IRGAC 2006
COLOR SUPERCONDUCTIVITY and MAGNETIC FIELD:
Strange Bed Fellows in the Core of Neutron Stars?Vivian de la Incera
Western Illinois University
Barcelona, Spain, July 11-15, 2006
IRGAC 2006
B~ 1012 – 1014 G in the surface of pulsars
B~ 1015 – 1016 G in the surface of magnetars
Neutron Stars
20R km (12 miles)
Diameter:
Mass:
30 010 , 0.15 fm
Magnetic fields:
Density:
?
• Color Superconductivity
• Magnetic Field and Color
Superconductivity
• MCFL: Symmetry, gap structure, gap
solutions
• Conclusions and Outlook
E.J. Ferrer, V.I. and C. Manuel,
PRL 95, 152002 ; NPB 747, 88. IRGAC 2006
Outline
( )
High baryon density
quark deconfined matter
Attractive
one-gluon-exchange
interactions
Cooper instability
quark-quark pairing
IRGAC 2006Color Superconductivity
Bailin and Love ‘84
IRGAC 2006
Three flavors at very high density: CFL phase
C L R BSU( ) X SU( ) X S
U
Symmetry Br
( ) X U(
eaki
)3
n
g
3 3 1
:
Pairs: spin zero, antisymmetric in flavor and color
CFLib bi
j jia a
i
C+L+R( )S 3URapp, Schafer, Shuryak and Velkovsky, ‘98 Alford, Rajagopal and Wilczek, ‘98
Magnetic Field Inside a Color Superconductor
BB8G
In spin-zero color superconductivity a linear combination of the photon and one gluon remains massless (in-medium electromagnetic field). An external magnetic field penetrates the superconductor in the form of a “rotated” field (no Meissner effect)
u u ud d ds s s
0 0 -1 0 0 -1 1 1 0
- CHARGES
All -charged quarks have integer charges All pairs are -neutralQ
Q
Q
IRGAC 2006
IRGAC 2006
Color Superconductivity & B
Will a magnetic field reinforce color superconductivity?
CFL:SU(3)C X SU(3)L X SU(3)R
X U(1)B X U(1)e.m.
SU(3)C+L+R X U(1)e.m
Rapp, Schafer, Shuryak
and Velkovsky, PRL 81 (1998)
Alford, Rajagopal and Wilczek,
PLB 422 (1998)
1 2 331 2 31 2~j j
a a ab b b bi
aji i ij
MCFL:
SU(3)C X SU(2)L X SU(2)R
X U(1)B X U(1)e.m X U(-)(1)A
SU(2)C+L+R X U(1)e.m
Ferrer, V.I. and Manuel
PRL 95,152002
1 2 3
Dominant attractive interactions in 3-flavor QCD lead to a general order parameter of the form
1 2 3
IRGAC 2006
B = 0 B 0
0 0 01 1 1
( )0 ( )0 ( )0
5 0 5
5
0
( )[ ] ( ) ( )[ ] ( ) ( )[ ] ( )
1 [ ( ) ( ) ( ) ( )
2
( ) ( ) h.c.
{
]}
B
C
xy
MCFL MCFL
MCFL
C
C
x G y x G y x G y
x y x y
x
I
y
2 3 3
0 0
0
1 2 11 3 2( , , , , , , , , )
, ,
0
s s s d d d u u
Q
u
Q
0
0
' '
(1,1,0,1,1,0,0,0,1)
(0,0,0,0,0,0,1,1,0)
(0,0,1,0,0,1,0,0,0)
1
diag
diag
d
Q
iag
( )0
( )00
10
10
[ ] ( )
[ ] ( )
G i
G i
eA
Three-flavor NJL Theory
with Rotated Magnetic Field
MCFL ansatz including subdominant interactions
MCFL
and S A only get contributions from pairs of neutral quarks
IRGAC 2006
2 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 2 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 2
S S A S A
S A
S A
S A
S A S S A
S A
S
B B
B B
B B
B B
B BA
S A
S A S
B
BA S
B
B B B
and B BS A get contributions from pairs of neutral and pairs of
charged quarks
0 0 01 1 11
[ , ] [2
]xy
S SI S
00
0
C C C
where the Gorkov fields are defined by:
The mean-field action can be written as:
and the Gorkov inverse propagators are
IRGAC 2006
( )
( )
(
( )0
( )0
)
( )
(0
0
( )0
( )0
( )0
( 0
0)
( ) )0
1
1
1
1
1
1
1
1
10
[ ] ( , )
[ ] ( , )
[ ] ( , )
[ ] ( , )
[ ] ( , )
[ ] ( , )
G x y
G x y
G x y
G x y
G x y
G x y
S
S
S
00
0
C
C C
(0)
( )
( )
0 0MCFL
MCFL
MCFL
IRGAC 2006
Gap Equations
IRGAC 2006
2 3 2
2 3 2 22 22 23 (2 ) 3 (2 )( ) 2( ) ( ) ( )
B BB
B
AA
A AB
Aeg d g
q
Bq dq
q
For fields the gap equations can be reduced to
2Be
2
2 2
3
2 3 2 2
17 7
9 94 (2 ) ( ) ( ) 2( )B
A AA
A A
g d q
q q
2 3
2 3 2 22 218 (2 ) ( ) ( ) 2( )B
A AS
A A
g d q
q q
2 2 3
2 2 2 32 22 26 (2 ) 6 (2 )( ) ( ) ( ) 2( )
A A
A
BB
S
A
B
B B
g dq g d q
q
eB
q
IRGAC 2006
2
2
2
2 2
0.3, 0.2
~
~ B
1A8
A
g
=3/2, ,
1
eB
B 0
yx
=
yx
G
Gap Solutions
2 2
2 2
3 1exp( )
1 2( )
36 21 1 2 2exp 1
17 17 (1 ) 74
1 1
4A S
A
B B
S A
BA
eg
x
y
y
x y y
B
2
2
1exp( ),
(2 2 )
2 2
3
~
2
2
B
B 2
AB
G
= , = ,
g
N N
N N
Be
G
Ferrer, V.I. and Manuel, NPB 747, 88IRGAC 2006
IRGAC 2006
The magnetic field “helps” CS. The field reinforces the gap that gets contributions from pairs of -charged quarks.
Q
1exp( )~
(2 2 )AB
BG N N
The physics behind MCFL is different from the phenomenon of magnetic catalysis. In MCFL the field reinforces the diquark condensate through the modification of the density of state
CFL vs MCFL
• 9 Goldstone modes: charged and neutral.
• 5 Goldstone modes: all neutral
• Low energy similar to low density QCD. Schafer & Wilzcek’ PRL 82 (1999)
• Low energy similar to low density QCD in a magnetic field.
Ferrer, VI and Manuel, NPB’06
IRGAC 2006
C L+ R+ 3
( )SU
( ) , ( ) 2
8 1
8 1
R+C +L(2)
SU
221 12 2
221 12 2
( ) , ( ) ,
( )
3 4 1 1
3 4
( 81
81
)
B
B
B
A
A
A
A
A A
A
A
CONCLUSIONS and OUTLOOK
Neutron stars provide a natural lab to explore the effects of B in CS
Is MCFL the correct state at intermediate, more realistic, magnetic fields? Gluon condensates?
What is the correct ground state at intermediate densities; is it affected by the star’s magnetic field?
Explore possible signatures of the CS-in-B phase in neutron stars: neutrino cooling, thermal conductivity, etc.
IRGAC 2006