1. Color Superconductivity: CFL and 2SC phases
Introduction
Hierarchies of effective lagrangians
Effective theory at the Fermi surface (HDET)
Symmetries of the superconductive phases

2. Introduction
Ideas about CS back in 1975 (Collins & Perry-1975, Barrois-1977, Frautschi-1978).
Only in 1998 (Alford, Rajagopal & Wilczek; Rapp, Schafer, Schuryak & Velkovsky) a real progress.
The phase structure of QCD at high-density depends on the number of flavors with mass m < m.
Two most interesting cases: Nf = 2, 3.
Due to asymptotic freedom quarks are almost free at high density and we expect difermion condensation in the color channel 3*.

3. Consider the possible pairings at very high density
a, b color; i, j flavor; a,b spin
Antisymmetry in spin (a,b) for better use of the Fermi surface
Antisymmetry in color (a, b) for attraction
Antisymmetry in flavor (i,j) for Pauli principle

4. Symmetry breaking pattern
Only possible pairings
LL and RR
For m >> mu, md, ms
Favorite state for Nf = 3, CFL (color-flavor locking) (Alford, Rajagopal & Wilczek 1999)
Symmetry breaking pattern

5. Why CFL?

6. What happens going down with m? If m << ms, we get
3 colors and 2 flavors (2SC)
However, if m is in the intermediate region we face a situation with fermions having different Fermi surfaces (see later). Then other phases could be important (LOFF, etc.)

7. Difficulties with lattice calculations
Define eucldean variables:
Dirac operator with chemical potential
At m = 0
Eigenvalues of D(0) pure immaginary
If eigenvector of D(0), eigenvector with eigenvalue - l

8. For m not zero the argument does not apply and one cannot use the sampling method for evaluating the determinant. However for isospin chemical potential and two degenerate flavors one can still prove the positivity.
For finite baryon density no lattice calculation available except for small m and close to the critical line (Fodor and Katz)

9. 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

10. LHDET is the effective theory of the fermions close to the Fermi surface. It corresponds to the Polchinski description. The condensation is taken into account by the introduction of a mean field corresponding to a Majorana mass. The d.o.f. are quasi-particles, holes and gauge fields. This holds for energies up to the cutoff.
LGolds describes the low energy modes (E<< D), as Goldstone bosons, ungapped fermions and holes and massless gauge fields, depending on the breaking scheme.

11. Effective theory at the Fermi surface (HDET)
Starting point: LQCD
at asymptotic m >> LQCD,

12. Introduce the projectors:
and decomposing:
States y+ close to the FS
States y- decouple for large m

13. Field-theoretical version:
Choosing
4 d.of.
Separation of light and heavy d.o.f.

14. Separation of light and heavy d.o.f.

15. Momentum integration for the light fields
The Fourier decomposition becomes v1 v2
For any fixed v, 2-dim theory

16. In order to decouple the states corresponding to
Momenta from the Fermi sphere
substituting inside LQCD and using

17. Proof:
Using:
One gets easily the result.

18. Eqs. of motion:

19. At the leading order in m:
At the same order:
Propagator:

20. For the heavy d.o.f. we can formally repeat the same steps leading to:
Integrating out the heavy d.o.f.
For the heavy d.o.f. we can formally repeat the same steps leading to:
Eliminating the E- fields one would get the non-local lagrangian:

21. Decomposing
one gets
When integrating out the heavy fields

22. This contribution from LhD gives the bare Meissner mass
Contribute only if some gluons are hard, but suppressed by asymptotic freedom
This contribution from LhD gives the bare Meissner mass

23. HDET in the condensed phase
Assume
(A, B collective indices)
due to the attractive interaction:
Decompose

24. We define
and neglect Lint . Therefore

25. Nambu-Gor’kov basis

26. From the definition:
one derives the gap equation (e.g. via functional formalism)

27. Four-fermi interaction one-gluon exchange inspired
Fierz using:

28. In the 2SC case
4 pairing fermions

29. G determined at T = 0. M, constituent mass ~ 400 MeV
with
For
Similar values for CFL.

30. Gap equation in QCD

31. Hard-loop approximation
For small momenta magnetic gluons are unscreened and dominate giving a further logarithmic divergence
electric
magnetic

32. However condensation arises at asymptotic values of m.
Results:
from the double log
To be trusted only for but, if extrapolated at MeV, gives values for the gap similar to the ones found using a 4-fermi interaction.
However condensation arises at asymptotic values of m.

