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Angular momentum mixing in non-spherical color superconductorsPowerPoint Presentation

Angular momentum mixing in non-spherical color superconductors

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Angular momentum mixing in non-sphericalcolor superconductors

Collaborators: Bo Feng , Hai-cang Ren

Defu Hou

Central China Normal University, Wuhan

- Color Superconductor (CSC) & complex gap
- Angular momentum mixing in non-spher. CSC
- Ground state of single flavor CSC
- Summary and outlooks

- 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)

- Lattice calculation not reliable
- High density effective theory
- Complications due to charge neutrality and \beta equilibrium
- What is the ground state of dense QCD

- Deconfined quarks( )
- Pauli principle(s=1/2)

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

Cooper instability

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)

- BSC-like pairing
2SC: u_r, d_r, u_g, d_g

CFL: all flavor and color

- Non-BCS pairing
gapless CSC

LOFF

……

J=0:

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

J=1:

N_f=1

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)

…….

- Eliashberg theory: energy depend. With imaginary part

- QCD single-gluon exchange potential
- Gap is E depend. with an imaginary part

T

L

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

EQ of RP

EQ of IP:

:

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

CSC at moderate density:

- Beta EQL.
- Non-zero s quark mass
- Charge neutrality

Mismatch

- Spherical states
all mixed states

CSL

- Non-spherical states
polar, planar and A phases

in both transv. and long.

A. Schmitt, PRD 71, 054016 (2005)

Most stable state

- Helium_3
- QCD

Pairing potential:

Angular momentum mixing

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

The two-loop approximation

to \gamma_2

Order of g^2mu^4

Stationary points

Powers of T

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

Minimization of F

Free energy density

CJT action(II)

Energy density of normal phase

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

L-pairing:

T-pairing:

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

General form of gap:

2SC gap

angular depend. Funct.

Integral eqs of gap funct:

L:

T:

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

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

- Polar state

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

- A phase

Transv.

Long.

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

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

Polar

J=1

mixing

Long.：

Transv.：

The drop amount is small (few percent) and can not make the non-spherical

states more favored than CSL

A. Schmitt, PRD 71, 054016 (2005)

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

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

where:

- Integral equation for angle dependent function

- Transv. Planar phase

Angular momentum mixing lowered the free energy of transv. Planar phase

by 0.99 percent

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

Ground state of single flavor CSC

Transv. CSL is the most stable phase even including angular momentum

mixing: we have proved

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

- Typical magnetic field ～10^12G

A. Schmitt et al., PRL 91, 242301 (2003)

PRD 69, 094017 (2004)

- de Haas-van Alpen oscillation in CFL

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

How about single flavor

CSC? Determining the

critical magnetic field in

single flavor CSC!

k_d

BCS pairing

Angular momentum mixing in LOFF

- LOFF state
first investigated by Larkin and Ovchinnikov (Sov. Phys. JETP 20, 762 (1965) )and Fulde and Ferrell (Phys. Rev. 135. A550 (1964) )

- LOFF window

角动量混合

M. Alford, et al. Phys. Rev. D 63, 074016 (2001)

I. Giannakis, et al. Phys. Rev. D 66, 031501 (2002)

- 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

Symmetry structures of Spin-1 CSC

A. Schmitt, PRD 71, 054016 (2005)

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