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

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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

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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

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( )

( )

(

( )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

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Gap Equations

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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