+( v′)[33].→Sinceweareinterested
in the norm−al ph→ase onl−y the forward scattering opera- v = pF Nf1s, p =µ 1 Nf0s , (6)
tors are included in equ. (1). The matrix Γ determines F µ − 3 F (cid:18) − 2 (cid:19)
the spin, color and flavor structure of the operator and
Rl(v v′)isasetoforthogonalpolynomials. Equ.(1)also where N = NcNfµ2/π2 is the density of states on the
·
contains the hard dense loop effective action Fermi surface. Using equ. (3) and (6) gives equ. (5).
Finally, we can study loop corrections in the effective
m2 vαvβ
= Ga Ga , (2) theory, see Fig. 2. The collinear loop gives
LHDL − 2 µα(v D)2 µβ
Xv · C α Λ
F s
Σ(ω)= ωlog , (7)
which accounts for damping and screening caused by 3π (cid:18)ω(cid:19)
particle-hole pairs on the entire Fermi surface.
The parameters in the effective lagrangian are deter- where ω = p µ. We recently showed that this re-
0
−
minedbymatchingQCDGreenfunctionsneartheFermi sult does not receive large logarithmic corrections of the
surface. In order to determine the dispersionrelationwe form g2nlogn(ω) as ω 0 [34]. The tadpole diagram is
→
have to match forwardscattering amplitudes, see Fig. 1. linearly divergent in the effective field theory. A naive
To leading order in the coupling constant the forward estimate can be obtained by cutting the divergence off
scattering amplitude is the sum of a direct and an ex- at Λ µ. This gives Σ f0Nµ which agrees with the
∼ ∼ s
changeterm. Inthe effective fieldtheory the directterm result for δµ given above. We can now summarize the
3
electron capture (ec)
d
EF
u d u+e−+ν¯ , (9)
EF → e
u+e− d+ν . (10)
e
→
e
EF
It is straightforward to introduce weak interactions into
e u d
pF pF pF the effective theory. The charged current interaction is
d given by
EF
u
EF l = g2 cosθ ψ†τ±v W∓ψ (11)
c
L √2 ·
e
EF where cosθc is the Cabbibo angle and g2 is related to
the Fermi coupling by G /√2 = g2/(8M2 ). The de-
e u d F 2 W
pF pF pF pendence on the Cabbibo angle suppresses the processes
involving the strange quark [10]. Therefore we will ne-
glect the neutrino emission of strange quarks. We have
FIG. 3: Kinematics for the quark direct Urca process in the
seen in the previous section that almost collinear gluon
limit T E µ . Inafreequarkgas (upperpanel)energy
∼ ν ≪ e exchanges can generate large logarithmic corrections to
momentum conservation forces the quark and electron mo-
the fermionselfenergy. This raisesthe questionwhether
menta to be collinear. If Fermi liquid corrections are taken
gluon corrections to the weak interaction vertex have to
intoaccount(lowerpanel)theoutgoingquarkhasanon-zero
transverse momentum l⊥2 ∼αsµ2e. The dashed dispersion re- be taken into account. The problem of vertex correc-
lations give the HDL result whereas the solid lines show the tions was studied in [31, 33]. It was shown that the ver-
changewhennon-Fermiliquidcorrectionsareincluded. These tex receiveslargelogarithmiccorrectionsinthe time-like
lead to a flattening of the dispersion relation in the vicinity regime, but not in the space-like regime. The coefficient
of the Fermi surface. of the logarithm in the time-like regime is exactly equal
to the logarithmic term in the fermion self energy. In
the fermion self energy problem the vertex is space-like
results obtained in this section. The dispersion relation
and does not have to be renormalized. However, since
is
the wave function renormalization is logarithmically di-
vergent the effective coupling constant goes to zero near
ω+Σ(ω)=v l+δµ, (8)
F the Fermi surface. In the neutrino emission problem the
vertex is time-like and receives logarithmic corrections.
where l is the longitudinal momentum, Σ(ω) is given in
However,since the vertex correction is equal to the field
equ. (7) and v ,δµ given in equ. (5).
F
renormalization the effective coupling constant does not
change.
