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arXiv:hepph/0601179v1 20 Jan 2006
Dense Quark Matter in a Magnetic Field
Efrain J. Ferrer, Vivian de la Incera∗
Western Illinois University, USA
Email: ejferrer@wiu.edu, vincera@wiu.edu
Cristina Manuel
Instituto de Fisica Corpuscular (CSICU. de Valencia), Spain
Email: Cristina.Manuel@ific.uv.es
We explore the effects of an applied strong external magnetic field in the structure and magni
tude of the color superconducting diquark condensate of a three massless flavor theory . The
longrangecomponentof the B field that penetrates the superconductorenhances the condensates
formed by quarks charged with respect to this electromagnetic field.
29th Johns Hopkins Workshop in Theoretical Physics: Strong Matter in the Heavens. Budapest, August 13,
2005
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons AttributionNonCommercialShareAlikeLicence.
http://pos.sissa.it/
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Dense Quark Matter in a Magnetic Field
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1. Introduction
It is well established that at high baryon density the combination of asymptotic freedom and
the existence of attractive channels in the color interaction between the quarks lying in the large
Fermi surface come together to promote the formation of quarkquark pairs, which in turn break the
color gauge symmetry giving rise to the phenomenon of color superconductivity. Atdensities much
higher than the masses of the u, d, and s quarks, one can assume the three quarks as massless and
the favored state results to be the socalled ColorFlavorLocking (CFL) phase [1], characterized
by a spin zero diquark condensate antisymmetric in both color and flavor.
The conditions of extremely large density and very low temperature required for color super
conductivity cannot be recreated inEarth’s labs. Fortunately, nature provides us with alaboratory to
probe color superconductivity, the cores of celestial compact objects. These compact stars typically
have very large magnetic fields. Neutron stars can have magnetic fields as large as B ∼ 1012−1014
G in their surfaces, while in magnetars they are in the range B∼ 1014−1015G, and perhaps as high
as 1016G [2] (for a recent review of magnetic fields in dense stars see [3]). Even though we do not
know yet of any suitable mechanism to produce more intense fields, the virial theorem [4] allows
the field magnitude to reach values as large as 1018−1019G. If quark stars are selfbound rather
than gravitationalbound objects, the upper limit that has been obtained by comparing the magnetic
and gravitational energies, could go even higher.
A natural question to ask is: What is the effect, if any, of the huge star’s magnetic field in the
color superconducting core? A complete answer to this question would require a rather involved
study of quark matter at the intermediate range of densities proper of neutron stars, where the
strange quark mass cannot be ignored, with the additional complication of an extra parameter, the
magnetic field. However, as a first, more tractable approach to this question, one can ignore the
strange quark mass effects and look for the consequences of an external magnetic field on the
superconducting phase, assuming that the quark matter is formed by three massless flavors. This
was the strategy followed in our recent paper [5], whose main results will be described in what
follows.
In this talk I will show the way a magnetic field affects the pairing structure and hence its
symmetry, ultimately producing a different superconducting phase that we have called Magnetic
ColorFlavorLocking (MCFL) phase.
In a conventional superconductor, since Cooper pairs are electrically charged, the electromag
netic gauge invariance is spontaneously broken, thus the photon acquires a Meissner mass that can
screen a weak magnetic field, the phenomenon of Meissner effect. In spinzero color superconduc
tivity, although the color condensate has nonzero electric charge, there is a linear combination of
the photon and a gluon that remains massless [1]. This new field plays the role of the "inmedium"
photon in the color superconductor, so the propagation of light in the color superconductor is dif
ferent from that in an electric superconductor.
