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Angular momentum mixing in non-spherical color superconductors

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Collaborators: Bo Feng , Hai-cang Ren

Defu Hou

Central China Normal University, Wuhan

Outlines

- 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)

Dense QCD

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

Color superconductivity

- 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)

Phase structure in CSC

- 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)

…….

Gap function

● Dispersion relation:

● BCS theory

Real gap function

Eliashberg theory

- Eliashberg theory: energy depend. With imaginary part

HDL Resummed Gluon Propagator

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

T

L

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]

Gap Equation

- 2SC gap eq.

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

Complex Gap Equation

EQ of RP

EQ of IP:

:

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

Single flavor of CSC(I)

CSC at moderate density:

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

Mismatch

J=1 pairing

Angular momentum mixing

- 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

Nonlinear gap equation:

- 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)

CJL effective action(I)

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)

NG Propagators

L. Propaga.:

T. Propag:

Gap equation

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)

Gap Equation

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)

Angular dependence

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)

Angular momentum mixing(II)

- Polar state

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

Angular momentum mixing(III)

- 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(IV)

- 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)

Mixing in planar phase(I)

- 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

Mixing in planar phase(II)

- 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

CSC in nature

- Profile of
neutron star

Webber, astro-ph/0407155

CSC inside a neutron star(I)

- Typical chemical potential 500MeV
- Nonzero strange quark mass

√

？

？

Magnetic field effect

- Typical magnetic field ～10^12G

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

PRD 69, 094017 (2004)

CSC inside neutron stars(III)

- 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_u

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)

Summary and outlook

- 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

Thank you!

A. Schmitt, PRD 71, 054016 (2005)