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Astrophysics, Vol. 50, No. 1, 2007
0571-7256/07/5001-0065 ©2007 Springer Science+Business Media, Inc.
MAGNETIC FIELD OF A NEUTRON STAR WITH A SUPERCONDUCTING QUARKCORE IN THE CFL-PHASE
D. M. Sedrakian, K. M. Shahabasyan, and M. K. Shahabasyan
The Ginzburg-Landau equations are derived for the magnetic and gluomagnetic gauge fields in the colorsuperconducting core of a neutron star containing a CFL-condensate of diquarks. The interaction of thediquark CFL-condensate with the magnetic and gluomagnetic gauge fields is taken into account. Thebehavior of the magnetic field in a neutron star is studied by solving the Ginzburg-Landau equations takingcorrect account of the boundary conditions, including the gluon confinement conditions. The magnetic fielddistribution in the quark and hadronic phases of a neutron star is found. It is shown that a magnetic fieldgenerated in the hadronic phase by the entrainment effect penetrates into the quark core in the form of quarkvortex filaments because of the presence of screening Meissner currents.Keywords: stars: neutron: magnetic field
1. Introduction
The possible existence of superdense quark matter in the cores of neutron stars has been examined over the last
three decades. Because of the attraction between quarks in a color antitriplet channel owing to the exchange of a single
gluon, it is expected that a superconducting diquark condensate should develop in this material. The quark pairs that
form the condensate have zero total angular momentum J [1-4]. It has been shown that in chiral quark models with a
nonperturbative four-point interaction caused by instantons [5] or with nonperturbative gluon propagators [6,7], the
anomalous quark coupling amplitudes are on the order of 100 MeV. Thus, we might suppose that a diquark condensate
would exist at densities above the deconfinement density and at temperatures below the critical temperature Tc (on the
order of 50 MeV).
Two types of condensate are possible: an isoscalar 2SC-phase [2] in which only “u” and “d”-quarks of two colors
are coupled and a CFL-phase in which massless “u”, “ d”, and “s”-quarks of all three colors are coupled [8-10]. The CFL-
phase with three flavors of massless quarks is the most stable in the limit of the weak interaction at T = 0 and near the
Original article submitted October 25, 2006. Translated from Astrofizika, Vol. 50, No. 1, pp. 87-98 (February 2007).
Erevan State University, Armenia,e-mail: [email protected], [email protected]
66
critical temperature Tc. It has been shown [11] that a diquark condensate in the 2SC-phase is a type II superconductor.
The existence of electric and color charges in the Cooper diquark pairs leads to the appearance of electrical and color
superconductivity in the 2SC- and CFL-phases. These two phenomena are not independent because the photon and gluon
gauge fields are coupled. One of the resulting mixed fields is massless, while the other field has mass [12].
The Ginzburg-Landau equations for the 2SC-phase have been derived in Ref. 13 with the presence of mixed fields
taken into account. These equations have been used [14] to study the effect of an external uniform magnetic field on
the superconducting quark core of a neutron star. It was shown that, in the absence of vortex filaments, the Meissner
currents in the core will screen the external magnetic field almost completely. A distribution of the magnetic field in
a neutron star with a superconducting 2SC-quark core has been found [15] such that a magnetic field is generated in a
hadronic npe-phase owing to the entrainment of superconducting protons by superfluid neutrons [16] and penetrates into
the quark core in the form of quark vortex filaments which form because of the presence of screening Meissner currents.
The Ginzburg-Landau free energy of a uniform superconducting CFL-phase has been found in Refs. 10 and 17. A gauge
invariant expression for the kinetic energy has also been obtained [18]. The Ginzburg-Landau equations for the ordering
parameters and mixed fields in the CFL-phase in the presence of an external field have been obtained in Refs. 19-21.