Because of the longrange "rotated" electromagnetic field, a spinzero color superconductor
may be penetrated by a rotated magnetic field? B. Although a few works [6] had previously ad
of these studies considered the modification produced by the field on the gap itself. However, as
we have recently shown [5], the gap structure gets modified due to the penetrating field. To un
dressed the problem of the interaction of an external magnetic field with dense quark matter, none
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Dense Quark Matter in a Magnetic Field
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derstand this, notice that, although the condensate is? Qneutral, some of the quarks participating
the gap equations through the Green functions of these? Qcharged quarks. Due to the coupling
colorflavor operator, and consequently the CFL order parameter splits in new independent pieces
giving rise to a new phase, the MCFL phase.
in the pairing are?Qcharged and hence can couple to the background field, which in turn affects
of the charged quarks with the external field, the colorflavor space is augmented by the?Qcharge
2. The MCFL Gap Structure
The linear combination of the photon Aµand a gluon G8
the spinzero color superconductor is given by [1, 7],
µthat behaves as a longrange field in
?Aµ= cosθAµ−sinθG8
µ,
(2.1)
while the orthogonal combination?G8
superconductor is mostly formed by the photon with only a small gluon admixture.
The unbroken U(1) group corresponding to the longrange rotated photon (i.e. the? U(1)e.m.)
our flavorspace ordering is (s,d,u). In the 9dimensional flavorcolor representation that we will
use in this paper (the color indexes we are using are (1,2,3)=(b,g,r)), the? Q charges of the different
s1
s2
s3
d1
d2
0000
µ= sinθAµ+cosθG8
µis massive. In the CFL phase the mix
ing angle θ is sufficiently small (sinθ ∼ e/g ∼ 1/40). Thus, the penetrating field in the color
is generated, in flavorcolor space, by?Q = Q×1−1×Q, where Q is the electromagnetic charge
generator. We use the conventions Q = −λ8/√3, where λ8is the 8th GellMann matrix. Thus
quarks, in units of ? e = ecosθ, are
d3

u1
+
u2
+
u3
0
(2.2)
In the presence of an external rotated magnetic field the kinetic part of the quarks’ Lagrangian
density must be rewritten using the covariant derivative
Lem
quarks= ψ(iΠµγµ)ψ ,
(2.3)
with
Πµ= i∂µ+? e? Q?Aµ.
?Q = Ω+−Ω−.
(2.4)
where
(2.5)
is the rotated charge operator. The charge projectors
Ω+= diag(0,0,0,0,0,0,1,1,0) ,
(2.6)
Ω−= diag(0,0,1,0,0,1,0,0,0) ,
(2.7)
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and
Ω0= diag(1,1,0,1,1,0,0,0,1) ,
(2.8)
obey the algebra
ΩηΩη′ = δηη′Ωη,
η,η′= 0,+,− .
(2.9)
Ω0+Ω++Ω−= 1 .
(2.10)
The rotated magnetic field naturally separates the quark fields according to their˜Q charge.
The fermion field in the 9×9 representation used above, ψT= (s1,s2,s3,d1,d2,d3,u1,u2,u3), can
then be written as the sum of three fields with zero, positive and negative rotated electromagnetic
charges,
ψ = ψ(0)+ψ(+)+ψ(−),
(2.11)
where the (0), (+/−)charged fields can be respectively written in terms of the charge projectors
as
ψ(0)= Ω0ψ ,
ψ(+)= Ω+ψ ,
ψ(−)= Ω−ψ .
