Color Superconductivity in High Density QCD Roberto Casalbuoni Department of Physics and INFN - Florence Bari,, September 9 October 1, 004 1
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 high µ Asymptotic region in µ fairly well understood: existence of a CS phase.. Real question: does this type of phase persists at relevant densities ( ~5-6 ρ 0 )?
Summary Mini review of CFL and SC phases Pairing of fermions with different Fermi momenta The gapless phases gsc and gcfl The LOFF phase 3
CFL and SC Study of CS back to 1977 (Barrois( 1977, Frautschi 1978, Bailin and Love 1984) based on Cooper instability: At T ~ 0 a degenerate fermion gas is unstable Any weak attractive interaction leads to Cooper pair formation Hard for electrons (Coulomb vs. phonons) Easy in QCD for di-quark formation (attractive channel ) 3 (3 3 = 3 6) 4
In QCD, CS easy for large µ due to asymptotic freedom At high µ,, m s, m d, m u ~ 0, 3 colors and 3 flavors Possible pairings: 0 ψ ψ 0 α ia β jb Antisymmetry in color (α,( β) for attraction Antisymmetry in spin (a,b( a,b) ) for better use of the Fermi surface Antisymmetry in flavor (i, j) for Pauli principle 5
s s Only possible pairings p p LL and RR Favorite state CFL (color-flavor locking) (Alford, Rajagopal & Wilczek 1999) α β α β αβc 0 ψalψbl 0 =- 0 ψarψbr 0 = ε εabc Symmetry breaking pattern SU(3) SU(3) SU(3) SU(3) c L R c+l+r 6
What happens going down with µ?? If µ << m s we get 3 colors and flavors (SC) 0 ψ α ψ β 0 = ε αβ3 ε al bl ab SU(3) c SU() L SU() R SU() c SU() L SU() R But what happens in real world? 7
M s not zero Neutrality with respect to em and color Weak equilibrium All these effects make Fermi momenta of different fermions unequal causing problems to the BCS pairing mechanism 8
Consider fermions with m 1 = M, m = 0 at the same chemical potential µ.. The Fermi momenta are p F1 = µ M p F = µ Effective chemical potential for the massive quark M eff M Mismatch: µ = µ µ µ M δµ µ 9
If electrons are present, weak equilibrium makes chemical potentials of quarks of different charges unequal: d ue ν µ µ =µ d u e In general we have the relation: ( µ =µ+ Q µ ) i Q µ = µ e Q N.B. µ e is not a free parameter 10
Neutrality requires: V = Q = 0 µ e Example SC: normal BCS pairing when But neutral matter for µ =µ n = n u d u d 1/3 1 nd nu µ d µ u µ e =µ d µ u µ u 0 4 Mismatch: d u pf pf µ d µ u µ e µ u δ µ = = = 0 8 11
Also color neutrality requires V V = T = 0, = T = 0 µ µ 3 8 3 8 As long as δµ 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 1
In a simple model with two fermions at chemical potentials µ+δµ,, µ δµ µ the system becomes normal at the Chandrasekhar-Clogston Clogston point. Another unstable phase exists. 0 1. 1 0.8 BCS CC δµ= BCS 0.6 0.4 unstable phase Ω ρ 0 0. 0. 0.4 0.6 0.8 1 1. δµ 0 0.4 Normal unstable phase 0. -0. -0.4 0. 0.4 0.6 0.8 1 1. BCS CC δµ 0 δµ = 1 BCS 13
The point δµ = is special. In the presence of a mismatch new features are present. The spectrum of quasiparticles is E δµ E(p) = δµ± (p µ ) + blocking region gapless modes Energy cost for pairing Energy gained in pairing δµ = 0 δµ = δµ > p For δµ <, the gaps are δµ and + δµ For δµ =, an unpairing (blocking) region opens up and gapless modes are present. E(p) = 0 p =µ± δµ begins to unpair δµ> 14
gsc 4x3 fermions: quarks ungapped q ub Same structure of condensates as in SC ub, q db 4 quarks gapped q ur, q ug, q dr, q dg General strategy (NJL model): (Huang & Shovkovy,, 003) 0 ψ α ψ β 0 = ε αβ3 ε al bl ab Write the free energy: V( µµ,, µ, µ, ) 3 8 e Solve: Neutrality V V V = = = 0 µ µ µ e 3 8 Gap equation V 0 = 15
For δµ > (δµ δµ=µ e /) ) gapped quarks become gapless. The gapless quarks begin to unpair destroying the BCS solution. But a new stable phase exists, the gapless SC (gsc) phase. It is the unstable phase which becomes stable in this case (and CFL, see later) when charge neutrality is required. 16
Q = 0 gsc V µ e = const. ( MeV) 00 175 150 15 100 75 50 5 gap equation solutions of the gap equation neutrality line µ e = 50 100 150 00 50 300 350 400 µ e ( MeV) 0 V( GeV/fm 3 ) - 0.081-0.08-0.083-0.084-0.085-0.086-0.087 Normal state µ e Normal state neutrality line µ =148 MeV e 0 40 60 80 100 10 ( MeV) 17