The purpose of this article is to study the magnetic field distribution in the quark CFL-phase and hadronic npe-
phase of a neutron star. We allow for the generation of a magnetic field in the hadronic phase owing to the entrainment
of superconducting protons by superfluid neutrons. The following boundary conditions are taken into account: continuity
of the components of the magnetic field and the condition for gluon confinement. We shall assume a sharp boundary
between the quark and hadronic phases, since the thickness of the diffusive transition layer is small, on the order of the
confinement radius l = 0.2 fm.
In Section 2 we derive the Ginzburg-Landau equations for the magnetic and gluomagnetic gauge fields in the color
superconducting core of a neutron star containing a diquark CFL-condensate. In Section 3 these equations are solved
for the potentials of the magnetic and gluomagnetic fields. The components of the magnetic field in the quark and
hadronic phases are found in Section 4, along with the components of the external dipole field of the star. The conclusions
are discussed in Section 5.
2. Ginzburg-Landau equations for the magnetic and gluomagnetic fields
A diquark condensate is characterized by a complex 3×3 gap matrix ( ) ( )rd i
r
r
α in flavor and color space [17], where
α and i are the color and flavor indices, respectively, which differ from the indices of the two colors and flavors
participating in Cooper pairing, and rr
is the coordinate of the center mass of the Cooper pair. A “new” charge operator
8TQQ~ η+= has been introduced [12] which acts on the gap matrix in the following way:
( ) ( ) . 0=α rdQ~
i
r
r
(1)
Here 32−=η [18], Q is the generator of quark electric charge in the flavor space u, d, s, and 8T is the generator of
the color group SU(3)c; these are given by
67
( ). 1 ,1 ,2diag32
1,
3
1 ,
3
1 ,
3
2diag 8 −−=
−−= TQ (2)
The definition of 8T also differs from the standard cyclical permutation of the indices 1, 2, and 3 by a change in sign
[20]. Condition (1) means that the charge Q~
of all the Cooper pairs in the condensate is zero. The gap matrix is defined
as follows in the presence of an external magnetic field [21]:
( ) ( )[ ] ( ) ( )
( )[ ] ( ) ( ),
3
22(exp
3
2(exp
888
88
rrriTT
rrriTd
i
ii
rr
r
rr
r
r
χ−Φϕ+
+χ+Φ
ϕ=
α
αα
(3)
where the functions ( )rrΦ and ( )r
rχ are given by
( ) ( ) ( )[ ] ( ) ( ) ( )[ ].32exp,3exp 88 rirrrirrrrrrrr ϕ−χ=χϕΦ=Φ (4)
An analysis of the gauge invariant derivative leads to the following expression for the mixed fields [18]:
, sincos, cossin 88 α−α=α+α= AAAAAA yx
rrrrrr
(5)
where 222cos egg η+=α , Ar
is the vector potential of the magnetic field, and 8Ar
is the vector potential of the
gluomagnetic field. Note that the mixed field xAr
is massive, while the field yAr
is massless. Here g is the force interaction
constant (( )142 ≈πg ) and e is the electromagnetic interaction constant ( 137142 ≈πe ), so that 101≈η≈α ge . The
Ginzburg-Landau free energy in the presence of an external field takes the following form [21]:
( ) ( ) ( )( ) ( ) ,
8
1
8
1222
2
2222
442
2221
22
yxxTxT BBAiqKAiqK
Frrrr
π+
π+χ+∇+Φ−∇+
+χ+Φβ+χ+Φβ+χ+Φα=
(6)
where 643 22 egq += is the “new” charge of the Cooper pair, xx ABrr
rot= is the induction of the massive field, and
yy ABrr
rot= is the induction of the massless field. The coefficients 21 , , ββα~ , and KT are given by
( ) ( ) ( )( )
( ),38
373, ln34
221 µπζ==β=βµ=α NT
