QCD and Instantons: 12 Years Later. Thomas Schaefer North Carolina State

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1 QCD and Instantons: 12 Years Later Thomas Schaefer North Carolina State 1

2 ESQGP: A man ahead of his time 2

3 Instanton Liquid: Pre-History 1975 (Polyakov): The instanton solution r 2 2 E + B A a µ(x) = 2 η aµνx ν x 2 + ρ 2, X=0 τ X= ( t Hooft): Fermion zero modes G a µν G a µν = 192ρ4 (x 2 + ρ 2 ) 4. u L d L u R d R L = G det f ( ψ L,f ψ R,g ) G = dρ n(ρ) violates U(1) A but preserves SU(2) L,R... and contributes to the η mass 3

4 Phenomenology: Vector Channels (ρ and a 1 ) α s π c 1 G 2 x 4 + c 2 qq 2 x u L 1 Π(x)/Π 0 (x) α s π c 1 G 2 x 4 c 2 qq 2 x 6 d L ρ Aleph a 1 Aleph ρ OPE a 1 OPE +/ (L< >R) x [fm] 4

5 Phenomenology: Scalar Channels (π and δ) 10 Π(x)/Π 0 (x) 1 π lattice δ lattice π OPE δ OPE π,δ! 1 + c α α s π + c 1 G 2 x 4 + c 2 qq 2 x 6 u d L R +/ (L< >R) x [fm] 5

6 Phenomenology: OZI Violation σ δ u L d R Π(x)/Π 0 (x) σ δ η π ω ρ 3 loop O(αs,α 2 s) 3 O(1/Nc ) ω ρ η π u R +/ (L< >R) d L x [fm] 6

7 Phenomenology: Summary Only small effects in ( LL ± RR) 2. Sign changes for ( LR + RL) ( LR RL). Sign changes for (ūd)(ūd) (ūu)( dd). L = G det f ( ψ L ψ R ) + (L R) 7

8 The Instanton Liquid ES (1982): Instantons provide a quantitative description of QCD correlations functions (a) (b) ρ = 0.3 fm N V = 1 fm t z t z (c) (d) S δs 1 S t z 5 10 t z 8

9 The Instanton Ensemble Instanton liquid described by partition function (one parameter) Z = 1 N I!N A! N I +N A I [dω I n(ρ I )] det(d/ ) exp( S int ) Quark propagator S(x, y) = IJ ψ I (x) ( 1 ) T + im IJ ψ J (y) + S NZM(x, y) Instantons in QCD, Rev. Mod. Phys (1998) 9

10 Meson Correlation Functions pion delta (a 0 ) rho Π(OZI) 0 + Π(OZI) 0 ++ Π(OZI) Π(x)/Π 0 (x) 3.0 Π(x)/Π 0 (x) x [fm] x [fm] m π = 140 MeV (f π = 71 MeV) m ρ = 795 MeV m a0 1 GeV m ρ m ω m σ 580 MeV m η 1 GeV 10

11 V A Correlation Functions υ 1 a ALEPH τ (V,A, I=1) ν τ parton model/perturbative QCD Π V-A / (2Π 0 ) ALEPH data Instanton OPE Mass 2 (GeV/c 2 ) 2 Aleph spectral function τ (V, A, I =1)ν τ 1e x [fm] coordinate space correlator OPE, instanton liquid, data 11

12 Instantons in QCD: 12 Years Later Chirality and zero modes on the lattice High density QCD SUSY, large N c, AdS/CFT, AdS/QCD 12

13 Chiral Symmetry Breaking on the Lattice ψ λ L R L R 1.5e e+05 n(x) Number λ<λ( crit) λ>λ( crit) 1 (L) X +1 (R) Χ H (x)/ω H (x) chirality distribution from T. Blum et al., [hep-lat/ ] 13

14 Instantons and Color Superconductivity [MeV] N f =2 (OGE) N f =2 (OGE+INST) N f =3 (OGE) N f =3 (OGE+INST) = µ [MeV] 100 MeV, T c 60 MeV RSSV (1998), ARW (1998) 14

15 A pqcd Instanton Plasma (µ Λ QCD ) Schematic phase diagram (Here: N f = N c = 2) T <qq> <qq> µ diquark condensate breaks U(1) B and U(1) A q L q L = ρ e i(χ+φ)/2 q R q R = ρ e i(χ φ)/2 Effective lagrangian for U(1) A Goldstone boson Son, Stephanov, Zhitnitsky φ [ L = f2 2 ( 0 φ) 2 v 2 ( i φ) 2] V (φ + θ) + L(ρ, χ) ρ V (φ + θ) vanishes in perturbation theory 15

16 η Mass at Large Baryon Density Instanton induced effective interaction for quarks with p p F 1 q q n(ρ, µ) = n(ρ,0) exp [ N f ρ 2 µ 2] u L u R L/R d L d R ρ µ 1 Λ 1 QCD Instanton contribution to vacuum energy L L = A cos(φ + θ) G I L R R A = C N Φ [ 2 log ( )] ( µ 4 Λ Λ µ ) 8 Λ 2 η mass satisfies Witten-Veneziano relation f 2 m 2 φ = A 16

17 Very dilute instanton gas R D ρ r IA R D ρ µ 1 r IA = A 1/4 R D = m 1 φ A is the local topological susceptibility A = χ top (V ) = Q2 top V V r 4 IA V R4 D Global topological susceptibility vanishes χ top = lim V Q 2 top V V = 0 (m = 0) 17

18 Instantons and Large N c n(ρ) N c =3 N c =4 N c =5 N c =6 µ ¾ ½ ρ [Λ 1 ] ¼ ¼ ½ ¾ ½¼ ½½ ½¾ B. Lucini, M. Teper qq N c χ top 1 m 2 η 1/N c 18

19 From Instanton to Monopoles Kraan, van Baal: Instantons with non-zero holonomy Monopole constituents with fractional top charge ( confinement?) New WCI calculation of gluino condensate 1 16π 2 Tr[ λλ] = Λ 3 exp(2πik/n c ) 19

20 AdS/CFT: N = 4 SUSY Yang Mills String/field theory duality (Maldacena) N = 4 SUSY YM IIB strings on AdS 5 S 5 λ = g 2 N (l s /R) 4 0 (g 2 0) (g s 0) String theory contains D-instantons characterized by location on AdS 5 S 5 field theory instantons d 4 x dρ ρ 5 dλ ab Charge k instanton amplitudes AdS 5 S 5 (AdS 5 S 5 ) k (AdS 5 S 5 ) k instantons in commuting SU(2) s (bound by fermions) 20

21 Instantons and AdS/QCD Add singlet field Y = Y e ia to AdS/QCD ( axion ) S = d 5 x { 1 g 2 DY 2 + κ 0 ( Y N f det(x) + h.c. ) } 2 Katz & Schwartz (2007) Topological charge correlator: Treat κa 2 as a perturbation Π P (Q) = 1 zm [ ] 2 dz 1 2N f z 5 κ 2 (Qz)2 K 2 (Qz), Compare to instanton result 0 AdS 5 measure (Bulk-to-boundary prop) 2 Π P (Q) = 2 dρ ρ 5 d(ρ) [ ] (Qρ)2 K 2 (ρq), instanton measure (F-trafo of G G I ) 2 21

22 Happy Birthday Edward!! ES ES 22