Color Superconductivity: CFL and 2SC phases

Transcript

1 Color Superconductivity: CFL and SC phases Introduction Hierarchies of effective lagrangians Effective theory at the Fermi surface (HDET) Symmetries of the superconductive phases 1

2 Introduction Ideas about CS back in 1975 (Collins & Perry-1975, Barrois-1977, Frautschi-1978). Only in 1998 (Alford, Rajagopal & Wilczek; ; Rapp, Schafer, Schuryak & Velkovsky) ) a real progress. The phase structure of QCD at high-density depends on the number of flavors with mass m < µ. Two most interesting cases: N f =, 3. 3 Due to asymptotic freedom quarks are almost free at high density and we expect difermion condensation in the color channel 3 *.

3 Consider the possible pairings at very high density 0 ψ ψ 0 α ia β jb α, β color; i, j flavor; a,b spin Antisymmetry in spin (a,b) for better use of the Fermi surface Antisymmetry in color (α, β) β for attraction Antisymmetry in flavor (i,j) for Pauli principle 3

4 s p s p Only possible pairings LL and RR For µ >> m u, m d, m s Favorite state for N f = 3, CFL (color-flavor locking) (Alford, Rajagopal & Wilczek 1999) 0 ψ ψ 0 =- 0 ψ ψ 0 ε ε α β α β αβc il jl ir jr ijc Symmetry breaking pattern SU(3) SU(3) SU(3) SU(3) c L R c+l+r 4

5 Why CFL? 0 ψ ψ 0 ε ε α β αβc il jl ijc 0 ψ ψ 0 ε ε α β αβc ir jr ijc 5

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 il jl ij SU(3) c SU() L SU() R SU() c SU() L SU() R However, if µ is in the intermediate region we face a situation with fermions having different Fermi surfaces (see later). Then other phases could be important (LOFF, etc.) 6

7 Difficulties with lattice calculations Define euclidean variables: x ix, x x 4 i i 0 E E γ γ, γ iγ 4 i i 0 E E Dirac operator with chemical potential D( µ ) = γ D +µγ µ µ 4 E E E At µ = 0 D(0) = -D(0) γ5d(0)γ 5 =-D(0) Eigenvalues of D(0) pure immaginary λ 5 If eigenvector of D(0), eigenvector with eigenvalue - λ γ λ 7

8 det[d(0)] = ( λ)( λ ) > 0 For µ not zero the argument does not apply and one cannot use the sampling method for evaluating the determinant. However for isospin chemical potential and two degenerate flavors one can still prove the positivity. For finite baryon density no lattice calculation available except for small µ and close to the critical line (Fodor and Katz) 8

9 Hierarchies of effective lagrangians Integrating out heavy degrees of freedom we have two scales. The gap and a cutoff, δ above which we integrate out. Therefore: two different effective theories, 9 L HDET and L Golds

10 L HDET is the effective theory of the fermions close to the Fermi surface. It corresponds to the Polchinski description. The condensation is taken into account by the introduction of a mean field corresponding to a Majorana mass. The d.o.f.. are quasi-particles, holes and gauge fields. This holds for energies up to the cutoff. L Golds describes the low energy modes (E<<( ), as Goldstone bosons, ungapped fermions and holes and massless gauge fields,, depending on the breaking scheme. 10

11 Effective theory at the Fermi surface (HDET) Starting point: L QCD 1 a aµν LQCD =ψidψ FµνF +µψγ0ψ, a = 1,,8 4 a a µ a λa Dµ = µ + igsaµ T, D =γ µ D, T = at asymptotic µ >> Λ QCD, (p +µγ ) ψ (p) = 0 (p +µ ) ψ (p) = α p ψ(p) (p +µ ) = p p = E ± = µ± p 11

12 Introduce the projectors: P =, v v = = = p ± 1±α vf E(p) p ˆ F p p p= p p= p and decomposing: µ v F p= µ v + F F Hψ + =α ψ+ H ψ = ( µ+α ) ψ States ψ + close to the FS States ψ decouple for large µ F 1

13 Field-theoretical version: µ µ µ p =µ v + = v +, = ( v)v µ µ 0 v = (0,v), v = 1, = (, ) 4 d.of. 0,,v 4 dp ip x (x) e (p) 4 ψ = ψ ( π) Choosing v p = 0 Separation of light and heavy d.o.f. light d.o.f. µ δ p µ+δ, δ + δ heavyd.o.f. p µ δ, p µ +δ, δ, +δ 13

