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. Hierarchies of effective lagrangians
Integrating out heavy degrees of freedom we have two scales. The gap D and a cutoff, d above which we integrate out. Therefore: two different effective theories, LHDET and LGolds

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

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

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

6. Generating functional:

7. At the lowest order:

8. Feynman rules
For each fermionic internal line
For each vertex a term iLint
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. +
Goldstone theorem:
Expanding in p/D:

10. CFL

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

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

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

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

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

16. Consider fermions at finite density:
as a gauge field A0
Invariant under:
Define:
Invariance under:

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

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

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

22. Mass spectrum
At zero momentum

23. At small momentum

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

25. For large values of ms:
and the masses of K+ and K0 are pushed down. For the critical value
masses vanish

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

27. Defining
with solution
and hypercharge density

28. breaking through the doublet as in the SM
Mass terms break original SU(3)c+L+R to SU(2)IxU(1)Y. Kaon condensation breaks this to U(1)
breaking through the doublet as in the SM
Only 2 NGB’s from K0, K+ instead of expected 3 (see Chada & Nielsen 1976)

29. Chada and Nielsen theorem: The number of NGB’s depends on their dispersion relation
If E is linear in k, one NGB for any broken symmetry
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 one NGB of type II, whereas for the antiferromagnet there are two NGB’s of type I

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

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

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

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 e = k+1, and magnetic permeability l =1

34. Broken gluons a
P00(0)
- Pij(0)
1-3
4-7
3mg2/2
mg2/2 8
3mg2
mg2/3

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

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

37. We find:
Numerically

38. Different quark masses
We have seen that for one massless flavors and a massive one (ms), the condensate may be disrupted for
The radii of the Fermi spheres are:
As if the two quarks had different chemical potential (ms2/2m)

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

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

41. Grand potential:
Also:
Favored solution

42. Also:
First order transition to the normal state at
For constant D, Ginzburg-Landau expanding up to D6