III. NEUTRINO EMISSIVITY The neutrino emissivity is given by the total energy
loss due to neutrino emission averaged over the initial
The dominant contribution to the emission of neutri- quark helicities and summed over the final state phase
nos is given by the quark analogs of β-decay (β) and space and helicities
d3p 1 d3p 1 d3p 1 d3p 1
d u e ν
ǫ N E (12)
≡ c Z (2π)32E Z (2π)32E Z (2π)32E Z (2π)32E ν
σuX,σd,σe d u e ν
M 2(2π)4δ(4)(p p p p )n(p )(1 n(p ))(1 n(p ))
β d u e ν d u e
·(cid:16)| | − − − − −
+M 2(2π)4δ(4)(p +p p p )n(p )n(p )(1 n(p ))
ec u e d ν u e d
| | − − − (cid:17)
The weak matrix element for the β and ec processes is given by
1 M 2 =64G2 cos2θ p2(v p )(v p ), (13)
2 | β/ec| F c F · e · ν
σuX,σd,σe
4
where p ,p are the momenta of the electron and the effective theory the weak decay is exactly collinear and
e ν
neutrino. Weak processes establish β equilibrium in the (v p )=(E v l )=0 up to terms of order O(T/µ ),
e e F e e
· −
star. Inthreeflavorquarkmatterwithamassivestrange see Fig. 3. If corrections to the dispersion relation are
quark the resulting electron chemical potential is small. taken into account we get
In the following we shall assume that (T E ) (µ
ν e
∼ ≪ ∼
E ) p . This assumption is appropriate in all cases
execep≪t duFring the first few seconds of the proto-neutron (v p ) (1 v )l + l⊥2 δµ δµ CFαsµ , (14)
e F e d u e
· ≃ − 2l ≃ − ≃ π
star evolution. e
In this case we can neglect the neutrino momentum
when applying the energy-momentum conservation rela- where l is the transverse momentum of the electron.
⊥
tion to the matrix element. As a consequence we find This result agreeswith Iwamoto’s result. Non-Fermi liq-
(v p ) E after averaging over the direction of the uid corrections only appear in the phase space integral.
ν ν
· ≃
outgoing neutrino. The matrix element is mainly deter- To leading order in the T/µ the sum of the rates for
mined by the factor (v p ). To leading order in the electron capture and β decay is given by
e
·
6G2 cos2θ ∞ ∞ ∞
ǫ= F cT6 dx dx dx x3n(x )n( x )n(x x +x )
π5 Z dZ uZ ν ν d − u u− d ν
−∞ −∞ 0
∂p(E )∂p(E )C α
d u F s
µ E E . (15)
e u d
·(cid:20) ∂E ∂E π (cid:21)
d u Ei→µi+Txi
The expression in the square brackets is determined by non-Fermi liquid corrections. The thermal evolution of
the quark dispersion relation given in equ. (8). Terms the star is governed by the neutrino emissivity, the spe-
of O(α (α log(T))n) with n=0,1,2 are independent of cificheatandthethermalconductivity. Non-Fermiliquid
s s
log(x ) and involve the integral corrections to the specific heat were initially considered
i
by Holstein et al. [27] in the case of QED. The calcula-
∞ ∞ ∞
dx dx dx x3n(x )n( x ) tionwasrecentlyrefinedandextendedtoQCDbyIppet
Z dZ uZ ν ν d − u al. [29]. They find
−∞ −∞ 0
457π6
n(x x +x )= . (16) N N C α Λ
· u− d ν 5040 cv = c3 fµ2qT (cid:18)1+ F3π s log(cid:18)T(cid:19)(cid:19), (18)
AtleadingorderinT/µthe neutrinoemissivityfromthe
quark direct Urca process is given by where the first term is the free gas result and the second
term is the non-Fermi liquid correction. Ipp et al. also
457 C α Λ 2 determinedthe scaleinsidethe logarithmaswellasfrac-
ǫ G2 cos2θ α µ2µ T6 1+ F s log .
≈ 630 F c s q e (cid:18) 3π (cid:18)T(cid:19)(cid:19) tional powers of T. From the complete O(αs) result we
findΛ 0.28mwherem2 =N α µ2/π is the dynamical
(17) f s
≃
mass.
