1. 1 NGB and their parameters  Gradient expansion: parameters of the NGB’s  Masses of the NGB’s  The role of the chemical potential for scalar fields: BE condensation  Dispersion relations for the gluons

2. 2 Hierarchies of effective lagrangians Integrating out heavy degrees of freedom we have two scales. The gap  and a cutoff,  above which we integrate out. Therefore: two different effective theories, L HDET and L Golds

3. 3 Gradient expansion: NGB’s parameters Recall from HDET that in the CFL phase and in the basis

4. 4 Propagator Coupling to the U(1) NGB: Invariant couplings

5. 5 Consider now the case of the U(1) B NGB. The invariant Lagrangian is:  At the lowest order in  generates 3-linear and 4-linear couplings

6. 6 Generating functional:

7. 7 At the lowest order:

8. 8 Feynman rules  For each fermionic internal line iL int  For each vertex a term iL int  For each internal momentum not constrained by momentum conservation:  Factor 2x(-1) from Fermi statistics and spin. A factor 1/2 from replica trick.  A statistical factor when needed.

9. 9 + Goldstone theorem: Expanding in p/ 

10. 10 CFL

11. 11 For the V NGB same result in CFL, whereas in 2SC 2SC With an analogous calculation:

12. 12 Dispersion relation for the NGB’s Different way of computing: Current conservation:

13. 13 Masses of the NGB’s QCD mass term:

14. 14 Calculation of the coefficients from QCD Mass insertion in QCD Effective 4-fermi Contribution to the vacuum energy

15. 15 Consider: Solving for as in HDET like chemical potential

16. 16 Consider fermions at finite density: as a gauge field A 0 Invariant under: Define: Invariance under:

17. 17 The same symmetry should hold at the level of the effective theory for the CFL phase (NGB’s), implying that The generic term in the derivative expansion of the NGB effective lagrangian has the form

18. 18 Compare the two contribution to quark masses: kinetic term mass insertion Same order of magnitude for since

19. 19 The role of the chemical potential for scalar fields: Bose-Einstein condensation  A conserved current may be coupled to the a gauge field.  Chemical potential is coupled to a conserved charge.  The chemical potential must enter as the fourth component of a gauge field.

20. 20 Complex scalar field: negative mass term breaks C Mass spectrum: m For  < m

21. 21  = m At  = m, second order phase transition. Formation of a condensate obtained from: Charge density Ground state = Bose-Einstein condensate

22. 22 Mass spectrum At zero momentum

23. 23 At small momentum

24. 24 Back to CFL. From the structure First term from “chemical potential” like kinetic term, the second from mass insertions

25. 25 For large values of m s: and the masses of K + and K 0 are pushed down. For the critical value masses vanish

26. 26 For larger values of m s these modes become unstable. Signal of condensation. Look for a kaon condensate of the type: (In the CFL vacuum,  = 1 ) and substitute inside the effective lagrangian negative contribution from the “chemical potential” positive contribution from mass insertion

27. 27 Defining with solution and hypercharge density

28. 28 Mass terms break original SU(3) c+L+R to SU(2) I xU(1) Y. Kaon condensation breaks this to U(1) breaking through the doublet as in the SM Only 2 NGB’s from K 0, K + instead of expected 3 (see Chada & Nielsen 1976)

29. 29 Chada and Nielsen theorem: The number of NGB’s depends on their dispersion relation I. If E is linear in k, one NGB for any broken symmetry II. If E is quadratic in k, one NGB for any two broken generators In relativistic case always of type I, in the non-relativistic case both possibilities arise, for instance in the ferromagnet there is a NGB of type II, whereas for the antiferromagnet there are to NGB’s of type I

30. 30 Dispersion relations for the gluons The bare Meissner mass The heavy field contribution comes from the term

31. 31 Notice that the first quantized hamiltonian is: Since the zero momentum propagator is the density one gets spin

32. 32 Gluons self-energy Vertices from Consider first 2SC for the unbroken gluons:

33. 33  Bare Meissner mass cancels out the constant contribution from the s.e.  All the components of the vacuum polarization have the same wave function renormalization Dielectric constant  = k+1, and magnetic permeability =1

34. 34 Broken gluons a   (0)-  ij (0) 1-300 4-73m g 2 /2m g 2 /2 83m g 2 m g 2 /3

35. 35 But physical masses depend on the wave function renormalization Rest mass defined as the energy at zero momentum: The expansion in p/  cannot be trusted, but numerically

36. 36 In the CFL case one finds: Recall that from the effective lagrangian we got: from bare Meissner mass implying and fixing all the parameters.

