1. Color Superconductivity in High Density QCD
Roberto Casalbuoni
Dept. of Physics, University of Florence
INFN - Florence
Budapest, August 1-3, 2005
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

2. R. Casalbuoni:Superconductivity in high density QCD
Outline of the talk
Introduction
Pairing fermions with different Fermi momenta
The gapless phases g2SC and gCFL
The LOFF phase
Preliminary results about the LOFF phase with 3 flavors
Conclusions and outlook
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

3. R. Casalbuoni:Superconductivity in high density QCD
Introduction
Motivations for the study of high-density QCD
Understanding the interior of CSO’s
Study of the QCD phase diagram at T~ 0 and moderate m
Asymptotic region in m fairly well understood: existence of a CS phase. Real question: does this type of phase persist at relevant densities (~5-6 r0)?
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

4. Pairing fermions with different Fermi momenta
Ms not zero
Neutrality with respect to em and color
Weak equilibrium
no free energy cost in neutral singlet, (Amore et al. 2003)
All these effects make Fermi momenta of different fermions unequal causing problems to the BCS pairing mechanism
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

5. Effective chemical potential for the massive quark
Consider 2 fermions with m1 = M, m2 = 0 at the same chemical potential m. The Fermi momenta are
Effective chemical potential for the massive quark
M2/2 effective chemical potential
Mismatch:
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

6. R. Casalbuoni:Superconductivity in high density QCD
If electrons are present, weak equilibrium makes chemical potentials of quarks of different charges unequal:
From this: with
N.B. me is not a free parameter: neutrality requires:
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

7. R. Casalbuoni:Superconductivity in high density QCD
Example 2SC: normal BCS pairing when
But neutral matter for
Mismatch
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

8. R. Casalbuoni:Superconductivity in high density QCD
As long as dm is small no effects on BCS pairing, but when increased the BCS pairing is lost and two possibilities arise:
The system goes back to the normal phase
Other phases can be formed
Notice that there is also a color neutrality condition
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

9. R. Casalbuoni:Superconductivity in high density QCD
In a simple model without supplementary conditions, with two fermions at chemical potentials m1 and m2, the system becomes normal at the Chandrasekhar - Clogston point. Another unstable phase exists.
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

10. R. Casalbuoni:Superconductivity in high density QCD
The point |dm| = D is special. In the presence of a mismatch new features are present. The spectrum of quasiparticles is
For |dm| < D, the gaps are D - dm and D + dm
For |dm| = D, an unpairing (blocking) region opens up and gapless modes are present
Energy cost for pairing
begins to unpair
Energy gained in pairing
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

11. R. Casalbuoni:Superconductivity in high density QCD
g2SC
Same structure of condensates as in 2SC (Huang & Shovkovy, 2003)
4x3 fermions:
2 quarks ungapped qub, qdb
4 quarks gapped qur, qug, qdr, qdg
General strategy (NJL model):
Write the free energy
Solve
Neutrality
Gap equation
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

12. R. Casalbuoni:Superconductivity in high density QCD
For |dm| > D (dm = m e/2) 2 gapped quarks become gapless. The gapped quarks begin to unpair destroying the BCS solution. But a new stable phase exists, the gapless 2SC (g2SC) phase.
It is the unstable phase (Sarma phase) which becomes stable in this case (and in gCFL, see later) when charge neutrality is required.
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

13. R. Casalbuoni:Superconductivity in high density QCD
g2SC
(Huang & Shovkovy, 2003)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

14. R. Casalbuoni:Superconductivity in high density QCD
However, evaluation of the gluon masses (5 out of 8 become massive) shows an instability of the g2SC phase. Some of the gluon masses are imaginary (Huang and Shovkovy 2004). Connected with gapless modes? (Alford & Wang, 2005).
4,5,6,7 8
But this instability occurs also in the 2SC phase (no gapless modes)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

15. gCFL Generalization to 3 flavors
(Alford, Kouvaris & Rajagopal, 2005)
Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

16. R. Casalbuoni:Superconductivity in high density QCD-1 +1 ru gd bs rd gu rs bu gs bd
Gaps in gCFL
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

17. R. Casalbuoni:Superconductivity in high density QCD
Strange quark mass effects:
Shift of the chemical potential for the strange quarks:
Color and electric neutrality in CFL requires
The transition CFL to gCFL starts with the unpairing of the pair gs-bd having (close to the transition)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

18. R. Casalbuoni:Superconductivity in high density QCD
It follows:
Energy cost for pairing
begins to unpair
Energy gained in pairing
Again, by using NJL model (modelled on one-gluon exchange):
Write the free energy:
Solve:
Neutrality
Gap equations
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