KTTNc
Tc (7)
with ( ) ( )( )22 3213 µπ=µN being the density of states on the Fermi surface, ( )3ζ is the Riemann zeta function, and
m is the chemical potential. Minimizing the free energy (6), we obtain the equations for the ordering parameter
( ) ( ) , 02222 22
221
2=ΦΦβ+Φχ+Φβ+Φα+Φ−∇− xT AiqK
r
(8)
( ) ( ) , 022 22
221
2=χχβ+χχ+Φβ+χα+χ+∇− xT AiqK
r
(9)
and Maxwell’s equations for the mixed fields
68
( ) ( )[ ]( ) , 416
28rotrot222
xT
Tx
AqK
qiKAr
r
χ+Φπ−
−χ∇χ−χ∇χ−Φ∇Φ−Φ∇Φπ= ∗∗∗∗
(10)
. 0rotrot =yAr
(11)
The following equations for the magnetic field Ar
and the gluomagnetic field 8Ar
can be derived from Eqs. (10) and (11):
( ) ( )[ ]( )
, cossin42
2sinsinrotrot 8
22
22 αα−χ+Φ
χ∇χ−χ∇χ−Φ∇Φ−Φ∇Φα=α+λ∗∗∗∗
Aq
iAAq
rrr
(12)
( ) ( )[ ]( )
, cossin42
2coscosrotrot
22
2882 αα−χ+Φ
χ∇χ−χ∇χ−Φ∇Φ−Φ∇Φα=α+λ∗∗∗∗
Aq
iAAq
rrr
(13)
where qλ is the penetration depth of the gauge fields Ar
and 8Ar
, which equals
( ) . 44 22 χ+Φπ=λ Tq Kq (14)
For the CFL-phase, Ak=χ=Φ 2 . Equations (12) and (13) can be rewritten in the form
, cossin6
sinsinrotrot 8822 αα−
ϕ∇α=α+λ A
qAAq
rrr
(15)
. cossin6
coscosrotrot 82882 αα−
ϕ∇α=α+λ A
qAAq
rrr
(16)
Here Ak and qλ are, respectively, equal to [19]
( ) , 64, 621
TAqA Kqkk π=λβα−= (17)
where 321 β+β=β . Note that Ak minimizes the free energy (6) in the absence of a field.
3. Solution of the Ginzburg-Landau equations for the potentials
Assuming that the CFL-condensate is a homogeneous type II superconductor, we rewrite Eqs. (15) and (16) in the
form
, cossinsinsinrotrot 822 αα−α=α+λ AfAAq
rrrr
(18)
, cossincoscosrotrot 2882 αα−α=α+λ AfAAq
rrrr
(19)
where ( ) qrf 68r
r
ϕ∇= . Note that the expression ( )rr
8rotgradϕ is proportional to the constant average density of vortical
filaments. Thus, the function fr
obeys the equation
. 0rotrot =fr
(20)
69
Note that Eqs. (18) and (19), as well as the corresponding equations for xAr
and yAr
in Ref. 21, are the same as the
equations in Refs. 12 and 13 when xAr
and 8Ar
are replaced by xAr
− and 8Gr
− , so these changes lead to the same
definitions of the mixed fields.
Let us introduce the new vector potentials ∗Ar
and ∗8Ar
in the form
, cos2
, sin2
88
α−=
α−= ∗∗ f
AAf
AA
r
rr
r
rr
(21)
and rewrite Eqs. (18) and (19) in the form
, cossinsinrotrot 822 αα−=α+λ ∗∗∗ AAAq
rrr
(22)
. cossincosrotrot 2882 αα−=α+λ ∗∗∗ AAAq
rrr
(23)
Using Eq. (22) we find ∗8Ar
:
. cossin
sinrotrot 228
ααα+λ
−=∗∗
∗ AAA
q
rr
r
(24)
Equations (23) and (24) yield the following equation:
. rotrotctgrotrot 8 ∗∗ α= AArr
(25)
Let us now consider the equivalent system of Eqs. (24) and (25) instead of the system of Eqs. (22) and (23). Substituting
∗8Ar
from Eq. (24) in Eq. (25), for the case const=α we obtain
, 0rotrot2 =+λ ∗∗ MMq
rr
(26)
where
. rotrot ∗∗ = AMrr
(27)
Thus, we can determine the function ∗Ar
by solving the system of Eqs. (26) and (27). Then we can find the electromagnetic
potential Ar
and the gluomagnetic potential 8Ar
from Eqs. (24) and (21). In order to find the distribution of the potentials
Ar
and 8Ar
inside a superconducting quark core lying in an external magnetic field, we require that the following
conditions be satisfied at the boundary between the quark core and the hadronic phase: continuity of the magnetic field
and vanishing of the gluomagnetic field ( 08 =Ar
) owing to the gluon confinement condition. Thus, the magnetic
induction Br
and the gluomagnetic potential 8Ar
must be finite in their domains of existence.