ThefirsttermisthestandardresultbyIwamoto[10],and
The thermal conductivity of degenerate quark matter
the logarithmic terms are non-Fermi liquid corrections.
was studied by Pethick and Heiselberg [38]. Their re-
We note thatthese termshaveto be includedbecause at
sultsuggeststhatequilibrationisfastandthatthequark
very low temperature α log(T)becomes large compared
s
phase is isothermal. In this case the cooling behavior is
to one. We also note that if the scale inside the loga-
governedby
rithmisontheorderofthescreeningscale,Λ gµ,then
∼
α (µ)log(Λ/T) stays finite in the limit µ at fixed
s → ∞ ∂u ∂u ∂T ∂T
T. = =c (T) = ǫ(T), (19)
v
∂t ∂T ∂t ∂t −
where u is the internal energy, t is time and we have
IV. COMPACT STAR COOLING assumedthatthereisnosurfaceemission. Without non-
Fermi liquid effects we have ǫ T6 and c T. In this
v
∼ 1 ∼
In this section we wish to study the impact of non- case the temperature scales as T 1/t4. With loga-
∝
Fermi liquid effects on the cooling history of an isolated rithmic corrections included there is no simple analytic
quarkphase. Ouraim is notto providea thoroughanal- solution and we have studied equ. (19) numerically.
ysis of the cooling behavior of an actual quark or hybrid We take the quark chemical potential to be µ =500
q
star, but to give a numerical estimate of the size of the MeV corresponding to densities ρ 6ρ where ρ is
B 0 0
≈
5
1.´1031
c erg
v 109Kcm3
(cid:2) (cid:3)
1.´1027
1.´1023
1.´1019
ǫ erg
ycm3
h i
1.´1015
0.001 0.005 0.01 0.05 0.1 0.5 1
T9
FIG. 4: Neutrino emissivity ǫ and specific heat c of quark matter. The dashed line shows the Fermi liquid results and the
v
dotted lines show the anomalous corrections. The solid line gives the sum of the two contributions and the gray band shows
an estimate of theuncertainties.
nuclear matter saturation density. We note that both c the electron chemical potential is given by
v
and ǫ are proportional to µ2 and the main dependence
of the cooling behavior on µ cancels. We evaluate the µ m2s 1 4αs log 2pF . (21)
e
strong coupling constant using the one loop renormal- ≃ 4pF (cid:18) − π (cid:18) ms (cid:19)(cid:19)
ization group solution at a scale µ. We take the scale
For the values of the parameters given above this equa-
parameter to be Λ =250 MeV which gives α 1
QCD s
≃ tiongivesanegativeelectronchemicalpotentialµ 16
at µ = 500 MeV. It is clear that the naive use of per- e≈−
MeV. In this case the quark phase contains a Fermi sea
turbation theory is in doubt if the coupling is this large.
of positrons and the quark direct Urca process is
In practice we estimate the uncertainty by varying α
s
between 1 and 0.4 which is the value used by Iwamoto
u d+e++ν , d+e+ u+ν¯. (22)
[10]. We take the scale of non-Fermi liquid effects to be → →
Λ=0.28masexplainedaboveandassesstheuncertainty The neutrino emissivity is again governed by equ. (17)
by varying Λ within a factor of 2. Finally, we took the where µ has to be replaced by µ . We observe that
e e
initial temperature to be T0=15 MeV. despite the large correction to µ−e the emissivity is not
The electron chemical potential is determined by the stronglyaffected. Thelargevariationinµ whenpertur-
e
requirementsofchargeneutralityandβ-equilibrium. Ina bativecorrectionsareincludedimplies,however,thatthe
non-interacting quark gas we find µ m2/(4p ). With electron chemical potential is very uncertain. In partic-
e ≃ s F
a strange quark mass m =150 MeV this relation gives ular, there is a possibility that µ is much smaller than
s e
µ 11 MeV. This result, however, is very sensitive to m2/(4p ). If α µ < T then the neutrino emissivity is
e≈ s F s e
interactions. To first order in α the chemical potential no longer proportional to α µ T6 but to T7 [25].
