Angular momentum mixing in non spherical color superconductors
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Angular momentum mixing in non-spherical color superconductors. 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

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Angular momentum mixing in non-spherical color superconductors

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Angular momentum mixing in non spherical color superconductors

Angular momentum mixing in non-sphericalcolor superconductors

Collaborators: Bo Feng , Hai-cang Ren

Defu Hou

Central China Normal University, Wuhan


Angular momentum mixing in non spherical color superconductors

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)


Qcd phase diagram

QCD Phase diagram


Angular momentum mixing in non spherical color superconductors

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


Angular momentum mixing in non spherical color superconductors

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)


Angular momentum mixing in non spherical color superconductors

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)

…….


Angular momentum mixing in non spherical color superconductors

Gap function

● Dispersion relation:

● BCS theory

Real gap function


Angular momentum mixing in non spherical color superconductors

Eliashberg theory

  • Eliashberg theory: energy depend. With imaginary part


Angular momentum mixing in non spherical color superconductors

HDL Resummed Gluon Propagator

  • QCD single-gluon exchange potential

  • Gap is E depend. with an imaginary part

T

L


Angular momentum mixing in non spherical color superconductors

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]


Angular momentum mixing in non spherical color superconductors

Gap Equation

  • 2SC gap eq.

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


Angular momentum mixing in non spherical color superconductors

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);


Angular momentum mixing in non spherical color superconductors

Single flavor of CSC(I)

CSC at moderate density:

  • Beta EQL.

  • Non-zero s quark mass

  • Charge neutrality

Mismatch


Angular momentum mixing in non spherical color superconductors

J=1 pairing


Angular momentum mixing in non spherical color superconductors

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


Angular momentum mixing in non spherical color superconductors

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)


Angular momentum mixing in non spherical color superconductors

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)


Angular momentum mixing in non spherical color superconductors

NG Propagators

L. Propaga.:

T. Propag:


Angular momentum mixing in non spherical color superconductors

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)


Angular momentum mixing in non spherical color superconductors

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 momentum mixing in non spherical color superconductors

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 in non spherical color superconductors

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 in non spherical color superconductors

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 in non spherical color superconductors

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)


Angular momentum mixing in non spherical color superconductors

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


Angular momentum mixing in non spherical color superconductors

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


Angular momentum mixing in non spherical color superconductors

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


Angular momentum mixing in non spherical color superconductors

CSC in nature

  • Profile of

    neutron star

Webber, astro-ph/0407155


Angular momentum mixing in non spherical color superconductors

CSC inside a neutron star(I)

  • Typical chemical potential 500MeV

  • Nonzero strange quark mass


Angular momentum mixing in non spherical color superconductors

Magnetic field effect

  • Typical magnetic field ~10^12G

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

PRD 69, 094017 (2004)


Angular momentum mixing in non spherical color superconductors

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!


Angular momentum mixing in non spherical color superconductors

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)


Angular momentum mixing in non spherical color superconductors

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


Angular momentum mixing in non spherical color superconductors

Thank you!


Symmetry structures of spin 1 csc

Symmetry structures of Spin-1 CSC

A. Schmitt, PRD 71, 054016 (2005)


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