We discuss the Meissner effect for a color superconductor formed by cold dense quark matter. Though color and ordinary electromagnetism are broken in a color superconductor, there is a linear combination of the photon and a gluon that remains massless. Consequently, a color superconducting region may be penetrated by an external magnetic field. We show that at most a small fraction of the magnetic field is expelled, and if the screening distance is the smallest length scale in the problem there is no expulsion at all. We calculate the behavior of the magnetic field for a spherical geometry relevant for compact stars. If a neutron star contains a quark matter core, this core is a color superconductor. Our results demonstrate that such cores admit magnetic fields without restricting them to quantized flux tubes. Such magnetic fields within color superconducting neutron star cores are stable on time scales longer than the age of the universe, even if the spin period of the neutron star is changing.

## 1 Introduction

Conventional superconductors result from a condensate of Cooper pairs of electrons. Because the Cooper pairs have nonzero electric charge, electromagnetic gauge invariance is spontaneously broken: the photon gets a mass and weak magnetic fields are expelled by the Meissner effect. In this paper we show that the same is not in general true for the color superconducting state that is formed by cold dense quarks [1, 2, 3, 4, 5, 6, 7], even though here also the Cooper pairs have nonzero electric charge. The reason is that a color superconductor is not quite an electric superconductor: it makes the gluons massive (there is a color Meissner effect) but does not simply make the photon massive. Rather, one linear combination of the photon and a gluon becomes massive, but the orthogonal combination remains massless. Thus a region of color superconductor can allow itself to be penetrated by the component of an external magnetic field that corresponds to the unbroken generator. As we will see below, in the limit in which the screening length is the shortest length scale in the problem, the magnetic field within the color superconductor has the same strength as the applied external magnetic field. There is no Meissner effect. Though the interior field is not diminished in strength, it is “rotated” relative to the external field: it is associated with the symmetry which is unbroken within the superconductor, not with the of ordinary electromagnetism. If the penetration length is not smaller than the thickness of the boundary of the color superconducting region (to be defined below), there is a partial Meissner effect. In this case, the strength of the field which penetrates the superconductor depends on details of the geometry, the relative sizes of the screening length and the boundary thickness, and the relative strengths of electromagnetism and the color force.

The most likely place to find superconducting quark phases in nature is in the core of neutron stars. If neutron stars achieve sufficient central densities that they feature quark matter cores, these cores must be color superconductors: because there are attractive interactions between pairs of quarks which are antisymmetric in color, the quark Fermi surfaces in cold dense quark matter are unstable to the formation of a condensate of quark Cooper pairs. Present theoretical methods are not sufficiently accurate to determine the density above which a quark matter description becomes appropriate, and thus cannot answer the question of whether quark matter, and hence color superconductivity, occurs in the cores of neutron stars. What theory can do is analyze the physical properties of dense quark matter and, eventually, make predictions for neutron star phenomenology, thus allowing astrophysical observation to settle the question.

There are a number of avenues which may allow observations of neutron stars to answer questions about the presence or absence of color superconductivity. (Examples we do not pursue in this paper include analysis of cooling by neutrino emission [8] and analysis of r-mode oscillations [9].) Since neutron stars have high magnetic fields ( to Gauss in typical pulsars [10]; perhaps as high as Gauss in magnetars [11]) one prerequisite to making contact with neutron star phenomenology is to ask how the presence of a superconducting core would affect the magnetic field. We will see in this paper that the strength of the magnetic field within the superconducting core is hardly reduced, and the magnetic flux within the color superconductor is not restricted to quantized flux tubes. The latter constitutes a qualitative difference between conventional neutron stars and those with quark matter cores. We will see in Section 5 that, unlike in conventional neutron stars, the magnetic field within a color superconducting core does not vary even if the spin period of the neutron star is changing.

### 1.1 A fiducial example

To give the reader some sense for typical scales in the problem, we now describe a fiducial example, which we will use in particular in Sect. 5. The numbers in this paragraph make the crude assumption that the quarks are noninteracting fermions, and so should certainly not be construed as precise. Consider quark matter with quark chemical potential , made of massless up and down quarks and strange quarks with mass . ( is a density dependent effective mass; this adds to the uncertainty in its value.) If the strange quark were massless, quark matter consisting of equal parts , and would be electrically neutral. In our fiducial example, on the other hand, electric neutrality requires a nonzero density of electrons, with chemical potential . Charge neutrality combined with the requirement that the weak interactions are in equilibrium determine all the chemical potentials and Fermi momenta:

(1.1) |

The baryon number density is 5 times nuclear matter density.

