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The three flavor LOFF phase of QCD

The three flavor LOFF phase of QCD. N. D. Ippolito University and INFN, Bari, Italy. HISS : Dense Matter in HIC and Astrophysics, Dubna, 2006. Very high densities (  >> m quark ) and low temperature ( T  0 ). CFL superconductive phase

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The three flavor LOFF phase of QCD

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  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( >> mquark)and low temperature( T  0 ) CFL superconductive phase (Color Flavor Locking; Alford, Rajagopal and Wilczek 1999) ( Nf = 3 )

  3. Form of the CFL condensate Note the presence of just one gap parameter  for all the pairs. (Neglecting the condensation in the symmetric6channel)

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

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

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

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

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

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

  10. Ptot = p1+ p2 = 2q  0 Simplest ansatz for the condensate (one plane wave) ~ ei2q•r(r) In general, more plane waves: LOFF : In a window 1 < < 2  0.754 BCS it can be energetically favourable to form pairs with non zero total momentum

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

  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-ms2/2 ;3= 8=0 ; (recently justified by Casalbuoni, Ciminale, Gatto, Nardulli, Ruggieri; June 2006) The chemical potential term in the Lagrangean has the form

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

  15. High Density Effective Theory Large component Small 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 themean field approximation where with is the pairing ansatz.

  18. Let’s change the basis for the spinor fields (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

  19. Gap Equation Electrical neutrality Ginzburg-Landau expansion

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

  21. Results The favorite structure has 1=0, 2 = 3and q2,q3 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 Ms2/ It is a serious candidate for being the true vacuum at intermediate densities

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