14 Separation of light and heavy d.o.f. heavy light light δ heavy 14

15 Momentum integration for the light fields 4 +δ + dp µ dvµ d dω d d = ( π) ( π) 4 π π ( π) δ The Fourier decomposition becomes dv i v x ψ (x) = e µ ψ v( x) 4π µ d i x ψv(x) = e ψ v( ) π ( π) ψ () v ψ(p) ( ) For any fixed v, -dim theory v 1 v 15

16 In order to decouple the states corresponding to to E Momenta dv µ i v x ψ (x) = e [ ψ (x) +ψ (x)] + 4π µ d ψ ψ = ψ π ( π) i x (x) P v(x) P e ± ± ± v( ) substituting inside LQCD and using Momenta from the Fermi sphere 4 µ dv d dx ψ (x) ψ (x) = ψ v() ψv() π 4 π ( π) 16

17 Proof: 4 µ dv F dv ' F d d ' dx ψ (x) ψ (x) = i π 4π 4 π ( π) ( π) i( π) δ ' +µ v' µ v ψ ( ') ψ ( ) ( ) 4 4 v' v Using: 4 4 ( ) ' v' v π δ +µ µ = ( ) ( ) ' 4 v' v = π δ πδ One gets easily the result. ( ) ( ) π µ 17

18 4 dv d x ψ (id +µγ0) ψ ψ+ iv D ψ + +ψ (µ+ iv D) ψ + ( ψ+ id ψ + h.c.) 4π µ µ V = (1,v), V = (1, v) γ = γ γ γ = γ γ µ 0 µ µ µ (,(v )v), ψ γψ = V ψ γψ = V ψψ µ µ ψψ µ µ ψ γψ =ψγψ µ µ + + ψ γψ =ψγψ µ µ + + Eqs.. of motion: 0 iv Dψ + + iγ D ψ = 0 0 (µ + iv D) ψ + iγ D ψ + = 0 18

19 At the leading order in µ: iv Dψ = 0 ψ = + 0 At the same order: L D dv = ψ iv Dψ 4π + + Propagator: 1 V p +µγ = (p +µ ) p V (T( ψψ)) 0 1 (p 0 +µ ) γ p γ V 1 µ 0 0 V = γ (1 α v) = P + γ ( T( ψ ψ) ) 19

20 Integrating out the heavy d.o.f. For the heavy d.o.f.. we can formally repeat the same steps leading to: 4 dv d x ψ (id +µγ0) ψ ψ+ iv D ψ + +ψ (µ+ iv D) ψ + ( ψ+ id ψ + h.c.) 4π Eliminating the E - fields one would get the non-local lagrangian: dv 1 = ψ ψ ψ ψ 4π µ+ iv D µν µν 1 µ ν ν µ P = g V V + V V µν LD + iv D + P + Dµ Dν + 0

21 Decomposing ψ =ψ +ψ h one gets h h LD = LD + LD + LD dv LD = ψ+ iv Dψ+ 4π h dv h µν 1 h LD = + iv D + P + Dµ Dν + h 4 ψ ψ ψ ψ + π iv D µ+ h dv h h µν h 1 h LD = ψ+ iv Dψ+ P ψ+ Dµ Dν ψ+ 4π µ+ iv D When integrating out the heavy fields 1

22 Contribute only if some gluons are hard, but suppressed by asymptotic freedom This contribution from L h D gives the bare Meissner mass

23 HDET in the condensed phase Assume A B ψ Cψ = (A, B AB (A, B collective indices) due to the attractive interaction: G A B C D LI = εabε V ab ABCDψaψbψa ψ b 4 V = V, V = V = V * ABCD CDAB ABCD BACD ABDC Decompose 3

24 L I = L + cond G CD* AT B G Lcond = VABCDΓ ψ Cψ VABCDΓ 4 4 G Lint = VABCD ψ ψ Γ ψ ψ 4 We define L int Cψ AB C D* ( AT C B AB )( C C D* +Γ CD* ) G G = V Γ, = V Γ CD * * CD* AB CDAB AB CDAB 1 1 L = ψ Cψ ψ Cψ * AT B A B* cond AB AB ψ and neglect L int. Therefore 4