A variety of estimates suggest that the gaps at the Fermi surfaces resulting from quark-quark pairing are about , resulting in critical temperatures above which the color superconductivity is destroyed which are about . Neutron stars have temperatures , and if they have quark matter cores these cores are certainly in the superconducting phase. Similar estimates for and are arrived at either by using phenomenological models, with parameters normalized to give reasonable vacuum physics [3, 4, 5, 6, 12, 13, 14, 15] or by using weak coupling methods [16, 17, 18, 19, 20, 21, 22, 23], valid for where the QCD coupling does become weak. Neither strategy can be trusted quantitatively for , where , but it is pleasing that both strategies agree qualitatively.

### 1.2 The CFL and 2SC phases of quark matter

There are, in fact, two different superconducting phases possible, which have very different symmetry properties. If is large compared to the differences among the three Fermi momenta in (1.1), , and condensates all form. Chiral symmetry is broken by color-flavor locking [6] in this phase. This is the favored phase at large where the differences between Fermi momenta decrease [24, 25]. This CFL phase has the same symmetries as baryonic matter which is itself sufficiently dense that the hyperon and nucleon densities are all similar, and there need not be a phase boundary between CFL matter and baryonic matter [26, 24, 25]. The properties of the CFL phase have been further investigated in Refs. [27, 28].

Now, imagine starting with CFL matter at very large and reducing . Because of the nonvanishing strange quark mass, as decreases the differences among the Fermi momenta increase. It may happen that before has decreased so far that a quark matter description ceases to be valid, one may find a phase of quark matter in which only two flavors ( and ) and two colors (chosen spontaneously) of quarks pair. Chiral symmetry is restored in this two-flavor superconductivity (2SC) phase, which is also the phase which arises in QCD with no strange quarks at all [1, 2, 3, 4, 5]. Because nature chooses a strange quark which can neither be treated as very light nor as very heavy, present theoretical analyses are not precise enough to determine whether quark matter at densities typical of neutron star interiors is in the CFL phase, or whether in this range of the 2SC phase is favored. For example, in (1.1) the splitting between and Fermi momenta is MeV, of the order of typical gaps. Current theoretical methods are not reliable enough to determine whether the quark-quark interactions in QCD are strong enough to generate a condensate larger than MeV, as required in the CFL phase, or are somewhat weaker, admitting only the 2SC pairing.

In discussing conventional superconductors in a magnetic field,
one normally begins by distinguishing between Type I and
Type II superconductors. Color superconductors are Type I
at asymptotically high densities [19, 20, 23], but they
may be Type I or Type II at neutron star densities.
However, this distinction will
not be of importance. First of all, the thermodynamic critical field
required to destroy the color superconductivity is
on the order of Gauss.
If color superconductors
exhibited the conventional Meissner effect, whether
or not they were Type I or Type II they would
simply exclude neutron star magnetic fields,
which are in fact weak for our purposes.
Second of all, we shall see that the presence of
an unbroken rotated electromagnetism changes the
story completely.^{1}^{1}1
See [2, 29] for early attempts to analyze
magnetic fields in color superconductors. These authors
neglect the existence of the unbroken rotated electromagnetism.
In the next three sections,
we solve the problem of how a sphere of
color superconducting matter, which can admit
a rotated magnetic field, responds to an applied
ordinary magnetic field. In the final section,
we discuss the consequences of our findings for
neutron stars.

## 2 Rotated electromagnetism

### 2.1 The new photon

The fundamental fields in the theory are the quarks (with color index and flavor index ) and the gauge fields: the photon and the gluons , . At low temperature and high density, the quarks form Cooper pairs, associated with a Higgs field . This field gets a vacuum expectation value which takes the form

(2.2) |

in the 2SC phase, with and running over and only and and running over two of the colors only. In the CFL phase, quarks of all three flavors and colors pair and the expectation value of the field takes the form

(2.3) |

with all indices now taking on three values. In both the CFL and 2SC phases, the condensate leaves an unbroken generated by

(2.4) |

where for CFL, and for 2SC. is the conventional electromagnetic charge generator, and is associated with one of the gluons. In the representation of the quarks,

(2.5) |

As is conventional, we have taken . The -charge of all the Cooper pairs which form the condensates are zero.