Suppose that a neutron star of radius R has a spherical core of radius a consisting of a color superconducting quark
material surrounded by a spherical layer of hadronic matter of thickness R - a. Because of the symmetry of the problem,
AMrr
,∗, and 8A
r
have only ϕ components in spherical coordinates ( ϕϑ , ,r ): ( ) ( )ϑϑ ϕ∗ϕ , , , rArM and ( )ϑϕ ,8 rA . To solve
Eq. (26) we make the substitution ( ) ( ) ϑ=ϑ ϕ∗ϕ sin , rMrM . Then Eq. (26) can be rewritten in the form
( ) ( )( ) . 0
122222
2
=
λ+−+ ϕ
ϕϕ rMrdr
rdM
rdr
rMd
q(28).
The solution of Eq. (28) can be written as
70
( ) . 111
212
λ
+′+
λ
−′= λ−
λϕ
r
q
r
q
er
cer
cr
rM (29)
From the condition that ( )rMϕ equals zero at the center of the quark core, we have 21 cc ′−=′ , so that
( ) . chsh21
λλ
−λ
=ϕqqq
rrr
r
crM (30)
Substituting the solution (30) in Eq. (27), for ∗Ar
we obtain the following solution:
( ) ( ) . sin , , 0 ϑ′+ϑ=ϑ ∗ϕ
∗ϕ rcrMrA (31)
On substituting the solution (31) in the definition (21) of the potential we have the following expression for the
electromagnetic potential:
( ) ( )( )
. sin2
,sin , , 0 α
ϑ+ϑ′+ϑ=ϑ ϕ∗
ϕϕrf
rcrMrA (32)
Solving Eq. (20), we find the function ..... . Using Eqs. (21) and (24), we obtain an expression for the gluomagnetic
potential of the form
( ) ( ) . sincos2
tgctg , 00
8 ϑ
α+α⋅′−α=ϑ ϕϕ
rcrcrMrA (33)
We determine the constant 0c′ from the gluon confinement condition at the surface of the quark core, ( ) 0 ,8 =ϑϕ aA . For
0c′ we then have
( ).
sin2ctg 02
0 α+α=′ ϕ c
a
aMc (34)
Substituting 0c′ in Eqs. (32) and (33), we obtain the final expressions for the electromagnetic and gluomagnetic potentials:
( ) ( ) ( ) , sinsin
ctg , 02 ϑ
α+α+=ϑ ϕϕϕ
rcaM
a
rrMrA (35)
( ) ( ) ( ) . sinctg8 ϑα
−= ϕϕϕ aM
a
rrMrA (36)
Note that the electromagnetic potential in the hadronic phase of the neutron star is obtained by replacing the London
penetration depth qλ in the solution (29) by the corresponding depth pλ for the hadronic matter.
4. The components of the magnetic field of a neutron star
The components of the magnetic field in the quark and hadronic phases can be found in the form ABrr
rot= using
the vector potentials. In spherical coordinates we have the following expressions for the field components:
71
( )( ) ( )( ). ,1
, sin ,sin
1rrA
rBrA
rBr ϑ
ϑ∂∂−=ϑϑ
ϑ∂∂
ϑ= ϕϑϕ (37)
On substituting the expression (35) for ( )ϑϕ ,rA in these formulas, we obtain the following components (for ar ≤ ) of
the magnetic field for the quark core:
( ) ( ), cos
sin
2ctg
22 02 ϑ
α
+α+= ϕϕ c
a
aM
r
rMBq
r (38)
( )( ) ( ). sin
sin
2ctg
21 02 ϑ
α
+α+−= ϕϕϑ
c
a
aMrrM
dr
d
rBq
(39)
where ( )rMϕ is given by Eq. (30).
