1. 1 Angular momentum mixing in non-spherical color superconductors Defu Hou Central China Normal University, Wuhan Collaborators: Bo Feng, Hai-cang Ren

2. 2 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)

3. 3 QCD Phase diagram

4. 4 Lattice calculation not reliable High density effective theory Complications due to charge neutrality and \beta equilibrium What is the ground state of dense QCD Dense QCD

5. 5 (i)Deconfined quarks( ) (ii)Pauli principle(s=1/2) (i)Effective models( ) (ii)One-gluon exchange( ) Cooper instability 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)

6. 6 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)

7. 7 ● Dispersion relation: ● BCS theory Real gap function Gap function

8. 8 Eliashberg theory: energy depend. With imaginary part Eliashberg theory

9. 9 QCD single-gluon exchange potential Gap is E depend. with an imaginary part TL HDL Resummed Gluon Propagator

10. 10 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]

11. 11 2SC gap eq. Gap Equation R. Pisarski and D. Rischke, PRD (2000)

12. 12 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 );

13. 13 CSC at moderate density: Beta EQL. Non-zero s quark mass Charge neutrality Mismatch Single flavor of CSC(I)

14. 14 J=1 pairing

15. 15 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)

16. 16 Helium_3 QCD Pairing potential: Nonlinear gap equation: Angular momentum mixing W. Brown, J. Liu and H-c Ren, PRD 61, 114012 (2000); PRD 62, 054013 (2000); PRD 62, 054016 (2000)

17. 17 The two-loop approximation to \gamma_2 Powers of T Stationary points Order of g^2mu^4 CJL effective action(I) D. Rischke Prog. Part. Nucl. Phys. 52 197 (2004)

18. 18 L. Propaga.: T. Propag: NG Propagators

19. 19 Energy density of normal phase Free energy density Gap equation Minimization 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)

20. 20 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)

21. 21 General form of gap: 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)

22. 22 Polar state Angular momentum mixing(II) BF D-f Hou and H-c Ren, NPB 796, 500 (2008); NPB 813, 408 (2009)

23. 23 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.

24. 24 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-spherical states more favored than CSL

25. 25 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

26. 26 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 phase by 0.99 percent

27. 27 Transv. CSL is the most stable phase even including angular momentum mixing: 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

28. 28 Profile of neutron star CSC in nature Webber, astro-ph/0407155

29. 29 Typical chemical potential 500MeV Nonzero strange quark mass √ ？ ？ CSC inside a neutron star(I)

30. 30 Typical magnetic field ～ 10^12G Magnetic field effect A. Schmitt et al., PRL 91, 242301 (2003) PRD 69, 094017 (2004)

31. 31 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)

32. 32 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 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

33. 33 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

34. 34

35. 35 A. Schmitt, PRD 71, 054016 (2005) Symmetry structures of Spin-1 CSC