To see exactly which gauge field remains unbroken, look at the covariant derivative of the Higgs field:

(2.6) |

From (2.4), we see that the kinetic term will give a mass to one gauge field

(2.7) |

but the orthogonal linear combination

(2.8) |

will remain massless. The denominators arise from keeping the gauge field kinetic terms correctly normalized, and we have defined the angle ,

(2.9) |

which specifies the unbroken . At neutron star densities the gluons are strongly coupled (), and of course the photons are weakly coupled (), so is small: in the 2SC phase and in the CFL phase. In both phases, the “rotated photon” consists mostly of the usual photon, with only a small admixture of the gluon.

We can now see that the electron couples to with charge which is less than , so the new photon is slightly more weakly coupled to electrons than the old one. It can also be shown that in the CFL phase, the quarks have -charges which are integer multiples of the -charge of the electron [6].

### 2.2 The Meissner effect

The main purpose of this paper is to study the Meissner effect in a high density region, which we will assume to be a sphere, like the core of a neutron star, with radius . In this region, color superconductivity occurs. The unbroken generator is associated with a rotated electromagnetic field , with rotation angle , as described above. The broken generator gives no long-range gauge field. For simplicity we will treat the region outside the core not as nuclear matter, but as vacuum. As we will explain in Sect. 5, this does not seriously affect our conclusions. In the external region, then, ordinary electromagnetism is unbroken, and the color generator is confined. We assume that currents at infinity create a uniform applied -magnetic field in the -direction. We want to know to what degree the flux is expelled from the inner region.

To study this situation, we need only work in the two-dimensional space of gauge symmetry generators spanned by and . It is natural to take (the angle by which the unbroken is rotated relative to ordinary electromagnetism, see (2.9)) to vary with radius, with in the high density (inner) region, and outside. So in the basis,

(2.10) |

To determine what fraction of the magnetic field is expelled from the color superconducting region, we must consider what happens at the boundary between the two phases. There are three relevant distance scales:

(2.11) |

We will always assume that . We will treat two limiting scenarios: “sharp” boundary and “smooth” boundary. The sharp boundary, , corresponds to a sudden step-function-like change in . This is what we would expect to find if there were a first-order phase transition between the low and high density regions, with phase boundary thickness less than the screening length. The smooth boundary, , corresponds to a gradual change in . This applies to the situation where there is no first-order phase transition between nuclear and quark matter (e.g. for low strange quark mass, where we expect no phase transition at all [26, 24, 25]), or where there is a first-order phase transition with phase boundary thicker than the screening length. We will see that the behavior of the magnetic field is quite different depending on whether the boundary is sharp or smooth.

What we are interested in is the behavior of the magnetic field at macroscopic distance scales of order , not at the microscopic scale . In other words, we will always be interested in the unbroken and unconfined gauge fields, which obey the Maxwell equations. In order to treat all gauge fields in a single formalism it will therefore be convenient to take into account screening not by introducing gauge boson masses into the field equations, but rather by introducing monopoles and supercurrents to describe the screening of the confined and Higgsed gauge fields. By “monopoles” we mean whatever gauge field configurations are responsible for terminating magnetic field lines of confined gauge fields; we do not require them to be solutions to any classical field equations. We will ignore the details of how the supercurrents (and “monopoles”) arrange themselves on distance scales of order , as all we care about is that they screen (terminate) the higgsed (confined) magnetic fields incident upon them.

## 3 Sharp boundary

In the sharp boundary case, changes quickly from to 0 at over a boundary region with thickness (Fig. 1). Just inside the boundary, in the region , the gauge field is Higgsed, so there is a density of -supercurrents that screen out the parallel component of the -flux, leaving only -flux at . Just outside the boundary, in the region , the gauge field is confined, so there is a density of -monopoles that terminate any perpendicular component of the -flux, leaving only -flux at .

To obtain the boundary conditions on the magnetic fields associated with the unbroken generators, write the magnetic field in the basis (2.10),

(3.12) |

then the Maxwell equations are

(3.13) |

Since we are only interested in the behavior of the fields on distance scales much greater than , we integrate the Maxwell equations (3.13) over , and obtain boundary conditions that relate the fields at to those at . We follow the standard derivation (Ref. [30], sect. I.5). We find a discontinuity in the normal compoment due to the surface density of -monopoles, and in the parallel component due to surface -supercurrents,

(3.14) |

From this we immediately obtain the boundary conditions on the flux:

(3.15) |

Thus, just inside the interface we find a whose component parallel (perpendicular) to the interface is weakened (strengthened) relative to that of just outside the interface.