The magnetic field in the hadronic phase is found using the solution (29) and taking into account the fact that
the protonic vortical filaments generate a homogeneous average magnetic field of amplitude B, parallel to the axis of
rotation of the star [22,23]. The components of the magnetic field in the hadronic phase, Bp ( Rra ≤≤ ), are given by
( ), cos
2ϑ
+= ϕ B
r
rABp
r (40)
( )( ) , sin1 ϑ
+−= ϕϑ BrrA
dr
d
rBp
(41)
where
( ) . 1123
22 pp r
p
r
p
er
r
ce
r
r
crA
λ−λϕ
λ
++
λ
−= (42)
The external magnetic field of the neutron star, eB ( Rr ≥ ), is dipole in character, with components
, sin, cos2
33ϑ=ϑ= ϑ
rB
rB ee
rÌÌ
(43)
where M is the total magnetic moment of the star. The constants c0, c
1, c
2, c
3, and Ì in Eqs. (38)-(43) are determined
from the continuity conditions at r = a and r = R and from the condition
, 3
821 Ì
π=+ BVVBq(44)
where qB is the z component of the magnetic field in the quark phase of volume V1, while V
2 is the volume of the hadronic
phase. We assume that the magnetic field in both phases is constant and parallel to the axis of rotation of the star, z.
As we shall see below, this assumption is satisfied with great accuracy, since a, R, and R - a are much greater than pλ
and qλ .
The continuity conditions for the components of the magnetic field at r = a and r = R lead to the following
equations:
72
( ) ( ) ( ),
2
sin
2ctg
2202 B
a
aAc
a
aM
a
aM+=
α+α+ ϕϕϕ
(45)
( )( ) ( )( )( ) ,
1
sin
2ctg
21 02 BrrAdr
d
r
c
a
aMrrM
dr
d
r arar+=
α+α+
=ϕϕ
=ϕ (46)
( ),
223R
BR
RA Ì=+ϕ(47)
( )( ) . 1
3RBrrA
dr
d
r Rr
Ì−=+=ϕ (48)
Substituting ( )rMϕ and ( )rAϕ from Eqs. (30), (35), and (42) in the system of Eqs. (45)-(48), and solving this system
subject to the fact that the radii a and R and the thickness of the spherical layer R-a are much greater than the London
lengths pλ and qλ , we obtain expressions for the constants c0, c
1, c
2, c
3, and Ì:
,
shsin2
2
1
qp
qp
q
a
D
ac
λ
α+
λλλ
λ−=
(49)
,
shsin2 2
2
pp
q
Rp
aR
De
ac
p
λ−
α+
λλ
λ=
λ−
(50)
,
shsin2 2
3
pp
q
Rp
aR
De
ac
p
λ−
α+
λλ
λ−=
λ
(51)
.
sinsh2 2
3
α+
λλ
λ−
−=
p
q
p
DaR
a
RBRÌ
(52)
where
. sinsin2
30
23
α−α= acBa
D (53)
To determine the constant c0 we shall assume that the induction B is constant in the hadronic phase. B originates in the
protonic vortical filaments generated by entrainment currents [23].