25 dv 1 L = ψ iv Dψ +ψ iv Dψ + ψ Cψ ψ Cψ π A B A B * AT B A B D + + AB + AB + 4 AB ψ (x) ψ ( ± v,x) ± + Nambu-Gor kov basis L ψ A A 1 + χ = A* Cψ dv iv D = χ χ π A AB AB B D * 4 AB iv D AB S( ) 1 V = (V )(V ) V (, = 0) 5

26 From the definition: G = V ψ C ψ * C D* AB ABCD one derives the gap equation (e.g. via functional formalism) * dv µ d * 1 AB = igv ABCD CE 4 ππ ( π) D ED 1 1 D (V )(V ) AB = AB 6

27 Four-fermi interaction one-gluon exchange inspired Fierz using: L I 3 G 16 a µ a = ψγµ λ ψψγ λ ψ 8 a a ( λ ) αβ( λ ) δγ = (3 δαγδβδ δαβδγδ) a= 1 3 µ ( σµ ) ab ( σ ) dc = εac εbd µ µ σ = (1, σ), σ = (1, σ) G α β γ δ LI = V( αi)( βj)( γk)( δ) ψiψψ j k ψ 4 V = (3 δ δ δ δ ) δ δ ( αi)( βj)( γk)( δ) αδ βγ αγ βδ ik j 7

28 In the SC case = =ε ε ( αi)( βj) αβ3 ij dv d 4iG 4 π π ( π ) µ 0 δ G µ = ρ d ξ, ρ= 4 0 ξ + π 4 pairing fermions 8

29 G determined at T = 0. 0 M,, constituent mass ~ 400 MeV Λ 3 dp 1 1= 8G ( π ) p + M 0 3 with Λ=µ+δ For µ= MeV, Λ= 800 MeV, M = 400 MeV = MeV SC Similar values for CFL. 9

30 Gap equation in QCD 3 1 g cos θ (p 0) = dq 0 d(cos θ ) + 1π 1 cos θ+ G /( µ ) cos θ (q ) + 1 cosθ+ F/( µ ) q + (q )

31 Hard-loop approximation F= m D electric G 5/ 4 5 b = 56π g Nf π q0 = md 4 q magnetic q q 0 µ (p ) = dq log 0 g µ m N π D = f For small momenta magnetic gluons are unscreened and dominate giving a further logarithmic divergence g b (q 0) π p0 q 0 q 0 + (q 0) 31

32 Results: g b (p 0) 0sin log 3 π p 0 = bµ e 0 3π g g 1 (log( δ/ )) δe c c/g 5 µ>10 GeV from the double log To be trusted only for but, if extrapolated at MeV,, gives values for the gap similar to the ones found using a 4-fermi interaction. However condensation arises at asymptotic values of µ. 3

33 Symmetries of superconducting phases Consider again the 3 flavors, u,d,s and the group theoretical structure of the two difermion condensate: α ia ψ ψ β jb [(3,3 ) (3,3 )] = (3,3 ) (6,6 ) * * c L(R) c L(R) S c L(R) c L(R) 33

34 implying in general α β αβi α β β α il jl iji 6 i j i j ψ ψ = ε ε + ( δ δ +δ δ ) = α β β α α β β α i j i j 6 i j i j = ( δ δ δδ ) + ( δ δ +δδ ) = α β β α 6 i j 6 i j = ( + ) δ δ + ( ) δ δ In NJL case with cutoff 800 MeV, constituent mass 400 MeV and µ = 400 MeV = 85.3MeV, = 1.3MeV 6 34

35 Original symmetry: GQCD = SU(3) c SU(3) L SU(3) R U(1) B U(1) A broken to G = SU(3) Z Z CFL c+ L+ R anomalous anomalous # Goldstones = = massive 8 give mass to the gluons and 8+1 are true massless Goldstone bosons 35

36 Notice: Breaking of U(1) B makes the CFL phase superfluid The CFL condensate is not gauge invariant, but consider X =ε ε ( ψ ψ ), Y =ε ε ( ψ ψ ) k ijk α β k ijk α β γ αβγ il jl γ αβγ ir jr Σ = (Y X) = (Y ) X i i j * i j j α α α Σ is gauge invariant and breaks the global part of G QCD. Also det(x), det(y) break U(1) U(1) Z Z break B A 36

