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Angular momentum mixing in non-sphericalcolor superconductors

Defu Hou Central China Normal University, Wuhan

Collaborators: Bo Feng , Hai-cang Ren

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• Color Superconductor (CSC) & complex gap

• Angular momentum mixing in non-spher. CSC

• Ground state of single flavor CSC

• Summary and outlooks

Outlines

• B. Feng, D-f Hou J-r Li and H-c Ren, Nucl.Phys. B 754, 351 (2006)• B. Feng, D-f Hou and H-c Ren, Nucl.Phys. B 796, 500 (2008)• B. Feng, D-f Hou and H-c Ren, Nucl.Phys. B 813, 408 (2009)• B. Feng, D-f Hou and H-c Ren, J. Phys. G 36, 045005 (2009)

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QCD Phase diagram

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• Lattice calculation not reliable • High density effective theory• Complications due to charge neutrality and

\beta equilibrium • What is the ground state of dense QCD

QCDT

Dense QCD

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(i) Deconfined quarks( )(ii) Pauli principle(s=1/2)

(i) Effective models( )(ii) One-gluon exchange( )

Cooper instability

Color superconductivity

5( ) 0Ci j

QCD

QCD QCD

Color superconductivity

Ground state of dense quark matter is CSC

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)

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• BSC-like pairing

2SC: u_r, d_r, u_g, d_g

CFL: all flavor and color

• Non-BCS pairing

gapless CSC

LOFF

……

Phase structure in CSC

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

Shovkovy and M. Huang, PLB 546, 205 (2003)M. Alford et al., PRL 92, 222001 (2004)M. Alford et al., PRD 63, 074016 (2001)…….

J=0:

J=1: N_f=1

T. Schaefer, PRD 62, 094007 (2000)A. Schmitt, PRD 71, 054016 (2005)

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● Dispersion relation:

2 2( )kE k

● BCS theory

Real gap function

Gap function

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• Eliashberg theory: energy depend. With imaginary part

Eliashberg theory

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• QCD single-gluon exchange potential

• Gap is E depend. with an imaginary part

T L

HDL Resummed Gluon Propagator

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Gap function

[Son 1999; Schafer,Wilczek 2000; Hong et al. 2000; Pisarski,Rischke 2000; Brown et al 2000; Bron, Liu,Ren 2000, Schmitt,Wang,Rischke 2003]

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• 2SC gap eq.

Gap Equation

R. Pisarski and D. Rischke, PRD (2000)

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0Im ( ) 0,

EQ of RP

EQ of IP:

:

Complex Gap Equation

BF, D-f Hou, J-r Li and H-c Ren NPB (2006), P. Reuter, PRD (2006);

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CSC at moderate density:

• Beta EQL.

• Non-zero s quark mass

• Charge neutrality

ed u e

eu e d d u e

8 0Qn n

Mismatch

Single flavor of CSC(I)

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J=1 pairing

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• Spherical states

all mixed states

CSL• Non-spherical states

polar, planar and A phases

in both transv. and long.

Most stable state

Angular momentum mixing

A. Schmitt, PRD 71, 054016 (2005)

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• Helium_3

• QCD

11( ) (cos ) (2 1) (cos ) 3l

l ll

V k k V l V P V k k

Pairing potential:

Nonlinear gap equation:

Angular momentum mixing 0 0 1ˆ ˆ(0, ) ( ) (cos )p f p P

W. Brown, J. Liu and H-c Ren, PRD 61, 114012 (2000); PRD 62, 054013 (2000); PRD 62, 054016 (2000)

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1 1 1 12

1[ , ] { ln ( 1) ln ( ) 2 [ , ]

2D S Tr D Tr D D Tr S Tr S S D S

The two-loop approximation to \gamma_2

Powers of TStationary points

Order of g^2mu^4

CJL effective action(I)

D. Rischke Prog. Part. Nucl. Phys. 52 197 (2004)

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L. Propaga.:

