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QCD@Work 2003 International Workshop on Quantum Chromodynamics Theory and Experiment

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  1. QCD@Work 2003 International Workshop on Quantum Chromodynamics Theory and Experiment Conversano (Bari, Italy) June 14-18  2003 Inhomogeneous color superconductivity Roberto Casalbuoni Department of Physics and INFN – Florence& CERN TH Division - Geneva

  2. Summary • Introduction • Effective theory of CS • Gap equation • The inhomogeneous phase (LOFF): phase diagram and crystalline structure • Phonons • LOFF phase in compact stellar objects • Outlook

  3. Introduction • mu, md,ms << m: CFL phase • mu, md << m << ms : 2SC phase

  4. In this situation strange quark decouples. But what happens in the intermediate region of m? The interesting region is for (see later) m ~ ms2/D Possible new inhomogeneous phase of QCD LOFF phase

  5. Effective theory of Color Superconductivity

  6. (cutoff) (gap) Relevant scales in CS Fermi momentum defined by The cutoff is of order wD in superconductivity and > LQCD in QCD

  7. Hierarchies of effective lagrangians LQCD Microscopic description p – pF >> d pF + d Quasi-particles (dressed fermions as electrons in metals). Decoupling of antiparticles (Hong 2000) LHDET d >> p – pF >> D D << d << pF pF + D Decoupling of gapped quasi-particles. Only light modes as Goldstones, etc. (R.C. & Gatto; Hong, Rho & Zahed 1999) LGold D p – pF << D pF

  8. 4-fermi attractive interaction ismarginal (relevant at 1-loop) Physics near the Fermi surface Relevant terms in the effective description (see:Polchinski, TASI 1992, also Hong 2000; Beane, Bedaque & Savage 2000, also R.C., Gatto & Nardulli 2001)

  9. SM gives risedi-fermion condensation producing a Majorana mass term. Work in theNambu-Gorkovbasis: Near the Fermi surface

  10. Dispersion relation At fixed vF onlyenergy and momentum along vFare relevant v1 v2 Infinite copies of 2-d physics

  11. Gap equation

  12. For TT0 At weak coupling density of states

  13. With G fixed by cSB at T = 0, requiring Mconst ~ 400 MeV and for typical values of m ~ 400 – 500 MeV one gets Evaluation from QCD first principles at asymptotic m(Son 1999) Notice the behavior exp(-c/g) and not exp(-c/g2) as one would expect from four-fermi interaction For m ~ 400 MeV one finds again

  14. The inhomogeneous phase (LOFF) • In many different situations the “would be” pairing fermions belong to Fermi surfaces with different radii: • Quarks with different masses • Requiring electrical neutrality and/or weak equilibrium

  15. Consider 2 fermions with m1 = M, m2 = 0 at the same chemical potential m. The Fermi momenta are To form a BCS condensate one needs common momenta of the pair pFcomm Grand potential at T = 0 for a single fermion

  16. Pairing energy Pairing possible if The problem may be simulated using massless fermions with different chemical potentials (Alford, Bowers & Rajagopal 2000) Analogous problem studied by Larkin & Ovchinnikov, Fulde & Ferrel 1964. Proposal of a new way of pairing. LOFF phase

  17. LOFF: 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. • Very difficult experimentally but claims of observations in heavy fermion superconductors (Gloos & al 1993) and in quasi-two dimensional layered organic superconductors (Nam & al. 1999, Manalo & Klein 2000)

  18. or paramagnetic impurities (dm ~ H) give rise to an energy additive term Gap equation Solution as for BCS D = DBCS, up to (for T = 0)

  19. According LOFF, close to first order line, possible condensation with non zero total momentum More generally fixed variationally chosen spontaneously

  20. blocking region Simple plane wave:energy shift Gap equation: For T T0

  21. The blocking region reduces the gap: Possibility of a crystalline structure (Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002) see later The qi’s define the crystal pointing at its vertices. The LOFF phase is studied via a Ginzburg-Landau expansion of the grand potential

  22. (for regular crystalline structures all the Dq are equal) The coefficients can be determined microscopically for the different structures (Bowers and Rajagopal (2002))

  23. Gap equation • Propagator expansion • Insert in the gap equation

  24. We get the equation Which is the same as with The first coefficient has universal structure, independent on the crystal. From its analysis one draws the following results

  25. Small window. Opens up in QCD?(Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu & Ren 2002)

  26. Results of Leibovich, Rajagopal & Shuster (2001)

  27. Along the critical line Single plane wave Critical line from

  28. General analysis(Bowers and Rajagopal (2002)) Preferred structure: face-centered cube

  29. Phonons In the LOFF phase translations and rotations are broken phonons Phonon field through the phase of the condensate (R.C., Gatto, Mannarelli & Nardulli 2002): Introduce:

  30. + Coupling phonons to fermions (quasi-particles) trough the gap term It is possible to evaluate the parameters of Lphonon (R.C., Gatto, Mannarelli & Nardulli 2002)

  31. Cubicstructure

  32. Using the symmetry group of the cube one gets: Coupling phonons to fermions (quasi-particles) trough the gap term

  33. This because the second order invariant for the cube and for the rotation group are the same! we get for the coefficients One can also evaluate the effective lagrangian for the gluons in the anisotropic medium. For the cube one finds Isotropic propagation

  34. LOFF phase in CSO Why the interest in the LOFF phase in QCD?

  35. In neutron stars CS can be studied at T = 0 For LOFF state from dpF ~ 0.75 DBCS Orders of magnitude from a crude model: 3 free quarks

  36. rn.m.is the saturation nuclear density ~ .15x1015 g/cm3 • At the core of the neutron star rB ~ 1015 g/cm3 Choosing m ~ 400 MeV Right ballpark (14 - 70 MeV)

  37. Glitches: discontinuity in the period of the pulsars. • Standard explanations require: metallic crust + superfluide inside (neutrons) • LOFF region inside the star might provide a crystalline structure + superfluid CFL phase • New possibilities for strange stars

  38. Outlook • Theoretical problems:Is the cube the optimal structure at T=0? Which is the size of the LOFF window? • Phenomenological problems: Better discussion of the glitches (treatment of the vortex lines) • New possibilities: Recent achieving ofdegenerate ultracold Fermi gasesopens up new fascinating possibilities of reaching the onset of Cooper pairing of hyperfine doublets. Possibility of observing theLOFF crystal?