37 U(1) is broken by the anomaly but induced by a 6-6 A fermion operator irrelevant at the Fermi surface. Its contribution is parametrically small and we expect a very light NGB (massless at infinite chemical potential) Spectrum of the CFL phase Choose the basis: 1 ψ = λ ψ 9 α α i ( A) i A= 1 A λ A = 1,,8 Gell Mann matrices A λ 9 =λ 0 = 1 3 [ ] Tr λ λ = δ A B AB 37

38 Inverting: 1 1 [ ] A i α ψ = ( λa) αψ i = Tr λaψ αi 1 = λ λ ε ε = ε ( ) A I B I ψψ A B i j αβi T ( A) α( B) β iji Tr λελ I I ε ε = We get T Ig I g Tr[g] A B ψψ quasi-fermions ( ε I) αβ =ε αβ I for any 3x3 matrix g = δ A1,,8 = = A A AB 9 ε (p) = (v ) + A A = A = 9 = A 38

39 gluons Expected m g F g NG coupling constant but wave function renormalization effects important (see later) NG bosons Acquire mass through quark masses except for the one related to the breaking of U(1) B 39

40 NG boson masses quadratic in m q since the approximate invariance (Z ) L (Z ) R: ψl(r) ψl(r) and quark mass term: ψ Mψ + L R h.c. M M Notice: anomaly breaks (Z ) L (Z ) R through instantons,, producing a chiral condensate (6 fermions -> > ), but of order (Λ QCD /µ) 8 40

41 In-medium electric charge D a µ ψ= µ ψ igµ Taψ iψqa µ The condensate breaks U(1) em but leaves invariant a combination of Q and T 8. CFL vacuum: Define: SU(3) c X = Y =δ i i i α α α 1 1 QSU(3) T c 8 = diag,, + = Q Q = Q 1 1 Q= Q 1 1 Q QX XQ Q δ δ Q = 0 αβ βi αj ji leaves invariant the ground state 41

42 Eigs(Q) = 0, ± 1 Integers as in the Han-Nambu model A = A cosθ G sinθ g 8 µ µ µ µ = A sin θ+ G cos θ µ µ new interaction: rotated fields g g T 1+ ea 1 Q eqa + ggt 8 s µ 8 µ µ s µ e gs tan θ=, e = ecos θ, gs = 3 g cosθ s 3 ( ) ( T = cos θ Q 1+ sin θ) 1 Q 4

43 The rotated photon remains massless, whereas the rotated gluons acquires a mass through the Meissner effect A piece of CFL material for massless quarks would respond to an em field only through NGB: bosonic metal For quarks with equal masses, no light modes: transparent insulator For different masses one needs non zero density of electrons or a kaon condensate leading to massless excitations 43

44 In the SC case, new Q and B are conserved Q = Q 1 1 T 8 =, 1 1 (1,, ) B = B T 8 =,,,, = (0,0,1) u α, α= 1, d α, α= 1, u d 3 3 Q B 1/ 0-1/

45 Spectrum of the SC phase Remember =ε ε ( αi)( βj) αβ3 ij No Goldstone bosons α ψi α= 1, gapped ψ ungapped 3 i SU(3) c SU() 8 3 = 5 massive gluons c (equal gap) Light modes: 3 ψ i + 3gluons(M = 0) 45

46 +1 flavors µ m, µ m,m It could happen s u d µ m,m m,m,m u d s <µ< m u d s Phase transition expected E =µ= p + M p = µ M F F F M > M 1 The radius of the Fermi sphere decrases with the mass 46

47 Simple model: pf = µ m 1 s, pf =µ pf 1 3 pf 3 Ω = dp dp unpair p m p ( π) + µ + ( π) µ ( ) ( ) s pf comm p dp ( ) dp µ Ω pair = p + ms µ + ( p µ ) ( π) ( π) 4π 3 Fcomm Ω pair = 0 p =µ F ms µ comm pf 4 comm 1 Ωpair Ωunpair s µ 16π ( m 4 4 ) condensation energy Condensation if: m µ> s 47

48 Notice that the transition must be first order because for being in the pairing phase > m s 0 µ (Minimal value of the gap to get condensation) 48