37. 37 We find: Numerically

38. 38 LOFF phase  Different quark masses  LOFF phase  Phonons

39. 39 Different quark masses We have seen that for one massless flavors and a massive one (m s ), the condensate may be disrupted for The radii of the Fermi spheres are: As if the two quarks had different chemical potential ( m s 2 /2  )

40. 40 Simulate the problem with two massless quarks with different chemical potentials: Can be described by an interaction hamiltonian Lot of attention in normal SC.

41. 41  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 )

42. 42 H I changes the inverse propagator and the gap equation (for spin up and down fermions): This has two solutions:

43. 43 Grand potential: Also: Favored solution

44. 44 Also: First order transition to the normal state at For constant  Ginzburg-Landau expanding up to  

45. 45 LOFF phase In 1964 Larkin, Ovchinnikov and Fulde, Ferrel, argued the possibility that close to the first order-line a new phase could take place. According LOFF possible condensation with non zero total momentum of the pair More generally

46. 46 fixed variationally chosen spontaneously Gap equation: Non zero total momentum

47. 47 For T T 0 blocking region The blocking region reduces the gap:

48. 48 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 has been studied via a Ginzburg-Landau expansion of the grand potential see later

49. 49 (for regular crystalline structures all the  q are equal) Bowers and Rajagopal (2002) ) The coefficients can be determined microscopically for the different structures ( Bowers and Rajagopal (2002) )

50. 50  Gap equation   Propagator expansion  Insert in the gap equation General strategy

51. 51 We get the equation Which is the same aswith The first coefficient has universal structure, independent on the crystal. From its analysis one draws the following results

52. 52 Small window. Opens up in QCD? (Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu & Ren 2002)

53. 53 Single plane wave Critical line from Along the critical line

54. 54 Preferred structure: face-centered cube Bowers and Rajagopal (2002)

55. 55 In the LOFF phase translations and rotations are broken phonons R.C., Gatto, Mannarelli & Nardulli 2002 Phonon field through the phase of the condensate ( R.C., Gatto, Mannarelli & Nardulli 2002 ): introducing Phonons

56. 56 Coupling phonons to fermions (quasi-particles) trough the gap term L phonon R.C., Gatto, Mannarelli & Nardulli 2002 It is possible to evaluate the parameters of L phonon ( R.C., Gatto, Mannarelli & Nardulli 2002 ) +

57. 57 Cubic structure

58. 58

59. 59 Coupling phonons to fermions (quasi-particles) trough the gap term  (i) (x) O h. O(3)xO h ~ O h diag  (i) (x) transforms under the group O h of the cube. Its e.v. ~ x i breaks O(3)xO h ~ O h diag

60. 60 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!

61. 61 Compact stellar objects

62. 62 Compact stellar objects High density core of a compact star, a good lab for testing QCD at high density.

63. 63 Some features of a compact star For simplicity consider a gas of free massless fermions. Grand potential: Density: Eq. of state:

64. 64 For a non-relativistic fermion: More generally assumed For high densities inverse beta decay becomes important At the equilibrium

65. 65 From charge neutrality Neutron star Radius of a neutron star (Landau 1932)

66. 66 N fermions in a box of volume V. Number density Position uncertainty Uncertainty principle Gravitational energy per baryon

67. 67 E > 0 otherwise not bounded. This condition gives Maximum mass Chandrasekhar limit

68. 68 Typical neutron star density

69. 69 Neutron stars are a good laboratory to test hadronic matter at high density and zero temperature

70. 70 T ns ~10 5 K In neutron stars CS can be studied at T = 0 ( T ns ~10 5 K ) Orders of magnitude from a crude model: 3 free quarks  BCS Consider the LOFF state. From  BCS

71. 71 Weak equilibrium: Electrical neutrality:

72. 72    n.m. ~.15x10 15 g/cm    n.m. is the saturation nuclear density ~.15x10 15 g/cm   B ~ 10 15 g/cm  At the core of the neutron star  B ~ 10 15 g/cm  ~ 400 MeV Choosing  ~ 400 MeV M s = 200  p F = 25 M s = 300  p F = 50 Right ballpark (14 - 70 MeV)

73. 73 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 dipole emission

74. 74  In the superfluid phase there are vortices pinned to the crust. When the star slows down the vortices do not participate in the motion until an instability is produced. Then there is a release of angular momentum to the crust seen as a jump in the rotational frequency.  The presence of the LOFF phase might avoid the main objection against the existence of strange stars (made of u,d,s quarks in equal ratios) since they cannot have a crust.

75. 75 Conclusions  SC almost 100 years old, but still actual  Important technological applications  Source of inspiration for other physical theories (SM as an example)  Deep implications in QCD at very high density: very rich phase structure  Possible applications for compact stellar objects  Unvaluable theoretical laboratory