(2.12)
A strong magnetic field affects the flavor symmetries of QCD, as different quark flavors have
different electromagnetic charges. For three light quark flavors, only the subgroup of SU(3)L×
SU(3)Rthat commutes with Q, the electromagnetic generator, is a symmetry of the theory. Simi
larly, in the CFL phase a strong? B field should affect the symmetries of the theory, as? Q does not
that the condensate should retain the highest degree of symmetry, we proposed [5] the following
ansatz for the gap structure in the presence of a magnetic field
commute with the whole locked SU(3) group. Based on the above considerations, and imposing
∆ =
2∆
0
0
0
′
S
0
0
0
0
0
0
0
0
0
0
∆A+∆S
0
0
0
2∆
0
0
0
∆B
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∆B
A+∆B
0
0
0
∆B
0
0
0
2∆
S
∆S−∆A
0
0
0
0
0
0
0
∆B
S−∆B
0
0
0
0
0
0
A
∆S−∆A
0
0
0
0
0
∆A+∆S
0
0
0
∆B
′
SA+∆B
S
∆B
S−∆B
0
0
0
A
∆B
S−∆B
0
0
A
∆B
S−∆B
0
A
A+∆B
S
A+∆B
S
′′
S
(2.13)
We call the reader’s attention to the fact that despite the?Qneutrality of all the condensates,
quarks feel the field directly through the minimal coupling of the background field? B with the
of the field via treelevel vertices that couple them to charged quarks. The gaps ∆B
symmetric/symmetric combinations of condensates composed by charged quarks and condensates
they can be composed either by neutral or by charged quarks. Condensates formed by? Qcharged
quarks in the pair. A subset of the condensates formed by? Qneutral quarks, can feel the presence
A/Sare anti
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formed by this kind of neutral quarks. The gaps ∆A, as well as ∆S, ∆
antisymmetric and symmetric combinations of condensates formed by neutral quarks that do not
belong to the above subset. The only way the field can affect them is through the system of highly
nonlinear coupled gap equations. At zero field the CFL gap matrix is recovered since in that case
∆B
Although the symmetry of the problem allows for four independent symmetric gaps, the con
densates ∆
the previous paragraph, they are formed by neutral quarks that are not coupled to charged quarks,
so they belong to the same class as ∆S. Therefore, there is no reason to expect that they will differ
much from ∆S. Hence, in a first approach to the problem, we will consider ∆S≃ ∆
The order parameter (2.13) implies the following symmetry breaking pattern: SU(3)color×
SU(2)L× SU(2)R×U(1)B×U(−)(1)A×U(1)e.m.→ SU(2)color+L+R×? U(1)e.m.. The U(−)(1)A
axial currents [8]. The locked SU(2) corresponds to the maximal unbroken symmetry, and as such
it maximizes the condensation energy. Notice that it commutes with the rotated electromagnetic
group? U(1)e.m..
tells us that there are only five NambuGoldstone bosons. One is associated to the breaking of the
baryon symmetry; three Goldstone bosons are associated to the breaking of SU(2)A, and another
one associated to the breaking of U(−)(1)A. All the NambuGoldstone bosons are? Qneutral. The
physics of the phase. Since in her talk Cristina Manuel will address the lowenergy physics of the
MCFL phase, I will not extend on this topic in mine.
′
Sand ∆
′′
S, on the other hand, are
A= ∆Aand ∆B
S= ∆S= ∆
′
S= ∆
′′
S.
′
Sand ∆
′′
Sare only due to subleading color symmetric interactions, and as explained in
′
S≃ ∆
′′
S.
symmetry is connected with the current which is an anomalyfree linear combination of s,d and u
The counting of broken generators, after taking into account the AndersonHiggs mechanism,
number and properties of the lightest particles in the MCFL have implications for the lowenergy
3. Effective Action in the Presence of a Magnetic Field
Let us construct the effective action of the system in the presence of a magnetic field. With this
aim, we will use a NambuJonaLasinio (NJL) fourfermion interaction abstracted from onegluon
exchange [1]. Although this simplified treatment disregards the effect of the? Bfield on the gluon
field, it keeps the main attributes of the theory, thereby providing the correct qualitative physics.