19. CFL # gCFL 2nd order transition at Ms2/m ~ 2D, when the pairing gs-bd starts breaking
(Alford, Kouvaris & Rajagopal, 2005)
(0 = 25 MeV,  = 500 MeV)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

20. R. Casalbuoni:Superconductivity in high density QCD
gCFL has gapless quasiparticles, and there are gluon imaginary masses also in this phase (RC et al. 2004, Fukushima 2005).
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

21. This makes the LOFF phase very interesting
Instability cured by gluon condensation? Assuming artificially <A3> or <A8> not zero (of order 10 MeV) this can be done. See also a very recent paper (Gorbar, Hashimoto & Miransky, 2005) about a gluonic phase curing the chromomagnetic instability in 2SC.
Three recent results obtained by Giannakis & Ren:
Chromomagnetic instability of g2SC makes the crystalline phase (LOFF) with two flavors energetically favored (Giannakis & Ren 2004)
LOFF with two flavors without requiring electrical neutrality has no magnetic instability although it has gapless modes (Giannakis & Ren 2005)
Last week the same result obtained requiring color and electric neutrality in the weak coupling limit (Giannakis, Hou & Ren 2005)
This makes the LOFF phase very interesting
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

22. R. Casalbuoni:Superconductivity in high density QCD
LOFF phase
LOFF (Larkin, Ovchinnikov, Fulde & Ferrel, 1964): 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 producing a mismatch of the Fermi momenta
Studied also in the QCD context (Alford, Bowers & Rajagopal, 2000)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

23. R. Casalbuoni:Superconductivity in high density QCD
According to LOFF, close to first order point (CC point), possible condensation with non zero total momentum
More generally
fixed variationally
chosen spontaneously
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

24. an unpairing (blocking) region opens up and gapless modes are present
Single plane wave:
Also in this case, for
an unpairing (blocking) region opens up and gapless modes are present
More general possibilities include a crystalline structure (Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002)
The qi’s define the crystal pointing at its vertices.
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

25. R. Casalbuoni:Superconductivity in high density QCD
Small window. Opens up in QCD? (Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu & Ren 2002)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

26. Preliminary results about LOFF with three flavors
Recent study of LOFF with 3 flavors within the following simplifying hypothesis (RC, Gatto, Ippolito, Nardulli & Ruggieri, 2005)
Study within the Landau-Ginzburg approximation.
Only electrical neutrality imposed (chemical potentials 3 and 8 taken equal to zero).
Ms treated as in gCFL. Pairing similar to gCFL with inhomogeneity in terms of simple plane waves, as for the simplest LOFF phase.
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

27. R. Casalbuoni:Superconductivity in high density QCD
A further simplifications is to assume only the following geometrical configurations for the vectors qI, I=1,2,3 (a more general angular dependence will be considered in future work) 1 2 3 4
The free energy, in the GL expansion, has the form
with coefficients I, I and IJ calculable from an effective NJL four-fermi interaction simulating one-gluon exchange
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

28. R. Casalbuoni:Superconductivity in high density QCD
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

29. R. Casalbuoni:Superconductivity in high density QCD
We require:
At the lowest order in I
since I depends only on qI and i we get the same result as in the usual LOFF case:
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

30. Structure 4 dominates starting from about 30 MeV
(we have assumed the same parameters as in Alford et al. in gCFL, 0 = 25 MeV,  = 500 MeV)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

31. Comparison with other phases:
If no chromo-magnetic instability LOFF takes over gCFL at about 120 MeV and goes over to the normal phase at about 150 MeV (both first order transitions)
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

32. R. Casalbuoni:Superconductivity in high density QCD
Otherwise it should go over to the CFL phase at 75 MeV
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

33. R. Casalbuoni:Superconductivity in high density QCD
The behaviour of e in LOFF is pretty similar to the one in gCFL. If the same is true for 3 and 8 our assumption of their vanishing is not too bad for Ms2/ in the region where LOFF is the favored phase.
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD

34. R. Casalbuoni:Superconductivity in high density QCD
Conclusions
The problem of the QCD phases at moderate densities and low temperature is still open.
Various phases are competing, many of them having gapless modes. However, when such modes are present a chromomagnetic instability arises (but this happens also under different conditions, see 2SC).
Also the LOFF phase is gapless but the gluon instability does not seem to appear.
Our recent study of the LOFF phase with three flavors seems to suggest that this should be the favored phase after CFL, although this study is very much simplified and more careful investigations should be performed.
JHW Budapest, August 1-3, 2005
R. Casalbuoni:Superconductivity in high density QCD