33. Symmetries of superconducting phases
Consider again the 3 flavors, u,d,s and the group theoretical structure of the two difermion condensate:

34. implying in general
In NJL case with cutoff 800 MeV, constituent mass 400 MeV and m = 400 MeV

35. 8 give mass to the gluons and 8+1 are true massless Goldstone bosons
Original symmetry:
anomalous
broken to
anomalous
# Goldstones
massive
8 give mass to the gluons and 8+1 are true massless Goldstone bosons

36. Breaking of U(1)B makes the CFL phase superfluid
Notice:
Breaking of U(1)B makes the CFL phase superfluid
The CFL condensate is not gauge invariant, but consider
S is gauge invariant and breaks the global part of GQCD. Also
break

37. Spectrum of the CFL phase
is broken by the anomaly but induced by a 6-fermion operator irrelevant at the Fermi surface. Its contribution is parametrically small and we expect a very light NGB (massless at infinite chemical potential)
Spectrum of the CFL phase
Choose the basis:

38. Inverting:
for any 3x3 matrix g
We get
quasi-fermions

39. but wave function renormalization effects important (see later)
gluons
Expected
NG coupling constant
but wave function renormalization effects important (see later)
NG bosons
Acquire mass through quark masses except for the one related to the breaking of U(1)B

40. NG boson masses quadratic in mq since the approximate invariance
and quark mass term:
Notice: anomaly breaks through instantons, producing a chiral condensate (6 fermions -> 2), but of order (LQCD/m)8

41. In-medium electric charge
The condensate breaks U(1)em but leaves invariant a combination of Q and T8.
CFL vacuum:
Define:
leaves invariant the ground state

42. Integers as in the Han-Nambu model
rotated fields
new interaction:

43. “transparent insulator”
The rotated “photon” remains massless, whereas the rotated gluons acquires a mass through the Meissner effect
A piece of CFL material for massless quarks would respond to an em field only through NGB:
“bosonic metal”
For quarks with equal masses, no light modes:
“transparent insulator”
For different masses one needs non zero density of electrons or a kaon condensate leading to massless excitations

44. In the 2SC case, new Q and B are conserved
1/2
-1/2 1

45. Spectrum of the 2SC phase
Remember
No Goldstone bosons
(equal gap)
Light modes:

46. 2+1 flavors
It could happen
Phase transition expected
The radius of the Fermi sphere decrases with the mass

47. Simple model:
condensation energy
Condensation if:

48. Notice that the transition must be first order because for being in the pairing phase
(Minimal value of the gap to get condensation)

49. Effective lagrangians
Effective lagrangian for the CFL phase
Effective lagrangian for the 2SC phase

50. Effective lagrangian for the CFL phase
NG fields as the phases of the condensates in the representation
Quarks and X, Y transform as

51. The number of NG fields is
Since
The number of NG fields is
8 of these fields give mass to the gluons. There are only 10 physical NG bosons corresponding to the breaking of the global symmetry (we consider also the NGB associated to U(1)A)

52. Better use fields belonging to SU(3). Define
and
transforming as
The breaking of the global symmetry can be described by the gauge invariant order parameters

53. S, dX, dY are 10 fields describing the physical NG bosons. Also
shows that ST transforms as the usual chiral field.
Consider the currents:

54. Most general lagrangian up to two derivatives invariant under G, the space rotation group O(3) and Parity (R.C. & Gatto 1999)

55. Using SU(3)c gauge invariance we can choose:
Expanding at the second order in the fields
Gluons acquire Debye and Meissner masses (not the rest masses, see later)

56. Low energy theory supposed to be valid at energies << gap
Low energy theory supposed to be valid at energies << gap. Since we will see that gluons have masses of order D they can be decoupled

57. we get the gauge invariant result:
~ c-lagrangian

58. Effective lagrangian for the 2SC phase
Light modes: 2 ungapped fermions and 3 massless gluons of SU(2)c. Consider gluons:
Gauge invariance:
O(3) + Parity
Gluon velocity

59. Effective gauge coupling (l = 1, see later):
Since e >> 1 (see later), gluons move slowly and

60. Equivelently do the re-scaling
Since from
This defines the coupling at the matching scale D

61. grows going to lower energies and becomes ~ 1 at The coupling is small at the matching scale and we expect to be small. From one-loop beta-function
Numerical estimate very difficult, it depends on

62. For
For increasing m exponential decreasing
Confinement radius grows exponentially
Looking at fixed distance and increasing the density there is a crossover at which the color degrees of freedom are unconfined.