1. QCD at very high density
Roberto Casalbuoni
Department of Physics and INFN - Florence
Perugia, January 22-23, 2007

2. Summary Introduction and basics in Superconductivity Effective theory
Color Superconductivity: CFL and 2SC phases
Effective theories and perturbative calculations
LOFF phase

3. Introduction
Important to explore the entire QCD phase diagram: Understanding of
Hadrons
QCD-vacuum
Understanding of its modifications
Extreme Conditions in the Universe: Neutron Stars, Big Bang
QCD simplifies in extreme conditions:
Study QCD when quarks and gluons are the relevant degrees of freedom

4. Studying the QCD vacuum under different and extreme conditions may help our understanding
Neutron star
Heavy ion collision
Big Bang

6. Free quarks R Asymptotic freedom: q When nB >> 1 fm-3
free quarks expected

7. Free Fermi gas and BCS
(high-density QCD)
f(E) E

8. Trivial theory ? High density means high pF pF
Typical scattering at momenta of order of pF pF
No chiral breaking
No confinement
No generation of masses
Trivial theory ?

9. Grand potential unchanged:
Adding a particle to the Fermi surface
Taking out a particle (creating a hole)

10. In QCD attractive interaction (antitriplet channel)
For an arbitrary attractive interaction it is convenient to form pairs particle-particle or hole-hole (Cooper pairs)
In matter SC only under particular conditions (phonon interaction should overcome the Coulomb force)
In QCD attractive interaction (antitriplet channel)
SC much more efficient in QCD

11. Effective theory Field theory at the Fermi surface
The free fermion gas
One-loop corrections

12. Field theory at the Fermi surface
(Polchinski, TASI 1992, hep-th/ )
Renormalization group analysis a la Wilson
How do fields behave scaling down the energies toward eF by a factor s<1?
Scaling:

13. Using the invariance under phase transformations, construction of the most general action for the effective degrees of freedom: particles and holes close to the Fermi surface (non-relativistic description)
Expanding around eF:

14. requiring the action S to be invariant
Scaling:
requiring the action S to be invariant

15. corresponding to a Cooper-like interaction
The result of the analysis is that all possible interaction terms are irrelevant (go to zero going toward the Fermi surface) except a marginal (independent on s) quartic interaction of the form:
corresponding to a Cooper-like interaction

16. s-1+4
Quartic
s-4x1/2
sd ??
Scales as s1+d

17. Scattering:

18. irrelevant
marginal s0
s-1

19. Higher order interactions irrelevant
Free theory BUT check quantum corrections to the marginal interactions among the Cooper pairs

20. The free fermion gas
Eq. of motion:
Propagator:
Using:

21. Fermi field decomposition
or:
Fermi field decomposition

22. The following representation holds:
with:
The following representation holds:
In fact, using

23. The following property is useful:

24. Closing in the upper plane we get
One-loop corrections
Closing in the upper plane we get

25. d, UV cutoff
From RG equations:

26. BCS instability
Attractive, stronger for

27. Functional approach
Fierzing (C = is2)
Quantum theory

29. Evaluating the saddle point:
Since y* appears already in c we are double-counting. Solution: integrate over the fermions with the “replica trick”:
Evaluating the saddle point:

30. At T not 0, introducing the Matsubara frequencies
and using (f(E) is the Fermi particle density)

31. By saddle point:
Introducing the em interaction in S0 we see that Z is gauge invariant under
Therefore also Seff must be gauge invariant and it will depend on the space-time derivatives of D through

32. In fact, evaluating the diagrams (Gor’kov 1959):

33. got the result (with a convenient renormalization of the fields):
charge of the pair
This result gave full justification to the Landau treatment of superconductivity

34. The critical temperature
By definition at Tc the gap vanishes. One can perform a GL expansion of the grand potential
with extrema:
a and b from the expansion of the gap equation up to normalization

35. To get the normalization remember (in the weak coupling and relatively to the normal state):
Starting from the gap equation:
Integrating over D and using the gap equation one finds:
Rule to get the effective potential from the gap equation: Integrate the gap equation over D and multiply by 2/G

36. Integrating over x and summing over n up to N
Expanding the gap equation in D:
One gets:
Integrating over x and summing over n up to N

37. Requiring a(Tc) = 0
Also
and, from the gap equation