s s e
for a massive strange quark is [39, 40] Inthefollowingweshallusethevalueµ =16MeVcor-
e
responding to interacting quarks. In Fig. 4 we show the
2α 3m2 p +E0 temperature dependence of both the neutrino emissivity
µs =EF0s+ 3πs (cid:18)pFs− E0s log(cid:18) Fsm Fs(cid:19)(cid:19) (20) andthespecificheat(solid)comparedtotheFermiliquid
Fs s result(dashed). The graybandshowsanestimate ofthe
uncertainties which are dominated by the uncertainty in
where EF0s = p2Fs+m2s. The important point is that the value of the strong coupling. For both cv and ǫ the
the O(αsm2s) tperm is negative and enhanced by a large anomalous logarithmic terms (dotted) dominate in the
logarithm log(pF/ms). The sign is related to the fact relevant temperature range and exceed the Fermi liquid
that the correlation energy changes sign in going from result considerably. The cooling behavior is controlled
the relativistic to the non-relativistic limit. bytheratioǫ/c . Since ǫ log2(T)andc log(T)this
v v
∼ ∼
Equation(20) implies that the strangequarkchemical ratio is logarithmically enhanced. However, because the
potential can become equal to or even smaller than the temperature at late times scales roughly as the fourth
up quark chemical potential. To leading order in m2/p2 root of the numerical coefficient in ǫ/c this logarithmic
s F v
6
1
0.5
0.2
0.1
T9
0.05
0.02
0.01
1 10 100 1000 10000 100000.
t [y]
FIG. 5: Cooling behavior of ungapped quark matter. We show the temperature T9 in units of 109 K as a function of the age
of the star in years. The dashed line shows the Fermi liquid result whereas the solid line gives the result including non-Fermi
liquid effects with the estimated uncertainty range. Although the non-Fermi liquid corrections to both the specific heat and
the neutrino emissivity are significant, there is only a modest reduction in the temperature at late times. However, for both
casesthecoolingofquarkmatterisconsiderably fasterthanthecoolingofneutronmatterviathemodifiedUrcaprocessgiven
by thedotted line.
enhancement only translates into a modest reduction of phase emissivity specific heat
the temperature. This can be seen in more detail in normal phase ǫ α3T6log2(T) c α µ2Tlog(T)
Fig. 5. We observe that compared to the Fermi liquid ∼ s v ∼ s
CFL ǫ T3/2exp( m /T) c T3
result (dashed) the non-Fermi liquid effects (solid) lead ∼ − K v ∼
2SC ǫ α T6 c µ2T
to a reduction of the temperature at late times which is ∼ s v ∼
nearly independent of time. The magnitude of the effect 2SC+1SC ǫ αsT6 cv µ2T
∼ ∼
is on the order of 20%. For comparison, we also show [1SC]3 ǫ α T6 c µ2T
s v
∼ ∼
the cooling behavior of normal nuclear matter via the gCFL ǫ α T5.5 c µ2√∆T
s v
modified Urca process n+n n+p+e−+ν¯ [6]. We gCFLK ǫ∼ α T8 c∼ µ2T
havechosenthesamedensitya→ndinitialtemperatureand ∼ s v ∼
g2SC ǫ α T6 c µ2T
s v
the effective baryon masses given in [5]. We clearly see ∼ ∼
the difference between the fast T6 quark direct Urca
process and the slow T8 modi∼fied Urca process. TABLEI:Dominantcoolingratesindifferentphasesofquark
∼ matter. Thistableisasummaryofthediscussion inSect.V.
In the entry [1SC]3 we have assumed that ungapped modes
exist and the Urcaprocess is allowed. This is not truein the
V. SUMMARY AND DISCUSSION CSL phase. In the gCFL phase we have used the estimate
of Kouvaris et al. and in thegCFLK phase we have assumed
thatonlyslowprocesses suchasneutrinobremsstrahlungare
In this work, we have discussed the influence of non-
allowed. In the g2SC phase we haveassumed that the Fermi
Fermi liquid effects on the cooling behavior of compact
liquid direct Urca rate can be used. In both the gCFL and
starscontainingquarkmatterinthenormalphase. Non- g2SCphasewehaveignored possibleinstabilities signaled by
Fermiliquideffectsleadtoalogarithmicenhancementin a negative current-currentcorrelation function.
