1. QCD@Work 2003 International Workshop on Quantum Chromodynamics Theory and Experiment Conversano (Bari, Italy) June 14-18 2003 Inhomogeneous color superconductivity Roberto Casalbuoni Department of Physics and INFN – Florence & CERN TH Division - Geneva

2.  Introduction to color superconductivity  Effective theory of CS  Gap equation  The anisotropic phase (LOFF): phase diagram and crystalline structure  Phonons  LOFF phase in compact stellar objects  Outlook SummaryLiterature

3.  Reviews of color superconductivity:  T. Schaefer, hep-ph/0304281  K. Rajagopal and F. Wilczek, hep-ph/0011333  G. Nardulli, hep-ph/0202037  Original LOFF papers:  A.J. Larkin and Y. N. Ovchinnikov, Zh. Exsp. Teor. Fiz. 47 (1964) 1136  P. Fulde and R.A. Ferrel, Phys. Rev. 135 (1964) A550  Review of the LOFF phase:  R. Casalbuoni and G. Nardulli, hep-ph/0305069

4. Study of CS back to 1977 (Barrois 1977, Frautschi 1978, Bailin and Love 1984) based on Cooper instability: Study of CS back to 1977 (Barrois 1977, Frautschi 1978, Bailin and Love 1984) based on Cooper instability: At T ~ 0 a degenerate fermion gas is unstable Any weak attractive interaction leads to Cooper pair formation  Hard for electrons (Coulomb vs. phonons)  Easy in QCD for di-quark formation (attractive channel ) Introduction

5. CS can be treated perturbatively for large  due to asymptotic freedom At high , m s, m d, m u ~ 0, 3 colors and 3 flavors Possible pairings:  Antisymmetry in color (  ) for attraction  Antisymmetry in spin (a,b) for better use of the Fermi surface  Antisymmetry in flavor (i, j) for Pauli principle

6. Only possible pairings LL and RR Favorite state CFL (color-flavor locking) (Alford, Rajagopal & Wilczek 1999) Symmetry breaking pattern

7. What happens going down with  ? If  << m s we get 3 colors and 2 flavors (2SC) In this situation strange quark decouples. But what happens in the intermediate region of  The interesting region is for (see later)  m s 2 /  Possible new anisotropic phase of QCD

9. Effective theory of Color Superconductivity

10. Relevant scales in CS (gap) (cutoff) Fermi momentum defined by The cutoff is of order  D in superconductivity and >  QCD in QCD

11. Hierarchies of effective lagrangians Microscopic description L QCD Quasi-particles (dressed fermions as electrons in metals). Decoupling of antiparticles (Hong 2000) L HDET Decoupling of gapped quasi- particles. Only light modes as Goldstones, etc. (R.C. & Gatto; Hong, Rho & Zahed 1999) L Gold p – p F >>  p – p F <<   p F pFpFpFpF  p F +  p F + 

12. Physics near the Fermi surface Relevant terms in the effective description ( see: Polchinski, TASI 1992, also Hong 2000; Beane, Bedaque & Savage 2000, also R.C., Gatto & Nardulli 2001 ) Marginal term in the effective description and attractive interaction

13. The marginal term becomes relevant at 1 – loop BCS instability solved by condensation and formation of Cooper pairs S res is neglected in the mean field approximation

14. The first term in S M behaves as a Majorana mass term and it is convenient to work in the Nambu-Gorkov basis: Near the Fermi surface

15. Dispersion relation Infinite copies of 2-d physics v1v1v1v1 v2v2v2v2 At fixed v F only energy and momentum along v F are relevant

16. Gap equation

17. For T T 0 At weak coupling density of states

18. With G fixed by  SB at T = 0, requiring M const ~ 400 MeV and for typical values of  ~ 400 – 500 MeV one gets Evaluationd from QCD first principles at asymptotic  ( Son 1999 ) Notice the behavior exp(-c/g) and not exp(-c/g 2 ) as one would expect from four-fermi interaction For  ~ 400 MeV one finds again

19. The anisotropic phase (LOFF) In many different situations pairing may happen between fermions belonging to Fermi surfaces with different radius, for instance: Quarks with different masses Quarks with different masses Requiring electric neutrality Requiring electric neutrality

20. Consider 2 fermions with m 1 = M, m 2 = 0 at the same chemical potential . The Fermi momenta are To form a BCS condensate one needs common momenta of the pair p F comm Grand potential at T = 0 for a single fermion