T. Propag:

1 10( )

GS S

G

0

0

0

0

0

GS

G

NG Propagators

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1 1 10

1[ , ] [ ln ln ( 1)]

2

n

D S Tr D Tr S Tr S S

F

Energy density of normal phase Free energy density

1 1 1 1 10 0

1[ ln ln ln ln ( 1)]

2 nF Tr D Tr D Tr S Tr S Tr S S

1 1ln ln [ ], n n nTr D Tr D Tr D

0F

Gap equationMinimization of F

CJT action(II)

BF D-f Hou and H-c Ren, NPB 796, 500 (2008); NPB 813, 408 (2009); J. Phys. G 36, 045005 (2009)

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L-pairing:

T-pairing:

Gap Equation

BF D-f Hou and H-c Ren, NPB 796, 500 (2008); NPB 813, 408 (2009); J. Phys. G 36, 045005 (2009)

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General form of gap:

0(0,cos ) (cos )m mf

Polar state: m=0 A state: |m|=1

Angular dependence

Integral eqs of gap funct:

L:

T:

2SC gap angular depend. Funct.

T. Shaefer, PRD 62, 094007 (2000); A. Schimitt, 71, 054016 PRD (2005)

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• Polar state

Angular momentum mixing(II)

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

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• A phase

Angular momentum mixing(III)

BF D-f Hou and H-c Ren, NPB 813, 408 (2009); J Phys. G 36, 045005 (2009)

Long. Transv.

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• Angular momentum mixing lowered the free energy of the non-spherical states(compare with spin-one state)

J=1 mixing

Long. ：

Transv. ：

Polar

Angular momentum mixing(IV)

A. Schmitt, PRD 71, 054016 (2005) BF D-f Hou and H-c Ren, NPB 796, 500 (2008); NPB 813, 408 (2009)

The drop amount is small (few percent) and can not make the non-sphericalstates more favored than CSL

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• Planar phase contains two antisymmetric Gell-Mann matices( \lambda_5 and \lambda_7), therefore we have two gap functions

Mixing in planar phase(I)

where:

• Integral equation for angle dependent function

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• Transv. Planar phase

Mixing in planar phase(II)

BF D-f Hou and H-c Ren, in preparation

Angular momentum mixing lowered the free energy of transv. Planar phaseby 0.99 percent

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Transv. CSL is the most stable phase even including angular momentummixing: we have proved

Ground state of single flavor CSC

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

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• Profile of

neutron star

CSC in nature

Webber, astro-ph/0407155

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• Typical chemical potential 500MeV• Nonzero strange quark mass

0sm

√

？

？

CSC inside a neutron star(I)

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• Typical magnetic field ～ 10^12G

Magnetic field effect

A. Schmitt et al., PRL 91, 242301 (2003) PRD 69, 094017 (2004)

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• de Haas-van Alpen oscillation in CFL

How about single flavor CSC? Determining the critical magnetic field in single flavor CSC!

CSC inside neutron stars(III)

J. Noronha and I. Shovkovy, PRD 76, 105030 (2007)

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• LOFF statefirst investigated by Larkin and Ovchinnikov (Sov. Phys. JETP 20, 762 (1965) )and Ful

de and Ferrell (Phys. Rev. 135. A550 (1964) )

• LOFF window

k_d k_u BCS pairing

M. Alford, et al. Phys. Rev. D 63, 074016 (2001) I. Giannakis, et al. Phys. Rev. D 66, 031501 (2002)

角动量混合

Angular momentum mixing in LOFF

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• Imaginary part of Gap function

• Angular momentum mixing reduces the free energy of nonspherical pairing states

• Effect of a strong magnetic field? m_s effect?

• Angular momentum mixing in LOFF state?

• What is its consequency for compact star physics

Summary and outlook

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35A. Schmitt, PRD 71, 054016 (2005)

Symmetry structures of Spin-1 CSC