1 angular momentum mixing in non-spherical color superconductors defu hou central china normal...
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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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Symmetry structures of Spin-1 CSC