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Vivian de la Incera
University of Texas at El Paso
DENSE QUARK MATTER IN A MAGNETIC FIELD
CSQCD IIPeking University, Beijing
May 20-24, 2009
Color Superconductivity
CS in a Magnetic Field
Magnetic Phases: MCFL, PCFL
Conclusions
OUTLINE
The biggest puzzles lie in the
intermediate regions
RHIC
Crystalline CS, Gluonic Phases, other?
Magnetic Field
QCD Phases
?
4
At the core
Super-High Densities (~ 10 times nuclear density)
Relatively Low Temperatures (T < 10 MeV)
High Magnetic Fields (probably larger than B~ 1015–1016G for core of magnetars)
NEUTRON STARS
plus
Attractive interaction
s
Cooper instability
at the Fermi surface Asymptoti
c freedom
Formation of Quark-Quark Pairs: Color
Superconductivity
COLOR SUPERCONDUCTIVITY
Bailin & Love, Phys Rep. ‘84
Diquark condensate
O = O Dirac⊗ O flavor⊗ O color
1 2 3
Rapp, Schafer, Shuryak and Velkovsky, PRL’98
Alford, Rajagopal and Wilczek, PLB ’98
If density great enough, Ms can be neglected and
6
COLOR–FLAVOR LOCKED PHASE
7
All quark pair. No gapless fermions, no massless gluons.
Color superconductivity is more robust than conventional superconductivity (no need to resort to phonons). Hence is a high Tc superconductor.
Chiral symmetry is broken in an unconventional way: through the locking of flavor and color symmetries.
CFL MAIN FEATURES
d
s
u
d
d
u
u
s
s
A
8G
A
8cos sin A A G 8 8sin cos G A G
ROTATED ELECTROMAGNETISM
u u ud d ds s s
0 0 -1 0 0 -1 1 1 0
- CHARGES
All -charged quarks have integer chargesQ
QThe pairs are all -neutral, but the quarks can be neutral or chargedQ
ROTATED CHARGES
CFL SCALES
At very large densities
MAGNETISM IN COLOR SUPERCONDUCTIVITY
Can a magnetic field modify the Can a magnetic field modify the Pairing Pattern? Pairing Pattern?
Can the CS produce a back reaction Can the CS produce a back reaction of the magnetic field?of the magnetic field?
Can a color superconductor generate Can a color superconductor generate a magnetic field?a magnetic field?
Color Superconductivity & B
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 in a Rotated Magnetic Field
MCFL AnsatzMCFL
and S A only get contributions from pairs of neutral quarks
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 quarksFerrer, V.I. and Manuel, PRL’05, NPB’06
0 0 01 1 11
[ , ] [2
]xy
S SI S
00
0
C C C
where the Gorkov fields separate by their rotated charge as
and the corresponding
Gorkov inverse propagators
and
contain the gaps:
(0)
( )
( )
0 0MCFL
MCFL
MCFL
, NAMBU-GORKOV FIELDS IN NONZERO B
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
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
GAP EQUATIONS at LARGE MAGNETIC FIELD
2
2
2
2 2
0.3, 0.2
~
~ B
1A8
A
g
=3/2, ,
1
eB
B 0
yx
=
yx
G
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, PRL’05, NPB’06
GAP SOLUTIONS at LARGE MAGNETIC FIELD
CFL VS MCFL
• 9 Goldstone modes: charged and neutral.
• 5 Goldstone modes: all neutral
• Low energy CFL similar to low density hadronic matter. Schafer & Wilzcek, PRL’99
• Low energy MCFL similar to low density hadronic matter in a magnetic field.