We start from the meanfield effective action
?
x,y
+ψ(−)(x)[G+
+ψ(+)C(x)[G−
+1
dynamics and assumes the same NJL couplings for both the situation with and without magnetic
IB(ψ,ψ) =
{1
2[ψ(0)(x)[G+
(0)0]−1(x,y)ψ(0)(y)+ψ(+)(x)[G+
(+)0]−1(x,y)ψ(+)(y)
(−)0]−1(x,y)ψ(−)(y)+ψ(0)C(x)[G−
(+)0]−1(x,y)ψ(+)C(y)+ψ(−)C(x)[G−
2[ψ(0)C(x)∆+(x,y)ψ(0)(y)+h.c.]+1
(0)0]−1(x,y)ψ(0)C(y)
(−)0]−1(x,y)ψ(−)C(y)]
2[ψ(+)C(x)∆+(x,y)ψ(−)(y)
+ψ(−)C(x)∆+(x,y)ψ(+)(y)+h.c.]} ,
(3.1)
where the external magnetic field has been explicitly introduced through minimal coupling with the
? Q−charged fermions. The presence of the field is also taken into account in the diquark condensate
∆+= γ5∆, whose colorflavor structure is given by Eq.(2.13).
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In (3.1) symbols in parentheses indicate neutral (0), positive (+) or negative (−)˜Q−charged
quarks. Supraindexes + or − in the propagators indicate, as it is customary, whether it is the
inverse propagator of a field or conjugated field respectively. Then, for example, [G−
sponds to the bare inverse propagator of positively charged conjugate fields, and so on. The explicit
expressions of the inverse propagators are
(+)0]−1corre
[G±
(0)0]−1(x,y) = [iγµ∂µ−m±µγ0]δ4(x−y) ,
(3.2)
[G±
(+)0]−1(x,y) = [iγµΠ(+)
µ −m±µγ0]δ4(x−y) ,
(3.3)
[G±
(−)0]−1(x,y) = [iγµΠ(−)
µ −m±µγ0]δ4(x−y) ,
(3.4)
with
Π(±)
µ
= i∂µ±? e?Aµ.
(3.5)
Transforming the fielddependent quark propagators to momentum space can be performed
with the use of the Ritus’ method, originally developed for charged fermions [9] and recently
extended to charged vector fields [10]. In Ritus’ approach the diagonalization in momentum space
of charged fermion Green’s functions in the presence of a background magnetic field is carried out
using the eigenfunction matrices Ep(x). These are the wave functions of the asymptotic states of
charged fermions in a uniform magnetic field and play the role in the magnetized medium of the
usual planewave (Fourier) functions eipxat zero field. Below we introduce the basic properties of
this transformation.
The transformation functions E(±)
fields are obtained as the solutions of the field dependent eigenvalue equation
q (x) for positively (+), and negatively (−) charged fermion
(Π(±)·γ)E(±)
q (x) = E(±)
q (x)(γ · p(±)) ,
(3.6)
with p(±)given by
p(±)= (p0,0,±
?
2? e? Bk,p3) ,
E(±)
(3.7)
and
E(±)
q (x) =∑
σ
qσ(x)δ(σ) ,
(3.8)
with eigenfunctions
E(±)
pσ(x) = Nn(±)e−i(p0x0+p2x2+p3x3)Dn(±)(ρ(±)) ,
(3.9)
where Dn(±)(ρ(±)) are the parabolic cylinder functions with argument ρ(±)defined by
?
ρ(±)=
2? e? B(x1± p2/? e? B) ,
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(3.10)
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Dense Quark Matter in a Magnetic Field
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and index n(±)given by
n(±)≡ n(±)(k,σ) = k±
? e? B
2? e? Bσ −1
2,
n(±)= 0,1,2,...
(3.11)
k = 0,1,2,3,... is the Landau level, and σ is the spin projection that can take values ±1 only.
Notice that in the lowest Landau level, k = 0, only particles with one of the two spin projections,
namely, σ = 1 for positively charged particles, are allowed. The normalization constant Nn(±)is
Nn(±)= (4π? e? B)
1
4/
?
n(±)! .