Let us consider magnetic field distribution at distances r much greater than pλ and qλ . We also use the facts
that apq <<λ<<λ , Rpq <<λ<<λ and aRpq −<<λ<<λ . Then the components of the magnetic field in the quark
phase ( ar ≤ ) are given by
73
( ) ( ), cos
sin
2ctg
2
sin1
2 02
22
ϑ
α+α+
αλλ
+= ϕ
λ−− c
a
aMe
ar
DB
q
p
raqr
q
(54)
( ) ( ). sin
sin
2ctg
2
sin
02
2
ϑ
α+α+
α+λλλ
−= ϕλ−−
ϑc
a
aMe
ar
DB
p
q
ra
p
(55)
In the hadronic phase ( Rra ≤≤ ) the magnetic fields are given by
( ), cos
sin22
ϑ
+α+
λλ
=λ−−
Be
ar
DB
p
q
arpr
p
(56)
( ). sin
sin2 2
ϑ
+α+
λλλ
=λ−
ϑ Be
ar
DB
p
q
ar
p
pp
(57)
The components of the external field ( Rr ≥ ) can be written in the form
( ),
cos
sin
23
2
3
r
De
a
RBrB
p
q
aRer
p ϑ
α+λλ
−=λ−−
(58)
( ).
sin
sin2 3
2
3
r
De
a
RBRB
p
q
aRe
p ϑ
α+λλ
−=λ−−
ϑ (59)
As can be seen from Eqs. (54)-(57), the magnetic fields in the quark and hadronic phases depend on the coordinate r only
near the phase boundary r = a. Since the penetration depths pλ and qλ are small compared to a and r - a, the variable
terms in these fields are nonzero only in a thin layer pq λ+λ near the surface of the quark core, so that the magnetic
field in both phases is constant and parallel to the axis of rotation:
( ), cos
sin
2ctg
2 02 ϑ=ϑ
α
+α= ϕ cosBc
a
aMB qq
r (60)
74
( ), sinsin
sin
2ctg
202 ϑ−=ϑ
α+α−= ϕ
ϑqq B
c
a
aMB (61)
, cosϑ= BBpr (62)
. sinϑ−=ϑ BBp (63)
Similarly, it follows from Eqs. (58) and (59) that the total magnetic moment of the star 23BR=Ì . Substituting this
value of the magnetic moment in Eq. (44), we obtain
( ).
sin
2ctg
202 B
c
a
aMBq =
α+α= ϕ
(64)
Solving the system of Eqs. (53) and (64), we find the constants c0 and D to be
. 2
sin, 0 0
α== BcD (65)
Thus, the constant c0 in the expression for ( )ϑϕ ,rf is determined by the constant magnetic field B in the hadronic phase,
which is generated by protonic vortical filaments.
In this approximation the magnetic field Br
penetrates from the hadronic phase into the quark phase by means of quark
vortical filaments. The transition region is of thickness on the order of pq λ+λ , so that the constant D is a small quantity
of order ( ) apq λ+λ , and the condition D = 0 is well satisfied.
Note that in this approximation the gluomagnetic field is given by
( ) ( ), cosctg
2
sin1
2
22
ϑα
−
α+
= ϕλ−−
a
aMe
ar
DK
q
p
raqr
q
ë
ë (66)
( ) ( ). sinctg
2
sin1 2
ϑα
−
α
λλ
+λ
−= ϕλ−−
ϑ a
aMe
ar
DK
q
p
ra
q
(67)
In this approximation, Eqs. (30), (49), and (65) yield the following relation:
( ). 0
sin1 23
=
α
λλ
+=ϕ
q
pa
D
a
aM
(68)
Thus, the gluomagnetic fields vanish in the quark phase.
5. Conclusion
75
We have solved the Ginzburg-Landau equations for the magnetic and gluomagnetic gauge fields in a color
superconducting core of a neutron star containing a diquark CFL-condensate. The interaction of the diquark CFL-
condensate with the magnetic and gluomagnetic gauge fields has been taken into account in these equations. The problem
was solved subject to the correct boundary conditions: continuity of the components of the magnetic field and the
conditions for gluon confinement. We have also determined the distribution of the magnetic field in the hadronic phase
(the npe-phase), taking into account the fact that a magnetic field is generated by the entrainment of superconducting
protons by superfluid neutrons. We have shown that this field penetrates into the quark core by means of quark vortical
filaments owing to the presence of screening Meissner currents.
We acknowledge with thanks the financial support of CRDF/NFSAT, grant No. ARP2-3232-YE-04.
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