Color Superconductivity in High Density QCD

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1 Color Superconductivity in High Density QCD Roberto Casalbuoni Department of Physics and INFN - Florence Bari,, September 9 October 1, 004 1

2 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 )?

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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 = = =

12 Also color neutrality requires V V = T = 0, = T = 0 µ µ 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

13 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 BCS CC δµ= BCS unstable phase Ω ρ δµ Normal unstable phase BCS CC δµ 0 δµ = 1 BCS 13

14 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

15 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

16 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

17 Q = 0 gsc V µ e = const. ( MeV) gap equation solutions of the gap equation neutrality line µ e = µ e ( MeV) 0 V( GeV/fm 3 ) Normal state µ e Normal state neutrality line µ =148 MeV e ( MeV) 17

18 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

19 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

20 Gaps in gcfl Q ru gd bs rd gu rs bu gs bd ru gd bs rd gu rs bu gs bd

21 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

22 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

23 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 LOFF? -0 SC gcfl CFL M /µ [MeV] S M S /µ[mev] 3

24 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

25 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

26 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

27 P = P = 4 P = 6 1/ 0-1/ x P = 6 P = Crystalline structures in LOFF 3/ 0-3/ 0-7

28 The LOFF phase is studied via a Ginzburg-Landau expansion of the grand potential Ω = α + β 4 + γ (for regular crystalline structures all the q are equal) The coefficients can be determined microscopically for the different structures (Bowers( and Rajagopal (00)) 8

29 Gap equation Propagator expansion Insert in the gap equation 9

30 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

31 ρ ΩBCS Ωnormal = ( BCS δµ 4 ) Ω LOFF Ωnormal = 0. 44ρ( δµ δµ ) LOFF 1.15 ( δµ δµ ) δµ δµ 1 = BCS / BCS Small window. Opens up in QCD? (Leibovich, Rajagopal & Shuster 001; Giannakis,, Liu & Ren 00) 31

32 General analysis (Bowers Bowers and Rajagopal (00)) Preferred structure: face-centered centered cube 3

33 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

34 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

35 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 ] CFL unpaired SC 4 gcfl gsc LOFF? M /µ [MeV] S Ω ρ δµ octahedron cube 35

36 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