(3.12)
In (3.8) the spin matrices δ(σ) are defined as
δ(σ) = diag(δσ1,δσ−1,δσ1,δσ−1),
σ = ±1 ,
(3.13)
and satisfy the following relations
δ (±)†= δ (±) ,
δ (±)δ (±) = δ (±) ,
δ (±)δ (∓) = 0 ,
(3.14)
γ?δ (±) = δ (±)γ?,
γ⊥δ (±) = δ (∓)γ⊥.
(3.15)
In Eq. (3.15) the notation γ?= (γ0,γ3) and γ⊥= (γ1,γ2) was used.
The functions E(±)
p
are complete
?
∑
k
dp0dp2dp3E(±)
p (x)E(±)
p (y) = (2π)4δ(4)(x−y) ,
(3.16)
and orthonormal,
?
xE(±)
p′ (x)E(±)
p (x) = (2π)4Λkδkk′δ(p0− p′
0)δ(p2− p′
2)δ(p3− p′
3)
(3.17)
with the (4×4) matrix Λkgiven by
Λk=
?
δ(σ = sgn[eB])
I
for
for
k = 0,
k > 0.
(3.18)
The matrix structure Λkwas recently introduced in Ref. [11]. It had been previously omitted
in the orthonormal condition of the Ep(x) functions given in Refs. [9, 10, 12]. Nevertheless,
it should be underlined that this matrix only appears in the zero Landau level contribution, and
consequently it enters as an irrelevant multiplicative factor in the SchwingerDyson equations in
the lowest Landau level approximation. Thus, all the results obtained in the works [9, 10, 12]
remain valid. In Eqs. (3.16)(3.17) we introduced the notation E(±)
Under the Ep(x) functions, positively (ψ(+)), negatively (ψ(−)) charged fields transform ac
cording to
ψ(±)(x) =∑
k
p (x) = γ0(E(±)
p (x))†γ0.
?
dp0dp2dp3E(±)
p (x)ψ(±)(p) ,
(3.19)
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ψ(±)(x) =∑
k
?
dp0dp2dp3ψ(±)(p)E(±)
p(x) .
(3.20)
One can show that
[γµ(Π(+)µ±µδµ0)−m]E(+)
p (x) = E(+)
p (x)[γµ(p(+)
µ ±µδµ0)−m] ,
(3.21)
and
[γµ(Π(−)µ±µδµ0)−m]E(−)
The conjugate fields transform according to,
p (x) = E(−)
p (x)[γµ(p(−)
µ ±µδµ0)−m] .
(3.22)
ψ(+)C(x) =∑
k
?
?
dp0dp2dp3E(−)
p (x)ψ(+)C(p),
(3.23)
ψ(−)C(x) =∑
k
dp0dp2dp3E(+)
p (x)ψ(−)C(p) .
(3.24)
After transforming to momentum space one can introduce NambuGorkov fermion fields of
different˜Q charges. They are the˜Qneutral Gorkov field
Ψ(0)=
?
ψ(0)
ψ(0)C
?
,
(3.25)
the positive
Ψ(+)=
?
ψ(+)
ψ(−)C
?
,
(3.26)
and the negative one
Ψ(−)=
?
ψ(−)
ψ(+)C
?
.
(3.27)
Using them, the NambuGorkov effective action in the presence of a constant magnetic field? B
can be written as
IB(ψ,ψ) =1
2
?
?
(2π)4Ψ(−)(p)S−1
d4p
(2π)4Ψ(0)(p)S−1
d4p
(0)(p)Ψ(0)(p)
+1
2
?