Ferrer, VI and Manuel, PRL’05 NPB’06
SU(3)C × SU(3)L × SU(3)R × U(1)B SU(3)C × SU(2)L × SU(2)R × U(1)B × U(1)A
8 1 : 8( ) , ( ) 21
221 12 2
221 12 2
3 4 1 1 : 3 4( ) , ( ) ,
( )1
1
8
( ) 8
B
B
B
A
A A
A A
A
A
A
B = 0 B 0
LOW ENERGY CFL THEORY IN A MAGNETIC FIELD
Ferrer & VI, PRD’07
Showing that the charged Goldstone bosons acquire a magnetic-field-induced mass
The dispersion relations for the charged Goldstone bosons is
Ferrer & VI, PRD’07
LOW ENERGY THEORY IN A MAGNETIC FIELD
For a meson to be stable its mass should be less than twice the gap,
otherwise it could decay into a particle-antiparticle pair. Hence,
CFL MCFL crossover
HAAS-VAN ALPHEN OSCILLATIONS OF THE GAP AND MAGNETIZATION
Noronha and Shovkovy, PRD’07
Fukushima and Warringa, PRL’08
1G 3G
2G
Because of the modified electromagnetism, gluons are charged in the color superconductor
G G
I I
0 0 0 1 -1 1 -1 0
8G
Fields bigger than the square Meissner gluon mass induce an instability which is removed by the formation of a paramagnetic vortex state
Ferrer & VI, PRL’06
B~
PCFL
PARAMAGNETIC CFL
CFL:
SU(3)C SU(3)L SU(3)R U(1)B U(1)e.m. SO(3)rot
SU(3)C+L+R U(1)e.m SO(3)rot
MCFL:
SU(3)C SU(2)L SU(2)R U(1)B U(-)(1)A U(1)e.m SO(2)rot SU(2)C+L+R U(1)e.m SO(2)rot
8 1 : 8( ) , ( ) 21
221 12 2
221 12 2
3 4 1 1 : 3 4( ) , ( ) ,
( )1
1
8
( ) 8
B
B
B
A
A A
A A
A
A
A
PCFL: gluon condensate G4i iG5
i & induced
SU(3)C SU(2)L SU(2)R U(1)B U(-)(1)A U(1)e.m SO(2)rot
SU(2)C+L+R U(1)e.m
PHASES IN THREE-FLAVORS THEORY
Rapp, Schafer, Shuryak& Velkovsky, PRL’98 Alford, Rajagopal and Wilczek, PLB ‘98
Ferrer, V.I. and Manuel PRL’05; NPB ’06
Ferrer & V.I. PRL ’06
B~
Chromomagnetic
Instability
E.J. Ferrer and V.I. Phys.Rev.D76:045011,2007
MAGNETIC PHASES AT HIGH DENSITY PHASES AT HIGH DENSITY
Supernova remnants associated with magnetars should be an order of magnitude more energetic, but
Recent calculations indicate that their energies are similar.
When a magnetar spins down, the rotational energy output should go into a magnetized wind of ultra-relativistic electrons and positrons that radiate via synchrotron emission.
So far nobody has detected the expected luminous pulsar wind nebulae around magnetars.
Possible Alternatives:
B can be boosted (Ferrer&
VI, PRL’06) or even induced (Ferrer& VI, PRD’07; Son and
Stephanov, PRD’08) by a CS core
DIFFICULTIES OF THE STANDARD MAGNETAR MODEL
Neutron stars provide a natural lab to explore the effects of B in CS
What is the correct ground state at intermediate densities? Is it affected by the star’s magnetic field? Inhomogeneous Gluon Condensates, other field-related effects…
Explore possible signatures of the CS-in-B phase in neutron stars
CONCLUSIONS
SB and Dynamical Anomalous Magnetic Moment
Once the chiral symmetry is broken by the chiral condensate, any structure that breaks chiral symmetry is allowed in the full propagator. In the presence of a magnetic field, there will be then a dynamically generated mass and a dynamically generated anomalous magnetic moment.
Let us consider a chiral theory (Massless QED, NJL etc.)
Induced Zeeman Effect
MCFL AnsatzMCFL
and S A only get contributions from pairs of neutral quarks
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
It seems to be a profound connection between magnetism and color superconductivity. More work needs to be done to explore this association at a deeper level and to establish a link between theory and astrophysical observations.
Connections between MCFL/PCFL and Quark-Nova Mechanism?
Ouyed et al. (this conference)
OUTLOOK