QCD and Instantons: 12 Years Later Thomas Schaefer North Carolina State 1
ESQGP: A man ahead of his time 2
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=1 1976 ( 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
Phenomenology: Vector Channels (ρ and a 1 ) 1.5 1 + α s π c 1 G 2 x 4 + c 2 qq 2 x 6 1.25 u L 1 Π(x)/Π 0 (x) 0.75 1 + α s π c 1 G 2 x 4 c 2 qq 2 x 6 d L 0.5 0.25 ρ Aleph a 1 Aleph ρ OPE a 1 OPE +/ (L< >R) 0 0 0.25 0.5 0.75 1 x [fm] 4
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) 0.1 0 0.25 0.5 0.75 1 x [fm] 5
Phenomenology: OZI Violation 2 1.5 1 σ δ u L d R Π(x)/Π 0 (x) 0.5 0-0.5-1 -1.5 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 -2 0 0.2 0.4 0.6 0.8 1 x [fm] 6
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
The Instanton Liquid ES (1982): Instantons provide a quantitative description of QCD correlations functions (a) (b) 4 0.0505 3 15 0 15 ρ = 0.3 fm N V = 1 fm 4 2 5 10 t 15 20 5 10 z -0.05 5 10 t 15 20 5 10 z (c) (d) S 10 1 0.03 0.002 δs 1 S 0.0202 0.01 0 10 15 0.00101 0-0.001 10 15 5 10 t 15 20 5 z 5 10 t 15 20 5 z 8
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
Meson Correlation Functions 6.0 5.0 pion delta (a 0 ) rho 3.0 2.0 Π(OZI) 0 + Π(OZI) 0 ++ Π(OZI) 1 4.0 1.0 Π(x)/Π 0 (x) 3.0 Π(x)/Π 0 (x) 0.0 2.0 1.0 1.0 2.0 0.0 0.0 0.5 1.0 x [fm] 3.0 0.0 0.5 1.0 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
V A Correlation Functions υ 1 a 1 2.5 2 1.5 1 ALEPH τ (V,A, I=1) ν τ parton model/perturbative QCD Π V-A / (2Π 0 ) 1 0.1 0.01 ALEPH data Instanton OPE 0.5 0.001 0-0.5 0.0001-1 0 0.5 1 1.5 2 2.5 3 3.5 Mass 2 (GeV/c 2 ) 2 Aleph spectral function τ (V, A, I =1)ν τ 1e-05 0 0.5 1 1.5 x [fm] coordinate space correlator OPE, instanton liquid, data 11
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
Chiral Symmetry Breaking on the Lattice ψ λ L R L R 1.5e+05 01 000 111 000 111 00000 11111 00000 11111 00000 11111 000000 111111 0000000 1111111 000000000 111111111 0000000000000 1111111111111 0000000000000 1111111111111 01 000 111 000 111 00000 11111 00000 11111 00000 11111 000000 111111 0000000 1111111 000000000 111111111 0000000000000 1111111111111 0000000000000 1111111111111 1e+05 n(x) Number λ<λ( crit) 50000 λ>λ( crit) 1 (L) X +1 (R) 0 1 0.5 0 0.5 1 Χ H (x)/ω H (x) chirality distribution from T. Blum et al., [hep-lat/0105006] 13
Instantons and Color Superconductivity [MeV] 100 50 20 N f =2 (OGE) N f =2 (OGE+INST) N f =3 (OGE) N f =3 (OGE+INST) = 10 + 5 500 1000 1500 2000 2500 3000 µ [MeV] 100 MeV, T c 60 MeV RSSV (1998), ARW (1998) 14
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
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
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
Instantons and Large N c n(ρ) 1.0 10 5 N c =3 N c =4 N c =5 N c =6 µ ¾ 5.0 10 4 ½ 0.0 0 0.1 0.2 0.3 0.4 0.5 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
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
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
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) [ ] 2 1 2 (Qρ)2 K 2 (ρq), instanton measure (F-trafo of G G I ) 2 21
Happy Birthday Edward!! ES ES 22