But evaluation of the gluon masses (5 out of 8 become massive) shows an instability of the gsc phase. Some of the gluon masses are imaginary (Huang( and Shovkovy 004). Possible solutions are: gluon condensation, or another phase takes place as a crystalline phase (see later), or this phase is unstable against possible mixed phases. Potential problem also in gcfl (calculation not yet done). 18
Generalization to 3 flavors gcfl α β αβ1 αβ αβ3 0 ψalψbl 0 = ε 1 ε ab1 + ε ε ab + ε 3 εab3 Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist) CFL : 1 gsc : 0, = = gcfl: = = 3 > 3 1 > 3 1 = 0 19
Gaps in gcfl Q ru gd bs rd gu rs bu gs bd 0 0 0-1 +1-1 +1 0 0 ru gd bs rd gu rs bu gs bd 3 3 1 1 3 3 1 1 0
Strange quark mass effects: Shift of the chemical potential for the strange quarks: M s µ αs µ αs µ Color and electric neutrality in CFL requires M s µ 8 =, µ 3 =µ e = 0 µ gs-bd unpairing catalyzes CFL to gcfl 1 M δµ ( ) s bd gs = µ bd µ gs = µ 8 = µ M s δµ rd gu =µ e, δµ rs bu =µ e µ 1
It follows: M µ Energy cost for pairing Energy gained in pairing begins to unpair M > µ Again, by using NJL model (modelled( on one-gluon exchange): Write the free energy: Solve: Neutrality Gap equations V( µµ,, µ, µ,m, ) 3 8 e s i V V V = = = 0 µ µ µ e 3 8 V i = 0
CFL # gcfl nd order transition at M s /µ ~, when the pairing gs-bd starts breaking gcfl has gapless quasiparticles.. Interesting transport properties Energy Difference [10 6 MeV 4 ] Gap Parameters [MeV] 10 gsc unpaired 4 0-10 LOFF? -0 SC gcfl -30-40 CFL -50 0 5 50 75 100 15 150 M /µ [MeV] S 30 5 0 15 10 5 3 1 0 0 5 50 75 100 15 150 M S /µ[mev] 3
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 4
According to LOFF, close to first order point (CC point), possible condensation with non zero total momentum p 1 = k + q p = k + q More generally ψ(x) ψ(x) = e ψ(x) ψ (x) = m iq x m c e m iq m x p + p 1 = q q fixed variationally q / q chosen spontaneously 5
Simple plane wave: E(p) E( p q) (p ) µ ± + µ δµ µ + µ µ = δµ v F q Also in this case, for µ=δµ v q< a unpairing (blocking) region opens up and gapless modes are present Possibility of a crystalline structure (Larkin( & Ovchinnikov 1964, Bowers & Rajagopal 00 iq x ψ(x) ψ (x) = q = 1.δµ i e F 00) The q i s define the crystal pointing at its vertices. i 6
P = P = 4 P = 6 1/ 0-1/ 1 0-1 x P = 6 P = 8 1 0-1 Crystalline structures in LOFF 3/ 0-3/ 0-7
The LOFF phase is studied via a Ginzburg-Landau expansion of the grand potential Ω = α + β 4 + γ 3 6 + (for regular crystalline structures all the q are equal) The coefficients can be determined microscopically for the different structures (Bowers( and Rajagopal (00)) 8
Gap equation Propagator expansion Insert in the gap equation 9
We get the equation α + β 3 + γ 5 + = 0 Which is the same as α = β 3 = Ω = 0 with The first coefficient has universal structure, independent on the crystal. From its analysis one draws the following results γ 5 = 30
ρ ΩBCS Ωnormal = ( BCS δµ 4 ) Ω LOFF Ωnormal = 0. 44ρ( δµ δµ ) LOFF 1.15 ( δµ δµ ) δµ δµ 1 = BCS / 0.754 BCS Small window. Opens up in QCD? (Leibovich, Rajagopal & Shuster 001; Giannakis,, Liu & Ren 00) 31
General analysis (Bowers Bowers and Rajagopal (00)) Preferred structure: face-centered centered cube 3
Effective gap equation for the LOFF phase (R.C., R.C., M. Ciminale,, M. Mannarelli,, G. Nardulli,, M. Ruggieri & R. Gatto,, 004) For the single plane wave (P = 1) the pairing region is defined by { for (p, v F) PR eff = θ(e u) θ (E d) = 0 elsewhere E =± ( δµ v q) + (p µ ) + u,d F 33
For P plane waves we approximate the pairing region as thesum of the regions P k such that P = (p,v ) (p,v ) = k { } k F E F P (p,v ) = (p,v q ) E F eff F m m= 1 The result can be interpreted as having P quasi-particles each of one having a gap k, k =1,, P. The approximation is better far from a second order transition and it is exact for P = 1 (original FF case). 34
Multiple phase transitions from the CC point (M( s /µ µ = 4 SC ) up to the cube case (M s /µ ~ 7.5 SC ). Extrapolating to CFL ( SC ~ 30 MeV) ) one gets that LOFF should be favored from about M s /µ ~10 MeV up M s /µ µ 5 MeV Energy Difference [10 6 MeV 4 ] 10 0-10 -0-30 -40 CFL unpaired SC 4 gcfl gsc LOFF? -50 0 5 50 75 100 15 150 M /µ [MeV] S Ω ρ 0 0.05 0.7 0.8 0.9 1.1 1. 1.3 1.4 δµ -0.05 0-0.1 octahedron cube 35
Conclusions Under realistic conditions (M s not zero, color and electric neutrality) new CS phases might exist In these phases gapless modes are present. This result might be important in relation to the transport properties inside a CSO. 36