### 3.1 Solution for the sharp boundary

Since this is a magnetostatic problem we can write it in terms of a magnetic scalar potential . The potential is associated with the unbroken flux inside the sphere, and the unbroken -flux outside

(3.16) |

Maxwell’s equations become

(3.17) |

with boundary conditions (3.15)

(3.18) |

Expanding in Legendre polynomials (see [30] sect. 5.12) we find the solution

(3.19) |

In Fig. 2 we show the resultant field configuration for . In the real world is small, so the field is mostly converted into flux by the supercurrents and monopoles, and penetrates the interior. Only a weak field is excluded.

We can check that the solution (3.19) makes sense in two limits. (1) , so and . The unbroken in the color superconducting region is exactly the same as conventional electromagnetism: the magnetic field does not even notice the color superconductor. (2) , so and . The region is a conventional superconductor, breaking electromagnetism. The magnetic field is completely expelled from the superconducting region.

How large a field must we apply before we begin to destroy the color superconductivity in parts of the sphere? The magnitude of the -magnetic field just outside the sphere is largest at the equator of the sphere, where it is given by

(3.20) |

This means that the equator of the sphere sees an -magnetic field
given by times (3.20). If the color
superconductor is Type I, the color
superconductivity is destroyed in regions of the sphere near its
equator when the -magnetic field exceeds the thermodynamic
field Gauss.^{2}^{2}2If the material is Type II, flux tubes
begin to penetrate when the -magnetic field exceeds ,
which is of the same order of magnitude. This
requires

(3.21) |

We conclude that because most of the applied field can penetrate the superconductor in the form of -flux, while only a small fraction of the applied field must be excluded, the applied field at which the color superconductivity begins to be destroyed is significantly larger than the thermodynamic critical field Gauss.

## 4 Smooth boundary

For a smooth boundary, , which means that changes slowly relative to the screening length . In this case we assume that there is a continuous distribution of monopoles and supercurrents. At a given these produce flux in the that is (locally) broken. Since the screening length is small, the net flux at a given must only be in the unbroken . The Maxwell equations therefore take the form

(4.22) |

where are functions of position. Now

(4.23) |

Substituting these into (4.22) and taking the component, we find that the locally unbroken magnetic field obeys the sourceless Maxwell equations:

(4.24) |

We conclude that in this case the magnetic field always rotates to be locally unbroken, but is otherwise unaffected: it obeys the free Maxwell equations everywhere. There is no expulsion at all. (In the smooth boundary case, then, the effective critical magnetic field is infinite: an arbitrarily strong magnetic field is always rotated to be in the locally unbroken , and never destroys the color superconductivity.)

One way of understanding this result is to note that the smallness of the screening length means that broken gauge fields are always zero. Consequently, when we move from radius to , the gauge field is immediately projected onto the locally unbroken generator , which differs by an infinitesimal angle from . As we vary , we therefore have a sequence of such projections. In the limit of an infinite number of projections, each infinitesimally different from the last, the gauge field is simply rotated to follow the locally unbroken generator,

(4.25) |

This is analogous to the behavior of a sequence of polarizers each at an angle to the previous one. In the limit where , an incoming photon aligned with the first polarizer exits from the last polarizer with its polarization rotated by , and with no loss of intensity.

Finally, it is interesting to ask why the limit of our
result for the smooth boundary is not the same as our result for the
sharp boundary. The reason is that in the smooth case we assume
, so the monopoles and supercurrents are always in the
region where is changing. In the sharp case, by contrast, we
assume , and there is in effect no such region.
Taking the limit of the smooth case
() would result in a pile-up of currents and monopoles in
a shrinking region within which changes rapidly from
to 0. In this limit, (4.24) is maintained and no flux is
excluded from the sphere of color
superconductor.^{3}^{3}3Formally, taking the
limit while maintaining yields both monopoles and
supercurrents concentrated at a point where .
If one modifies the right hand side of (3.14)
to reflect this, the boundary conditions (3.15)
becomes , and
no flux is excluded.
The sharp boundary studied in Sect. 3 represents
different physics, in which is constant as . In
this limit all the monopoles are in the region on the confined
side of the boundary while all the supercurrents are in the
region on the Higgsed side and we obtain
the boundary condition (3.15) and
partial flux
exclusion as in (3.19).

