Page 1
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/
Page 2
Dense Quark Matter in a Magnetic Field
Vivian de la Incera
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
2
Page 3
Dense Quark Matter in a Magnetic Field
Vivian de la Incera
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)
3
Page 4
Dense Quark Matter in a Magnetic Field
Vivian de la Incera
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
4
Page 5
Dense Quark Matter in a Magnetic Field
Vivian de la Incera
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).
5