INDUCED MAGNETISM IN COLOR-SUPERCONDUCTING MEDIA

Efrain J. FerrerThe University of Texas at El Paso

OUTLINE

• Magnetized CS

• Gluon Vortices and Magnetic Antiscreening in CS

• Chromomagnetic Instabilities & CondensatesG B

EJF & de la Incera, PRL 97 (2006) 122301 EJF & de la Incera, PRD 76 (2007) 045011 EJF & de la Incera, PRD 76 (2007) 114012

Compact stars in the QCD phase diagram II (CSQCD II)May 20-24, 2009, Peking University, Beijing, China

Boson: Zero Spin and opposite momenta

Electric Charge

Broken Symmetry : U(1)em

2C

23 )*(

2*)TT(*

m̂2xd][F

ud

Electric Superconductivity Color Superconductivity

ee

Color Charge

Barrois ‘77; Frautschi ’78; Bailin and Love’84; Alford, Rajagopal and Wilczek ’98; Rapp, Schafer, Shuryak and Velkovsky ‘98

Broken Symmetry : SU(3)C, U(1)em

J. Bardeen, L. N. Cooper and J. R. Schrieffer

L. D. Landau V. L. Ginzburg

Cooper Pair Condensation

d

s

ud

d

u

u

s

sA

8G

A

8cos sin A A G 8 8sin cos G A G

In-Medium Magnetic Field

Diameter:

Mass:

Magnetic fields:

Temperature:

pulsar’s surface: B~ 1012–1014G magnetar’s surface: B~ 1015–1016G

Neutron Stars

Chromomagnetic

Instability

Magnetic Phases at High Density

B=0 CFL

SU(3)C × SU(3)L × SU(3)R × U(1)B × U(1)e.m

B 0 MCFL

SU(3)C×SU(2)L×SU(2)R ×U(1)B × U(-)(1)A× U(1)e.m.

C+L+R(2 ) (1)emSU U

(3) (1)C L R emSU U 9 GB

5 GB

Magnetic CFLMagnetic CFL

1 2 3

1 2 3 E.J. Ferrer, V.I. and C. Manuel, PRL ‘05; NPB’06

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

Gluon Mean-Field-Effective Action in the CFL Phase:

Magnetic Effects on the Gluon Sector EJF & de la Incera, PRL 97 (2006) 122301

1G 3G

2G

where

G G

I I

0 0 0 1 -1 1 -1 0

8G

Effective action for the charged gluons within CFL at asymptotic densities

Charged Gluons Effective Action

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

H2MH 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 Instability for Charged Spin-1 Fields

Gibbs Free-Energy:

Minimum Equations:

In the approximation:

Minimum Equations:

=0

Magnetic Antiscreening

+

H < Hc H ≥ Hc

H < Hc H ≥ Hc

Color Superconductor

Conventional Superconductor vs Color Superconductor

Conventional Superconductor

B

HHc

ismParamagnet

Variation of internal magnetic field (B) with applied magnetic field (H) for Type I, Tipe II and Color Superconductors

The linear minimum equation for G* around the critical field is

20 2 2

2 11(2

, )

M M

eH me m

where

The solution is given by

0

2k

kx

H

Solution of the Linear Condensate Equation

The vortex lattice induces a magnetic field that forms a fluxoid along the z-direction. The magnetic flux through each periodicity cell in the vortex lattice is quantized

Vortex lattice, First image 1967

From the experience with conventional type II superconductivity, it is known that the inhomogeneous condensate solutions prefer periodic lattice domains to minimize the energy. Then, putting on periodicity in the y-direction:

,...,, ,/ 2102 nbnkby

The periodicity is also transferred to the x-direction:

Then, the general solution is given by the superposition:

Vortex Solution

Neutrality Conditions

)(0

B

B

J

0

e

0

8

0

3

)8(08 )2/3( Gg

)3(03 )2/( Gg

• The optimal Cooper pairing occurs when

• Matter inside a star should be electrically neutral to guarantee the star stability and to minimize the system energy

• There are no enough electrons in β equilibrium to balance the charge deficit

Cooper pairing is consequently distorted by the Fermi sphere mismatch.