d4p
(2π)4Ψ(+)(p)S−1
(+)(p)Ψ(+)(p)+1
2
(−)(p)Ψ(−)(p) ,
(3.28)
where
S−1
(0)(p) =
[G+
(0)0]−1(p)
∆−
(0)
∆+
(0)
[G−
(0)0]−1(p)
,
,
(3.29)
S−1
(+)(p) =
[G+
(+)0]−1(p)
∆−
(+)
∆+
(+)
[G−
(+)0]−1(p)
(3.30)
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S−1
(−)(p) =
[G+
(−)0]−1(p)
∆−
(−)
∆+
(−)
[G−
(−)0]−1(p)
,
(3.31)
with
∆+
(+)= Ω−∆+Ω+,
(3.32)
∆+
(−)= Ω+∆+Ω−,
(3.33)
∆+
(0)= Ω0∆+Ω0,
(3.34)
Notice that to form the positive (negative) NambuGorkov field we used the positive (negative)
fermion field and the charge conjugate of the negative (positive) field. This is done so that the ro
tated charge of the up and down components in a given NambuGorkov field be the same. This way
to form the NambuGorkov fields is mandated by what kind of field enters in a given condensate
term, which in turn is related to the neutrality of the fermion condensate ?ψCψ? with respect to the
rotated˜Qcharge.
In momentum space the bare inverse propagator for the neutral field is
[G±
(0)0]−1(p) = [γµ(pµ±µδµ0)−m] ,
(3.35)
where the momentum is the usual p = (p0,p1,p2,p3) of the case with no background field.
For positively and negatively charged fields the bare inverse propagators are
[G±
(+)0]−1(p) = [γµ(p(+)
µ ±µδµ0)−m] ,
(3.36)
and
[G±
(−)0]−1(p) = [γµ(p(−)
µ ±µδµ0)−m]
(3.37)
respectively.
4. Gap Solutions
The main question we would like to address now is: Can we find a region of magnetic fields
where the gaps ∆Aand ∆B
the CFL phase anymore, but in the MCFL phase? To explore the possible answer to this question
we need to solve the gap equations derived from the NambuGorkov effective action (3.28).
In coordinate space the QCD gap equation reads
A, (or ∆Sand ∆B
S), differ enough from each other that the system is not in
∆+(x,y) = ig2
4λT
AγµS21(x,y)γνλBDAB
µν(x,y) ,
(4.1)
where S21(x,y) is the offdiagonal part of the NambuGorkov fermion propagator in coordinate
space and, for simplicity, we have omitted explicit color and flavor indices in the gap and fermion
propagator. Here DAB
µνis the gluon propagator.
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In a NJL model the gap equation can be obtained from Eq. (4.1) simply by substituting the
gluon propagator by
DAB
µν(x,y) =
1
Λ2gµνδABδ(4)(x−y) .
(4.2)
The NJL model is characterized by a coupling constant g and an ultraviolet cutoff Λ. The
ultraviolet cutoff should be much larger than any of the energy scales of the system, typically the
chemical potential. In the presence of a magnetic field we should also assume that Λ is larger than
the magnetic energy
˜ e˜B. In other studies of color superconductivity within the NJL model, the
values of g and Λ are chosen to match some QCD vacuum properties, thus hoping to get in such
a way correct approximated quantitative results of the gaps. We will follow the same philosophy
here, noticing however that this completely ignores the effect of the magnetic field on the gluon
dynamics.
To solve the gap equation (4.1) for the whole range of magneticfield strengths we need to use
numerical methods. We have found, however, a situation where an analytical solution is possible.
This corresponds to the case ? e? B ? µ2/2. Taking into account that the leading contribution to the
only the LLL (l = 0) contributes.
Using the approximation ∆B
equation for ∆B
?
?Λ
√
gap solution comes from quark energies near the Fermi level, it follows that for fields in this range
A≫ ∆B
S,∆A, and ∆A≫ ∆S, the gap equations decouple and the
Ais
∆B
A≈
g2
3Λ2
Λ
d3q
(2π)3
∆B
A
?
(q−µ)2+2(∆B
A)2
+g2? e? B
3Λ2
−Λ
dq
(2π)2
∆B
A
?
(q−µ)2+(∆B
A)2,
(4.3)
where thefirst/second term inther.h.s. ofEq.(4.3)corresponds tothe contribution of?Qneutral/charged
we are interested in the leading term.
The solution of Eq. (4.3) reads
?
with δ ≡ Λ−µ. It can be compared with the antisymmetric CFL gap [13]
?