## 5 Consequences for Neutron Stars and Outlook

If a neutron star features a core made of color superconducting quark matter, we have learned that this core exhibits (almost) no Meissner effect in response to an applied magnetic field. Even though the Cooper pairs of quarks have nonzero electric charge, there is an unbroken gauge symmetry , and the color superconducting region can support a -magnetic field. If the boundary layer of the color superconducting region is thick compared to the screening length, there is no Meissner effect: the -magnetic field within has the same strength as the applied -magnetic field. (This smooth boundary case applies if the color superconducting phase and the baryonic phase are not separated by a phase transition; it may also apply in the presence of a first order phase transition, if the phase boundary is thick enough.) If the thickness of the boundary of the superconducting region is less than the penetration length, there is a partial Meissner effect: the -magnetic field within is somewhat reduced relative to the applied -magnetic field. Because is small in nature, the overlap between -photons and -photons is large, and this partial Meissner effect occurs only at the few percent level.

The physics of magnetic fields in neutron stars with color superconducting cores is qualitatively different from that in conventional neutron stars. In conventional neutron stars, proton-proton pairing breaks and there is no unbroken . This means that magnetic fields thread the cores of conventional neutron stars in quantized flux tubes, within each of which there is a nonsuperconducting region. In contrast, the -magnetic field within a color superconducting neutron star core is not confined to flux tubes. This means that, as we discuss below, the enormous -electrical conductivity of the matter ensures that the -magnetic field is constant in time. In ordinary neutron stars the -magnetic flux tubes can be dragged about by the outward motion of the rotational vortices as the neutron star spins down [31, 32, 10, 33, 34], and can also be pushed outward if the gap at the proton fermi surface increases with depth within the neutron star [35]. One therefore expects the magnetic field of an isolated pulsar to decay over billions of years as it spins down [32, 10, 33, 34] or perhaps more quickly [35]. There is no observational evidence for the decay of the magnetic field of an isolated pular over periods of billions of years [10]; this is consistent with the hypothesis that they contain color superconducting cores which serve as “anchors” for magnetic field, because they support a -magnetic field which does not decay.

We now estimate the decay time of the -magnetic field for neutron stars with color superconducting cores, doing the calculation separately for cores which are in the 2SC and the CFL phase. To this point, the only difference between the two color superconducting phases in their response to applied magnetic fields has been a difference of a factor of two in the value of ; because in both phases, this difference is of no qualitative consequence. However, the 2SC and the CFL phase differ qualitatively in their symmetries and their low-energy excitations, and can therefore be expected to have quite distinct transport properties. As we will see below, the -electrical conductivity in the CFL phase is much larger than the conductivity in the CFL phase. However, even the “smaller” conductivity of the 2SC phase is so large that the timescale for the decay of a -magnetic field within a color superconducting neutron star core is long compared to the age of the universe.

The characteristic magnetic field decay time due to ohmic dissipation is [36]

(5.26) |

where is the radius of the color superconducting core
and is the -electric conductivity.
(We set throughout.)
We begin by estimating , and hence the
decay time, for a core
which is in the 2SC phase.
At keV temperatures, the
dominant carriers are the relativistic electrons and
those quarks of the 2SC phase which are ungapped, or which acquire gaps
so small as to be or less. In the 2SC phase
the up and down quarks of one color acquire gaps smaller than
or of order keV [3]. The strange quarks
of all three colors — which do not participate in
the dominant pairing characterizing the 2SC phase —
can be expected to have gaps which are similar in magnitude
or even smaller [24].^{4}^{4}4
The contribution from the quark quasiparticles with
gaps can be neglected.
In the CFL phase, all quark quasiparticles
have gaps which are . We discuss this situation
below.
To obtain a lower bound on , we assume that
all strange quarks, and up and down quarks of one color,
have gaps . These five quarks have -charges
-charge
of the electron , which we henceforth
take to be just since . The
sum of the squares of the -charge
of the ungapped quarks is therefore .
Following [37],
the electrical conductivity is given by in units of the