Cooper Pairing and Neutrality Conditions

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 2SC

Some Suggestions

Crystalline Superconductivity Alford, Bowers & Rajagopal, PRD 63 (2001) 074016

Phases with Additional Bose Condensates Bedaque & Schäfer, NPA 697 (2002) 802

Homogeneous Gluon Condensate Gorbar, Hashimoto & Miransky, PLB 632 (2006) 305

Inhomogeneous Gluon Condensate with an Induced Magnetic Field

Ferrer& Incera, PRD 76 (2007) 114012

EJF & de la Incera, PRD 76 (2007) 114012

Chromomagnetic Instabilities & G-B Condensates in 2SC

Neutrality Conditions

)(0

B

B

J

0

e

0

8

0

3

)8(08 )2/3( Gg

)3(03 )2/( Gg

Stable Phase:

/~ , ,0 883 e

Huang/Shovkovy, PLB 564 (2003) 205

Effective Action

- 8

At Tachyonic Mode of Charged Gluons

µ8

Weakly First-Order Phase transition

Free-Energy:

Minimum Equations:

Linear Equations

Gluon-Condensate Solution

Condensate Free-Energy

2

2

~200q

mF

M

g

23

222~

200

)1( 2

2

gs

Mq

g

gm

mF

Inhomogeneous Condensate:

Homogeneous Condensate:

0.1ξ

EJF & de la Incera, hep-ph/0705.2403

Gorbar/Hashimoto/Miransky, PLB 632 (2006) 305

Fukushima, PRD 72 (2005) 074002; Casalbouni et al, PLB 605 (2005) 362; Alford/Wang JPG 31 (2005) 719.

Meissner Masses & Chromomagnetic Instabilities in Gapless-Three Flavor Quark Matter

Origin of Stellar Magnetic Fields

G10B 24EW

G10B 4

G10B 8

)Bv(

dt

Bd

.Yrs billion 12~

0Sd)B(4

1

dt

dS

S

22 TRB

.Yrs billion 1~

Supernova remnants associated with magnetars should be an order of magnitude more energetic.

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.

Difficulties of the Standard Magnetar Model

Possible Alternative:

B induced by the CS core.

Conclusionsand Future Directions

Magnetism in Color Superconductivity is totally different from magnetism in Conventional Superconductivity

Magnetism is reinforced in Color Superconductivity

Magnetars CS Cores (?) Numerically solving the nonlinear equation, looking for the

realization of the vortex state

Exploring the possibility to induce a magnetic field in a three-flavor system: vortex state in gCFL

Three flavors at very high density: CFL phase

C L R BSU( ) S

Symme

U( ) S3 3 3

try Breaking:

1U( ) U( )

C+L+R( )S 3U

Diquark condensate

O = O Dirac⊗ O flavor⊗ O color

in the CFL is:

B=0

SU(3)C × SU(3)L × SU(3)R × U(1)B × U(1)e.m

B 0

SU(3)C×SU(2)L×SU(2)R ×U(1)B × U(-)(1)A× U(1)e.m.

C+L+R(2 ) (1)emSU U

(3) (1)C L R emSU U

9 GB

5 GB

3-Flavor QCD

3-Flavor QCD

B = 0

SU(3)C × SU(3)L × SU(3)R × U(1)B × U(1)e.m

B ≠ 0

SU(3)C × SU(2)L × SU(2)R × U(1)B × U(-)(1)A ×

U(1)e.m.

Miransky & Shovkovy, PRD 66(2002)

Meissner Screening Masses

& Chromomagnetic Instabilities

in Neutral Dense Two-Flavor Quark Matter

Huang/Shovkovy, PRD 70 (2004) 051501

At Tachyonic Mode of Charged Gluons

plus

Attractive interaction

s

Cooper instability

at the Fermi surface

Baryonic Matter at Very High Density & CSBaryonic Matter at Very High Density & CS

Asymptotic freedom

Color Superconductivity

Chromomagnetic

Instability

Haas-van Alphen Oscillations of the Gaps

Fukushima and Warringa, ArXiv: 0707.3785

Haas-van Alphen Oscillations of the Magnetization

Noronha and Shovkovy, ArXiv: 0708.0307

Magnetic Phases at High Density