Superconductivity in a Magnetic Field

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15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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The Single Flavor Color
Superconductivity in a Magnetic Field

Bo Feng

De
-
fu Hou

Hai
-
cang Ren

Ping
-
ping Wu

Institute of particle physics ,CCNU


1



Phys. Rev. Lett. 105, 042001 (2010)




Motivation to study single flavor CSC


Ground state for single flavor CSC


Ground state for single flavor CSC in a
magnetic field


Conclusion and outlook

2

Outlines

3

QCD
T

  
QCD at arbitrarily high density



No reliable lattice results at
finite density


No reliable effective models of
dense QCD


Complications due to charge
neutrality and equilibrium


Difficulties in determining
stable ground state


Ground state of dense quark matter

(i)
Deconfined quarks( )

(ii)
Pauli principle (s=1/2)

(i)
Effective models( )

(ii)
One
-
gluon exchange( )

Cooper instability

Color superconductivity

5
( ) 0
C
i j
 
 
 
QCD

 
QCD

 
QCD

 
Color superconductivity

B. Barrois, NPB 129, 390 (1977)

D. Bailin and A. Love, Phys. Rep. 107,325 (1984)

M. Alford et al., PLB 422, 247 (1998)

R. Rapp et al., PRL 81, 53 (1998)

5



1
f
N

,,,
r r g g
u d u d

BCS

type paring:


J=0:


2SC:


CFL:




J=1:






Non
-
BCS

type paring:


gapless

phases


LOFF


……

all quarks participate in the paring


Phase structure of CSC

M. Alford, K. Rajagopal and F. Wilczek, NPB 537, 443 (1999)

T. Schaefer, PRD 62, 094007 (2000)

A. Schmitt, PRD 71, 054016 (2005)

Shovkovy and M. Huang, PLB 546, 205 (2003)

M. Alford et al., PRL 92, 222001 (2004)

M. Alford et al., PRD 63, 074016 (2001)

…….

:Pairing two quarks with same flavor

6


equilibrium:





Non
-
zero strange quark mass:



Charge neutrality:

e
d u e


  
e
u e d


  
d u e
  
 
8
0
Q
n n
 
Fermi

momentum

Mismatch

The single flavor CSC (I)


CSC in moderate density:

M. Alford et al., Rev. Mod. Phys. 80, 1455 (2008)

Single flavor CSC show up

7


spherical


mixed

phase


CSL


non
-
spherical


longitudinal and transverse
pairing states of


polar ,planar ,A phase

The most stable phase

The single flavor CSC (II)

A. Schmitt, PRD 71, 054016 (2005)

Angular momentum mixing

The most stable state is the transverse CSL even
counting angular momentum mixing.

A. Schmitt, PRD 71, 054016 (2005)

BF D
-
f Hou and H
-
c Ren, NPB 796, 500 (2008); NPB 813, 408 (2009);

J Phys. G 36, 045005 (2009); in preparation

9

Neutron star

Webber, astro
-
ph/0407155

10



The nonzero strange quark mass



The typical density inside a neutron star is
about 500MeV



A magnetic field is present in a compact star
and could reach G in magnitude


CSC in a compact star

15
10
P. Reuter, PRD 74, 105008 (2006)

T. Schaefer, PRD 62, 094007 (2000)

In CSL, there is an electromagnetic Meissner effect.

In a nonspherical condensate , the Meissner effect is incomplete.

Ground state for single flavor CSC

In a magnitude field



3
4
l l
d r G T T
 
     
 
    
 

Only pick up the dominant pairing channel
----
the transverse pairing,

The Hamiltonian given by a NJL
-
like action in the ultra
-
relativistic limit:

1
2
l l
T


Under mean field approximation:

2
2 1
( ) ( )
2 9
ln 1 exp
4
4
ln 1 exp
K
K K
K
K
K
p k E k k
k
T
T
E
T
T
  

       
 
 
  
    
 
 
 
 
 
 
 
 
  
 
 

 
 
 






2
2 2
K
E k f
 
  
here



12


Given by the solution of the gap equation

0
p



M.Alford and G.Cowan,J.Phys.G.G:32,511
-
528(2006)

c c
C
 
 
The condensate of a diquark operator takes the form
:



f

The function

is given by:

For CSL

( ) 1
f


3
( ) sin
2
f
 

2
( ) 3cos
2
f





2
3
( ) 1 cos
4
f
 
 
For polar

For A

For planar

Introducing

2 2
0
2
( )
2
s s n s
p p p T




   
We have

(0) 1,(0) 0.98,(0) 0.88,(0) 0.65
CSL planar polar A
   
   
and

0
( ) 0,
E
s c c
e
T T



  
13

8
2
i
iq
e
 
 
 

2 3
4 3
 
 
 

For polar and A


For planar

tan 2 3 (/)
tan 2 3 (/)
q e g
q e g




For polar and A


For planar

8
tan
A A
 


8
tan
B B

 
14

0
v

The condensate
under U(1) transformation


is related to the electromagnetic field
A
and the 8
-
th component
of the color field in the normal phase through a
U
(1) rotation

8
A
G BH



2
8
2
1
1 1
2 2
l
l
B B p

   

2
2 2
1
2
1
cos
2
n
CSL
n i
G p H
G p
G p H

  
 
  
For normal phase


For CSL


For i=polar,A,planar

n A polar planar CSL
p p p p p
   
15

The Gibbs free energy density:

0
0
u
d
m
m


inf
s
m inity

0
s
m

By balancing the Gibbs free energy of different phases,

we obtain the phase diagram with respect to temperature

and magnetic field.

0
0
H




5
2 2
2
4
0
5
2 3 4 9
512 exp
8 2
2
f
N g
g
  

 
 

    
 
 
 
 
 
500
MeV


1
s


16

T. Schaefer, PRD 62, 094007 (2000); Phys.Rev.D60,114033(1999)

A.
Schmitt, PRD 71, 054016 (2005)

D.T.Son,Phys.Rev.D59,094019(1999)

R.D.Pisarski et al., Phys.Rev.D61,074017(2000)

W.E.Brown et al., Phys.Rev.D62,054016(2000)

we have explored the consequences of the incomplete
Meissner effect of the nonspherical CSC phase of single
flavor pairing.


We found that with the increasing magnetic field, the ground
state of single flavor CSC is no longer CSL phase, but the
phase combination of nonspherical phases.




Conclusions and outlooks

Possible phenomenological implications and the influence of
the nonzero strange quark mass for the ground state of
single flavor CSC are interesting topics deserving further
investigations.

Thanks!

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