(5.27) |

where and are the momentum relaxation times for electrons and quarks in the plasma, defined and calculated in Ref. [37]. For both the electrons and quarks, the dominant scattering process contributing to momentum relaxation is scattering off quarks. The five gapless quarks of interest yield [37]

(5.28) |

where and
is the Debye screening mass for the gluons,
neglecting relative to ,
and where we have assumed that .^{5}^{5}5Note
that we have worked directly from
equations (28) and (39) of Ref. [37],
and have not used the numerical factor
in equation (40) of Ref. [37],
which contains an error [38].
Similarly,

(5.29) |

where , and is the Debye screening mass for the -photon. We have neglected , used the fact that the average squared -charge of the nine quarks participating in screening is , and have assumed . Taking , we find first that the electrons dominate the conductivity, by a factor of about , and second that the time-scale for the decay of the magnetic field in a color superconducting 2SC core of radius is

(5.30) |

Thus, the magnetic field in the core of a neutron star which is made of matter in the 2SC phase decays only on a time-scale which exceeds the age of the universe.

We now turn to a color superconducting core in the CFL phase.
In contrast to the 2SC phase, the condensation in the
CFL phase produces gaps for quarks of all three colors
and all three flavors.^{6}^{6}6It may be possible for some
quark quasiparticles to be gapless even in the CFL phase, at densities just
above those where the 2SC phase is favored [27]. In
this nongeneric circumstance, the CFL phase conductivity would
be similar to that of the 2SC phase.
Thus there are no low-energy quark quasiparticle excitations.
Furthermore, the CFL condensate gives a mass to all eight gluons.
In this quark matter phase, the degrees of freedom
which are most easily excited are neither
quarks nor gluons.
Because of the spontaneous breakdown
of chiral symmetry in the CFL phase, there are
charged Nambu-Goldstone bosons, which would
be massless pseudoscalar excitations
if the quarks were massless. Once the nonzero
quark masses are taken into account, one finds
pseudoscalar masses which are small [13]
(in the sense that they are )
but which are still large compared to .
There is one remaining scalar Nambu-Goldstone excitation,
associated with the superfluidity of the CFL phase,
but this excitation is -neutral.
We thus discover that
all possible hadronic excitations which have nonzero -charge
do acquire a gap: their populations in thermal equilibrium
are suppressed exponentially by factors of the form ,
where is either a Fermi surface gap or a suitable
bosonic mass.
This first of all means that the only charge carriers which
could contribute to are the electrons. Second,
the scattering of these electrons off the positively charged
hadronic system in which they are immersed vanishes exponentially
for , because at low temperatures there are
no hadronic excitations off which to scatter.^{7}^{7}7We
are describing the conductivity in the linear response regime.
In particular, we are assuming that the current
is small enough that the momenta acquired by the electrons
due to the current is small compared
to the excitation energy for all charged hadronic
modes. If the current were increased beyond
the linear regime, the conductivity
would decrease and eventually the CFL condensate itself would
be destroyed. The nonlinear
regime is not relevant for our purposes.
The electrons can only scatter off other electrons. However,
such collisions do not alter the total electric
current [39].
We therefore conclude that although matter in the CFL phase is not a
-superconductor (it does not exclude -magnetic
field) it is a near-perfect conductor: the resistivity drops
exponentially to zero as . At typical neutron star
temperatures , the density of hadronic excitations, and therefore
the resistivity, is exponentially
suppressed relative to the 2SC phase. As a consequence, the decay time
for a -magnetic field in a CFL core is exponentially larger
than that (5.30) for a core in the 2SC phase, which was
already longer than the age of the universe.

It is clear that the -magnetic field is rigidly locked in the color superconducting core. It cannot decay with time, even if rotational vortices move through the core as the spin rate of the pulsar changes with time. Rotational vortices do exist in the CFL phase, because the CFL condensate spontaneously breaks a global . Instead of dragging magnetic flux tubes with them as they move, as occurs in a conventional neutron star, the rotational vortices can move freely through the CFL phase because there are no -flux tubes, only a -magnetic field. Thus, as the spin period of the neutron star changes and the rotational vortices move accordingly, there is no change at all in the strength of the -magnetic field in the core. The conclusion is the same in the 2SC phase, but the argument is even simpler because in this phase there is no spontaneously broken global , and therefore no rotational vortices.

