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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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