1. The three flavor LOFF phase of QCD N. D. Ippolito University and INFN, Bari, Italy HISS : Dense Matter in HIC and Astrophysics, Dubna, 2006

2. Very high densities (  >> m quark ) and low temperature ( T  0 ) CFL superconductive phase (Color Flavor Locking; Alford, Rajagopal and Wilczek 1999) ( N f = 3 )

3. one Note the presence of just one gap parameter  for all the pairs. Form of the CFL condensate (Neglecting the condensation in the symmetric 6 channel)

4. Going down with the density, we cannot still neglect the strange quark mass. The condition  >> m s is not more fulfilled ! m s  0 Color and electrical neutrality must be imposed Equilibrium under weak interactions Different gaps for pairs of different flavors

5. (Alford, Kouvaris, Rajagopal 2004) Gapless CFL phase (Alford, Kouvaris, Rajagopal 2004) Pairing ansatz  1 ~  ds   2 ~  us   3 ~  ud 

6. Results of gCFL phase Gap parametersFree energy ( Alford, Kouvaris, Rajagopal : hep-ph/0406137 ) BUT…

7. Imaginary Meissner masses Gluon 8 Gluons 1,2 Gluon 3 ( Casalbuoni, Gatto, Mannarelli, Nardulli, Ruggieri : hep-ph/0410401 ) Signal of instability of gCFL phase Problem not yet solved. Probably indicates that gCFL is not the true vacuum

8. LOFF phase inhomogeneous An inhomogeneous side of Superconductivity Larkin, Ovchinnikov 1964; Fulde, Ferrell 1964 ; Alford, Bowers, Rajagopal 2001; Casalbuoni, Nardulli 2004

9. In presence of a difference of chemical potentials  : Two flavor Superconductivity (not necessarily CSC) BCS survives until  up down For  >  1 it’s difficult to form pairs with zero total momentum

10. LOFF : In a window  1 <  <  2  0.754  BCS it can be energetically favourable to form pairs with non zero total momentum P tot = p 1 + p 2 = 2q  0 Simplest ansatz for the condensate (one plane wave)  ~  e i2qr  (r) In general, more plane waves:

11. LOFF phase in QCD with three flavors Casalbuoni, Gatto, NDI, Nardulli, Ruggieri. PLB 2005 Pairing ansatz with

12. Requirements and approximations  -equilibrated quark matter Non zero strange quark mass  3 =  8 =0 HDET(High Density Effective Theory) approximation Mean field approximation Ginzburg-Landau approximation for the free energy and the gap Imposition of electrical neutrality

13.  -equilibrium:  d =  u +  e ;  s =  u +  e ; Strange quark mass treated at the leading order in 1/  :  s   s -m s 2 /2  ;  3 =  8 =0 ; (recently justified by Casalbuoni, Ciminale, Gatto, Nardulli, Ruggieri; June 2006) The chemical potential term in the Lagrangean has the form

14. Explicitely we have strangedown up where; So the starting point is the free Lagrangean L=

15. High Density Effective Theory Large componentSmall residual momentum In four dimensions

16. In this way we can consider just the degrees of freedom near the Fermi surface, i.e. the residual component of quark momenta, and integrate only on a small region near it. Within HDET, the free Lagrangean reads

17. To this free Lagrangean we add a NJL coupling treated in the mean field approximation where is the pairing ansatz. with

18. (This change is performed by matrices that are combinations of Gell-Mann matrices) and introduce the Nambu-Gor’kov field. So the complete Lagrangean reads Let’s change the basis for the spinor fields

19. Ginzburg-Landau expansion Gap Equation Electrical neutrality

20. The norm of q I is fixed minimizing the Free Energy. At the first order in  q I  1.2  I  As to the directions of the q I, one should look for the energetically favored orientationsCrystallography In our work we consider just four structures, with the q I parallel or antiparallel to the same axis

21. Results The favorite structure has  1 =0,  2 =  3 and q 2,q 3 parallel The result of imposing electrical neutrality is just

22. Free energy diagram Loff phase with three flavors DOES NOT suffer of chromomagnetic instabilities! (Ciminale, Nardulli, Ruggieri, Gatto hep-ph/0602180)

23. Very good result, but recently other good news!! ( Rajagopal, Sharma hep-ph/0605316 )

24. Free energy

25. Conclusions Three flavor LOFF phase is chromomagnetically stable It has lower free energies than the normal phase and the homogeneous phases in a wide window of M s 2 /  It is a serious candidate for being the true vacuum at intermediate densities