In this approximation the remaining gap equations read
?
?Λ
quark propagators, respectively. For the last one, we dropped all Landau levels but the lowest, as
∆B
A∼ 2
?
δµ exp
−
3Λ2π2
µ2+? e? B
−3Λ2π2
2g2µ2
g2?
?
?
,
(4.4)
∆CFL
A
∼ 2
δµ exp
?
?
.
(4.5)
∆B
S≈ −g2
6Λ2
Λ
d3q
(2π)3
∆B
A
?
?
(q−µ)2+2(∆B
∆B
A
(q−µ)2+(∆B
A)2
+g2? e? B
6Λ2
−Λ
dq
(2π)2
A)2
,
(4.6)
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∆A≈
g2
4Λ2
?
Λ
d3q
(2π)3
?17
9
∆A
?
(q−µ)2+∆2
?
A
+7
9
∆A
?
(q−µ)2+2(∆B
A)2
,
(4.7)
and
∆S≈
g2
18Λ2
?
Λ
d3q
(2π)3
?
∆A
?
(q−µ)2+∆2
?
A
−
∆A
?
(q−µ)2+2(∆B
A)2
.
(4.8)
We express below the solution of these gap equations as ratios over the CFL antisymmetric
and symmetric gaps
?
where x ≡ g2µ2/Λ2π2, and y ≡ ? e? B/µ2, and
∆CFL
S
∆CFL
A
∆S
∆CFL
S
∆CFL
A
2
Note that our analytic solutions are only valid at strong magnetic fields. The lower value
? e? B ∼ µ2/2 corresponds to ? e? B ∼ (0.8−1.1)·1018G, for µ ∼ 350−400 MeV. For fields of this
estimated (see the details in Cristina Manuel’s talk in the proceedings) that the separation between
CFL and MCFL will take place already at fields ∼ 1016G.
All the gaps feel the presence of the external magnetic field. The effect of the magnetic field
in ∆B
of a BCS solution. The density of states appearing in (4.4) is just the sum of those of neutral and
charged particles participating in the given gap equation (for each Landau level, the density of
states around the Fermi surface for a charged quark is ? e? B/2π2).
are all equal. As two of these quarks have positive? Q charge, while the other two have it negative,
5. Conclusions
∆A
∆CFL
A
∼
1
2(7/34)exp
−36
17x+21
17
1
x(1+y)+3
2x
?
,
(4.9)
∆B
S
∼
∆B
A
?3
3
4+
?
9
2xln2
y−1
y+1
?
?
,
(4.10)
∼
∆A
1−
4
1+y
.
(4.11)
order and larger the ∆B
Agap is larger than ∆CFL
A
at the same density values. Nevertheless, we have
Ais to increase the density of states, which enters in the argument of the exponential as typical
All the? Qcharged quarks have common gap ∆B
the? Q neutrality of the medium is guaranteed without having to introduce any electron density.
A. Hence, the densities of the charged quarks
In this paper, we have shown that a magnetic field leads to the formation of a new colorflavor
locking phase, characterized by a smaller vector symmetry than the CFL phase. The essential
11
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Dense Quark Matter in a Magnetic Field
Vivian de la Incera
role of the penetrating magnetic field is to modify the density of states of charged quarks on the
Fermi surface. To better understand the relevance of this new phase in astrophysics we need to
explore the region of moderately strong magnetic fields ? e? B < µ2/2, which requires to carry out
total density of states around the Fermi surface for charged particles does not vary monotonically
with the number of Landau levels, we still expect to find a meaningful splitting of the gaps at these
fields and therefore a qualitative separation between the CFL and MCFL phases.
Acknowledgments
The work of E.J.F. and V.I. was supported in part by NSF grant PHY0070986, and C.M. was
supported by MEC under grant FPA200400996.
a numerical study of the gap equations including the effect of higher Landau levels. Because the
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