Vivian de la Incera

University of Texas at El Paso

THE ROLE OF MAGNETIC FIELDS IN DENSE QUARK MATTER

Color Superconductivity

Color Superconductivity in a Magnetic Field: Magnetic CFL

Magnetic-Field-Induced Gluon Condensate: Paramagnetic CFL

Chromomagnetic Instabilities at Intermediate Densities (unstable

gapped 2SC)

Solution to the CI in 2SC: Spontaneous Generation of GC and B

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 on the magnetic field?on 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

Charged Gluon Sector of Mean-Field Effective Action in CFL:

EJF & de la Incera, PRL 97 (2006) 122301

MAGNETIC EFFECTS ON THE GLUONS

Assuming that there is an external magnetic field in the z-direction, one mode becomes unstable when

H2

MH m

with corresponding eigenvector:

“Zero-mode problem” for non-Abelian gauge fields whose solution is the formation of a vortex condensate of charged spin-1 fields.

Nielsen & Olesen NPB 144 (1978)

Skalozub, Sov.JNP23 (1978);ibid 43 (1986)

Ambjorn & Olesen, NPB315 (1989)

1 2( , ) (1, )G G G i

MAGNETIC FIELD INDUCED INSTABILITY IN CHARGED SPIN-ONE FIELDS

Minimum Equations:

Magnetic Antiscreening

+

24

PARAMAGNETIC CFLPARAMAGNETIC CFL

H < Hc H ≥ Hc

H < Hc H ≥ Hc

Color Color SuperconductoSuperconducto

rr

Conventional Conventional SuperconductorSuperconductor

25

MCFLPCFL

DIFFERENT BEHAVIOR in a B

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

Color Neutrality and beta equilibrium

Unstable Gapped 2SC

a=1,2,3 masslessa=4,5,6,7 negativea=8 positive

Gapless 2SC

a=1,2,3 masslessa=4,5,6,7 negativea=8 negative

Stable Gapped 2SC

a=1,2,3 masslessa=4,5,6,7 positivea=8 positive

Gluons Masses

1 2 2

Huang/Shovkovy, PRD 70 (2004) 051501

CHROMOMAGNETIC INSTABILITIES IN 2SCCHROMOMAGNETIC INSTABILITIES IN 2SC

At Tachyonic Mode of Charged Gluons

µ8

CHROMOMAGNETIC INSTABILITIES IN 2SC

charged gluons

8th gluon

Huang/Shovkovy, PRD 70 (2004) 051501

- 8

EFFECTIVE ACTION for CHARGED GLUONS

GLUON CONDENSATE AND INDUCED MAGNETIC FIELD

Solutions:

The gluon condensate generates a magnetic field

E.J. Ferrer and V.I. , Phys.Rev.D76:114012, 2007 .

2g-1

2q2

2Mm

22 0q G B

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

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?

(CSQCD II conference)

OUTLOOK