As we have noted above, the data on isolated pulsars show no evidence for any decay of the observed magnetic field even as they spin-down over time [10]. This is consistent with the possibility of a color superconducting core within which the field does not decay. However, we must also ask whether and how the fact that the observed magnetic fields of accreting pulsars do change as they spin up [10] is consistent with the possibility of quark matter cores. There are several ways in which the surface magnetic field could decrease as a neutron star accretes and spins up, even though the magnetic field in the core remains constant. One possibility is that the accreting matter may bury the magnetic field [40, 10]. Another possibility is that as the magnetic flux tubes in the mantle and crust of the neutron star are pulled around by the rotational vortices, the north and south magnetic poles on the star’s surface may be pushed toward one another, reducing the observed dipole field even though the field deep within, in the color superconducting core, remains undiminished [33, 34].

The analysis of Section 3 was idealized in three ways relative to that appropriate for a neutron star, if there is a first order phase transition between baryonic and quark matter. First, we assumed that ordinary electromagnetism was unbroken outside the core. This is false: proton-proton superconductivity results in the restriction of ordinary -magnetic field to flux tubes. This means that our derivation of the boundary conditions in Section 3 only applies upon averaging over an area of the boundary that is sufficiently large that many -flux tubes are incident on it from the outside. A more microscopic description of the field configuration near the boundary would in fact require further work, but this is not necessary for our purposes.

Second, our assumption in Section 3 of a spherical boundary is oversimplified. If there is a first order phase transition between baryonic and quark matter, because there are two distinct chemical potentials and there will be a mixed phase region, with many boundaries separating regions of quark matter and baryonic matter which have complex shapes [41]. This complication of the geometry of the boundary evidently makes a complete calculation more difficult than the one we have done in Section 3, although if the boundary is thick then the conclusions of Section 4 are unaffected. Regardless, the qualitative conclusion that only a very small fraction of the flux will be excluded because is so small will not be affected by these complications.

Third, the configuration of Figure 2 cannot, in fact, be attained in a neutron star even though it is favored in the sharp boundary case. The core of a newborn neutron star is threaded with ordinary -magnetic field. Because of the enormous conductivity, the time it would take to accomplish the partial exclusion of flux seen in Figure 2 is exceedingly long [36]. This means that, instead, although the field within the core will be largely -magnetic field, there will in addition be a small fraction of -magnetic flux confined in quantized flux tubes. The sum of the - and -fluxes adds up to the original -flux. Over time, the motion of rotational vortices may move the -flux tubes around. The much larger -flux, which is not constrained in flux tubes, is frozen as described above.

We conclude that relaxing the simplifying assumptions which we have made would not change our qualitative conclusions. If neutron stars contain quark matter cores, those cores will exclude at most a very small fraction of any applied magnetic field. Instead, the flux penetrates (almost) undiminished. The only change in the flux within the color superconducting core is that it is a -magnetic field, associated with that linear combination of the photon and the gluon which is unbroken by the quark pair condensate. Most important for neutron star phenomenology, and in qualitative distinction from the results for conventional neutron stars, is the conclusion that this -flux does not form quantized flux tubes and is frozen over timescales long compared to the age of the universe.

We are confident that we have not said the last word on the effects of color superconducting quark matter cores within neutron stars on magnetic field evolution. More work is required to better understand pulsars which are accreting and spinning up. We can already conclude that if observational evidence were to emerge that an isolated pulsar loses its magnetic field as it spins down (in such a way that the field would vanish in the limit in which the spin vanishes), this would allow one to infer that such a pulsar does not have a quark matter core.

Acknowledgements

We are grateful to the Aspen Center for Physics, where much of this work was completed. We thank I. Appenzeller, M. Camenzind, D. Chakrabarty, H. Heiselberg, L. Hernquist, R. Jaffe, V. Kaspi, D. Psaltis, M. Ruderman, T. Schäfer, E. Shuster and F. Wilczek for helpful discussions. This work is supported in part by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818. The work of KR is supported in part by a DOE OJI Award and by the Alfred P. Sloan Foundation.