49 Effective lagrangians Effective lagrangian for the CFL phase Effective lagrangian for the SC phase 49

50 Effective lagrangian for the CFL phase NG fields as the phases of the condensates in the (3,3) representation k ijk α β * k ijk α β * γ =εαβγε ψilψ jl γ =εαβγε ψirψjr X, Y Quarks and X, Y transform as ψ e g ψ g, ψ e g ψ g i( α+β) T i( α β) T L c L L R c R R iα iβ c c L,R L,R B A g SU(3), g SU(3), e U(1), e U(1) X g Xg e, Y g Yg e T i( α+β ) T i( α β ) c L c R 50

51 Since X, Y U(3) The number of NG fields is #X + #Y = (1+ 8) + (1+ 8) = 18 8 of these fields give mass to the gluons. There are only 10 physical NG bosons corresponding to the breaking of the global symmetry (we consider also the NGB associated to U(1) A ) SU(3) L SU(3) R U(1) V U(1) A SU(3) L+ R Z Z 51

52 Better use fields belonging to SU(3). Define and d X = X = det(x) i( Xˆ e = transforming as e ϕ+θ) 6i( ϕ+θ) Xˆ g Xˆ d Y T c g L ϕ ϕ α = Y = i( ϕ θ) Ŷe det(y) = Ŷ g e Ŷ T c g R θ θ β 6i( ϕ θ) The breaking of the global symmetry can be described by the gauge invariant order parameters ˆ ˆ ˆ ˆ i (Y j )*X i Σ= Y + X, d, d j α α X Y Σ = 5

53 Σ, d X, d Y are 10 fields describing the physical NG bosons. Also Σ T R g L g Σ shows that Σ T transforms as the usual chiral field. Consider the currents: ˆ ˆ ˆ ˆ ˆ ˆ ˆ µ µ µ µ µ µ JX = XD X = X( X + X g ) = X X + g µ ˆ µ ˆ ˆ µ ˆ ˆ µ ˆ µ ˆ µ JY = YD Y = Y( Y + Y g ) = Y Y + g µ µ a g = igsgat J g J g µ µ X,Y c X,Y c 53

54 Most general lagrangian up to two derivatives invariant under G, the space rotation group O(3) and Parity (R.C. & Gatto 1999) P: (X Y,J J, ϕ ϕ, θ θ) X F ( ) ( ) T 0 0 FT L= Tr JX J Y α T Tr JX + J Y + ( ϕ 0 ) + ( θ 0 ) F ( ) ( ) S FS 0 0 v ϕ v θ + Tr JX J Y +α S Tr JX + J Y ϕ θ 4 4 F ( ) ( ) T FT L= Tr Xˆ 0X ˆ Yˆ 0Y ˆ αt Tr Xˆ 0X ˆ + Yˆ 0Y ˆ + g F ( ) ( ) S ˆ ˆ ˆ ˆ FS ˆ ˆ ˆ ˆ Tr X X Y Y S Tr + +α X X + Y Y + g v ϕ v θ + ( 0ϕ ) + ( 0θ) ϕ θ 54 Y

55 Using SU(3) c gauge invariance we can choose: ˆ = ˆ = X Y e i Π T /F T Expanding at the second order in the fields ( a v L ) 0Π + ( 0ϕ ) + ( 0θ) Π ϕ θ F v v a ϕ vθ S = FT Gluons acquire Debye and Meissner masses (not the rest masses, see later) m =α gf, m =α gf =α vgf D T s T s S s S S s T a a 55

56 Low energy theory supposed to be valid at energies << gap. Since we will see that gluons have masses of order they can be decoupled F ( ) ( ) T FT L= Tr Xˆ 0X ˆ Yˆ 0Y ˆ αt Tr Xˆ 0X ˆ + Yˆ 0Y ˆ + g F ( ) ( ) S ˆ ˆ ˆ ˆ FS ˆ ˆ ˆ ˆ Tr X X Y Y S Tr + +α X X + Y Y + g v ϕ v θ + ( 0ϕ ) + ( 0θ) ϕ θ 1 g X ˆ X ˆ Y ˆ Y ˆ µ = µ + µ ( ) 56

57 we get the gauge invariant result: ~ χ-lagrangian F ( ) T + + LNGB = Tr t t v Tr 4 Σ Σ Σ Σ 1 1 ϕ ϕ θ θ ( ) ( ( ) t ) v ϕ ( t ) v θ Σ = YX ˆ + ˆ 57