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Angular momentum mixing in non-spherical color superconductors

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

  2. 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)

  3. QCD Phase diagram

  4. 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

  5. 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)

  6. 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) …….

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

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

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

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

  12. 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);

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

  14. J=1 pairing

  15. 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

  16. 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)

  17. 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)

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

  19. 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)

  20. 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)

  21. 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)

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

  23. 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)

  24. 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)

  25. 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

  26. 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

  27. 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

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

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

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

  31. 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!

  32. 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)

  33. 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

  34. Thank you!

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

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