both the neutrino emissivity and the specific heat. The
net result of these two effects is a mild logarithmic en-
hancementinthe coolingrate. As ourrate is evenlarger
than the Iwamoto rate we confirm and sharpen earlier Letusnowdiscussthe importanceofnon-Fermiliquid
bounds on the existence of ungapped quark matter in effects in partially gappedcolorsuperconducting phases,
neutron stars [41, 42, 43, 44]. More quantitative state- see Table I. The simplest case is the 2SC phase. The
ments will require detailed studies of realistic models in 2SC phase can arise when the difference between the
whichthe quarkcoreis incontactwitha hadronicphase Fermi momenta of the strange quark and the up and
oranatmosphere. This isbeyondthe scopeofourinves- down quarks is too large for strange-non-strangepairing
tigation. to occur [45, 46]. There also regions in the phase dia-
7
gram where us (dSC) or ds (uSC) pairing might occur that will contribute to neutrino transport at MeV tem-
[47, 48, 49]. The 2SC phase is characterized by a par- peratures. In the g2SC phase there are gapless up and
tial Higgs mechanism. Color SU(3) is broken according down quark excitations and the direct Urca process is
to SU(3) SU(2) and five out of eight gluons acquire possible.
→
a mass. The gapless fermions of the third color inter- We should stress that our discussion of neutrino emis-
act via screened gauge bosons. As a consequence, there sion from the g2SC and gCFL phase ignores potential
are no non-Fermi liquid corrections in either the specific instabilities related to a negative current-current corre-
heat or the neutrino emissivity and the standard result lationfunction [52,53, 54]. Wuand Yipas wellas Gian-
of Iwamoto applies [10]. nakisandRen[52,55]havearguedthattheseinstabilities
If the 2SC phase occurs in quark matter with three indicate that the correct ground state is an inhomoge-
flavors then the unpaired quark flavor can form a spin neous superconductor of the type discussed by Larkin,
one condensate. Spin one condensates are interesting Ovchninkov, Fulde and Ferrell [56, 57, 58]. The LOFF
fromthepointofviewofcoolingbecausethetypicalgaps phase also contains gapless fermions and the calculation
are much smaller than the spin zero gap. In particular, of the neutrino emissivity from this phase in an impor-
we can have Tc Tnfl and non-Fermi liquid effects in tant problem for future studies.
≃
the specific heat are important. Whether or not gapless Note added in proof:
modes occur below Tc depends on the exact nature of Since the first version of this work the scale in the log-
the spin one condensate [20, 21]. In a phase with CSL arithm of equ. (7) has been computed by Gerhold and
pairing in both LL and LR chirality channels the order Rebhan [59] and the result Λ 8m turned out to be
Σ
parameter is isotropic and no gapless modes exist. In all substantially larger than the cor≈responding scale in the
other phases not all fermions are gapped and the order specific heat Λ 0.28m we have employed as an esti-
c
parameterhasnodesonFermisurface. Ifspinonepairing mate for the scale≈in the emissivity. This leads quantita-
takes place in all flavor sectors (referred to as [1SC]3 in tivelytoafurtherenhancementoftheneutrinoemissivity
TableI)these modeswilldominatethecoolingbehavior. but does not change the cooling behavior significantly.
The cooling behavior of the recently proposed gapless
CFL[22]and2SC[23]phasesisamoredifficultquestion.
Inthe gCFLphasetherearegaplesschargedandneutral
Acknowledgments
modes and the direct Urca process is possible. Alford
et al. [50] have argued that the (almost) quadratic dis-
persion relation of one of the gapless modes leads to an We would like to thank C. Kouvaris, M. Prakash,
enhancement of the cooling rate by a factor √T. A sim- K. Rajagopal, and S. Reddy for useful discussions and
ilar enhancement occurs in the specific heat [22]. Kry- Q. Wang for pointing out an error in equ. (12) of the
jevski and Scha¨fer studied quark modes in a kaon con- first versionof this manuscript. This work was begun at
densed CFL phase [51]. They find at most one gapless the Institute forNuclearTheoryduringthe workshopon
mode. As a consequence, the dominant cooling mecha- “QCD and Dense Matter: From Lattices to Stars”. We
nism is expected to be a slow process such as neutrino thank the INT for hospitality. This work was supported
bremsstrahlung. However, there are several light modes in part by US DOE grant DE-FG-88ER40388.
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