## References

- [1] B. Barrois, Nucl. Phys. B129 (1977) 390. S. Frautschi, Proceedings of workshop on hadronic matter at extreme density, Erice 1978.
- [2] D. Bailin and A. Love, Phys. Rept. 107 (1984) 325, and references therein.
- [3] M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422 (1998) 247.
- [4] R. Rapp, T. Schäfer, E. V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 81 (1998) 53.
- [5] J. Berges and K. Rajagopal, Nucl. Phys. B538 (1999) 215.
- [6] M. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B537 (1999) 443.
- [7] For recent reviews, see K. Rajagopal, to appear in Proceedings of Quark Matter ’99, hep-ph/9908360; and F. Wilczek, to appear in Proceedings of PANIC ’99, hep-ph/9908480.
- [8] D. Blaschke, T. Klähn and D. Voskresensky, astro-ph/9908334.
- [9] J. Madsen, Phys. Rev. Lett. 81 (1998) 3311 and references therein.
- [10] For a review, see D. Bhattacharya and G. Srinivasan, in X-Ray Binaries, W. H. G. Lewin, J. van Paradijs, and E. P. J. van den Heuvel eds., (Cambridge University Press, 1995) 495.
- [11] C. Thompson and R. C. Duncan, Astrophys. J. 473 (1996) 322.
- [12] G. Carter and D. Diakonov, hep-ph/9812445.
- [13] R. Rapp, T. Schäfer, E. V. Shuryak and M. Velkovsky, hep-ph/9904353.
- [14] N. Evans, S. D. H. Hsu and M. Schwetz, Nucl. Phys. B551 (1999) 275; Phys. Lett. B449 (1999) 281.
- [15] T. Schäfer and F. Wilczek, Phys. Lett. B450 (1999) 325.
- [16] R. Pisarski and D. Rischke, nucl-th/9903023.
- [17] D. T. Son, Phys. Rev. D59 (1999) 094019.
- [18] D. K. Hong, hep-ph/9812510; hep-ph/9905523.
- [19] D. K. Hong, V. A. Miransky, I. A. Shovkovy and L. C. R. Wijewardhana, hep-ph/9906478.
- [20] T. Schäfer and F. Wilczek, hep-ph/9906512.
- [21] R. Pisarski and D. Rischke, nucl-th/9907041.
- [22] W. Brown, J. Liu and H. Ren, hep-ph/9908248.
- [23] S. D. H. Hsu and M. Schwetz, hep-ph/9908310.
- [24] M. Alford, J. Berges and K. Rajagopal, hep-ph/9903502.
- [25] T. Schäfer and F. Wilczek, hep-ph/9903503.
- [26] T. Schäfer and F. Wilczek, hep-ph/9811473.
- [27] M. Alford, J. Berges and K. Rajagopal, hep-ph/9908235.
- [28] R. Casalbuoni and R. Gatto, hep-ph/9908227 and hep-ph/9909419.
- [29] D. Blaschke, D.Sedrakian and K. Shahabasyan, astro-ph/9904395.
- [30] J. D. Jackson, “Classical Electrodynamics”, second edition, John Wiley and Sons, New York, 1975.
- [31] J. Sauls, in Timing Neutron Stars, J. Ögleman and E. P. J. van den Heuvel, eds., (Kluwer, Dordrecht: 1989) 457.
- [32] G. Srinivasan, D. Bhattacharya, A. G. Muslimov and A. I. Tsyagan, Curr. Sci. 51 (1990) 31.
- [33] M. Ruderman, Astrophys. J. 366 (1991) 261; 382 (1991) 576; and ibid. p. 587.
- [34] M. Ruderman, T. Zhu and K. Chen, Astrophys. J. 492 (1998) 267. M. Ruderman, talk at Aspen Center for Physics, August, 1999.
- [35] S. Hsu, nucl-th/9903039.
- [36] G. Baym, C. Pethick and D. Pines, Nature, 224 (1969) 673 and ibid. p. 674.
- [37] H. Heiselberg and C. J. Pethick, Phys. Rev. D48 (1993) 2916.
- [38] H. Heiselberg, private communication.
- [39] E. Flowers and N. Itoh, Astrophys. J. 206 (1976) 218.
- [40] G. S. Bisnovatyi-Kogan and B. V. Komberg, Sov. Astr. 18 (1974) 217; R. E. Taam and E. P. J. van den Heuvel, Astrophys. J. 305 (1986) 235; R. W. Romani, Nature 347 (1990) 741.
- [41] N. K. Glendenning, Phys. Rev. D46 (1992) 1274; N. K. Glendenning, Compact Stars (Springer-Verlag, 1997).