21. Pairing energy Pairing possible if The problem may be simulated using massless fermions with different chemical potentials (Alford, Bowers & Rajagopal 2000) Analogous problem studied by Larkin & Ovchinnikov, Fulde & Ferrel 1964. Proposal of a new way of pairing. LOFF phase

22. p F2 =  p F1 =  – M 2 /2  p F c =  – M 2 /4  E 1 (p F c ) =    E F1 = E F2 =  E 2 (p F c ) =  M 2 /4 

23.  LOFF: ferromagnetic alloy with paramagnetic impurities.  The impurities produce a constant exchange field acting upon the electron spins giving rise to an effective difference in the chemical potentials of the opposite spins.  Very difficult experimentally but claims of observations in heavy fermion superconductors ( Gloos & al 1993 ) and in quasi-two dimensional layered organic superconductors ( Nam & al. 1999, Manalo & Klein 2000 )

24. or paramagnetic impurities (  H) give rise to an energy additive term Gap equation Solution as for BCS  BCS, up to (for T = 0)

25.  First order transition, since for  1,    For , usual BCS second order transition at T= 0.5669  BCS  Existence of a tricritical point in the plane (  T)

26. According LOFF possible condensation with non zero total momentum of the pair More generally fixed variationally chosen spontaneously

27. Simple plane wave: energy shift Gap equation: For T T 0 blocking region

28. The blocking region reduces the gap: Possibility of a crystalline structure (Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002) The q i ’s define the crystal pointing at its vertices. The LOFF phase is studied via a Ginzburg-Landau expansion of the grand potential see later

29. (for regular crystalline structures all the  q are equal) The coefficients can be determined microscopically for the different structures.

30.  Gap equation  Propagator expansion  Insert in the gap equation

31. We get the equation Which is the same as with The first coefficient has universal structure, independent on the crystal. From its analysis one draws the following results

32. Small window. Opens up in QCD? (Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu & Ren 2002)

33. Results of Leibovich, Rajagopal & Shuster (2001)  (MeV)    BCS (      BCS (LOFF)0.7540.047 4001.240.53 10003.632.92 Corrections for non weak coupling

34. Normal LOFF BCS weak coupling strong coupling

35. Single plane wave Critical line from Along the critical line

36. Preferred structure: face-centered cube

37. Tricritical point General study by Combescot and Mora (2002). Favored structure 2 antipodal vectors  At T = 0 the antipodal vector leads to a second order phase transition. Another tricritical point ? (Matsuo et al. 1998)  Change of crystalline structure from tricritical to zero temperature?

38. Two-dimensional case ( Shimahara 1998 ) Analysis close to the critical line

39. In the LOFF phase translations and rotations are broken phonons Phonon field through the phase of the condensate (R.C., Gatto, Mannarelli & Nardulli 2002): introducing Phonons

40. Coupling phonons to fermions (quasi-particles) trough the gap term It is possible to evaluate the parameters of L phonon (R.C., Gatto, Mannarelli & Nardulli 2002) +

41. Cubic structure

43. Coupling phonons to fermions (quasi-particles) trough the gap term  (i) (x) transforms under the group O h of the cube. Its e.v. ~ x i breaks O(3)xO h ~ O h diag. Therefore we get

44. we get for the coefficients One can evaluate the effective lagrangian for the gluons in tha anisotropic medium. For the cube one finds Isotropic propagation This because the second order invariant for the cube and for the rotation group are the same!

45. Why the interest in the LOFF phase in QCD? LOFF phase in CSO

46. In neutron stars CS can be studied at T = 0 Orders of magnitude from a crude model: 3 free quarks For LOFF state from  p F  BCS

47. Weak equilibrium: Electric neutrality:

48.   n.m. is the saturation nuclear density ~.15x10 15 g/cm  At the core of the neutron star  B ~ 10 15 g/cm Choosing  ~ 400 MeV M s = 200  p F = 25 M s = 300  p F = 50 Right ballpark (14 - 70 MeV)

49. Glitches: discontinuity in the period of the pulsars.  Standard explanation: metallic crust + neutron superfluide inside  LOFF region inside the star providing the crystalline structure + superfluid CFL phase

50.  Theoretical problems: Is the cube the optimal structure at T=0? Which is the size of the LOFF window?  Phenomenological problems: Better discussion of the glitches (treatment of the vortex lines)  New possibilities: Recent achieving of degenerate ultracold Fermi gases opens up new fascinating possibilities of reaching the onset of Cooper pairing of hyperfine doublets. However reaching equal populations is a big technical problem ( Combescot 2001 ). LOFF phase? Outlook