x(1 x) b 2 dζ 2 m 2 1 x dζ 2 d dζ 2 + V (ζ) ] + k2 + m2 2 1 x k2 + m2 1 dζ 2 + m2 1 x + m2 2 LF Kinetic Energy in momentum space Holographic Variable

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1 [ d dζ + V (ζ) ] φ(ζ) = M φ(ζ) m 1 de Teramond, sjb x ζ = x(1 x) b m b (1 x) Holographic Variable d dζ k x(1 x) LF Kinetic Energy in momentum space Assume LFWF is a dynamical function of the quark-antiquark invariant mass squared d dζ d dζ + m 1 x + m 1 x k + m 1 x + k + m 1 x March 8,

2 Hadronization at the Amplitude Level τ = x + e + γ q q g x, k Event amplitude generator Construct helicity amplitude using Light-Front Perturbation theory; coalesce quarks via LFWFs e 1 x, k p H Hard Wall Confinement: Capture if ζ = x(1 x)b > 1 Λ QCD i.e., M = k x(1 x) < Λ QCD ψ(x, k, λ i ) March 8,

3 Result: Soft-Wall LFWF for massive constituents ψ(x, k ) = 4πc κ x(1 x) e 1 κ k x(1 x) + m 1 x + m 1 x LF WF in impact space: soft-wall model with massive quarks ψ(x, b ) = c κ» 1 x(1 x) e κ x(1 x)b 1 m κ 1 π z ζ χ χ = b x(1 x) + 1 κ 4 [m 1 x + m 1 x ] «+ x m 1 x March 8,

4 ψ J/ψ (x, b) b[gev 1 ] J/ψ LFWF peaks at x i = where m i P n j m j m i = m + k minimum of LF energy denominator κ = GeV x m a = m b = 1.5 GeV March 8,

5 π + >= u d > K + >= u s > m u = MeV m d = 5 MeV m s = 95 MeV D + >= c d > η c >= c c > m c = 1.5 GeV B + >= u b > m b = 4. GeV b[gev 1 ] x η b >= b b > κ = 375 MeV 90

6 First Moment of Kaon Distribution Amplitude < ξ >= 1 1 dξ ξ φ(ξ) ξ = 1 x < ξ > K = 0.04 ± 0.0 ξ κ = 375 MeV Range from m s = 65 ± 5 MeV (PDG) < ξ > K = 0.09 ± 0.00 < ξ > K = 0.07 ± March 8, Donnellan et al. Braun et al. 91

7 m s = 65 ± 5 MeV (PDG) M ξ M ξ M π 0.5 K 0.04 ± 0.0 a 0.35 ± a D η c 0.0 B η b 0.00 π 0.8 ± 0.03 b K 0.09 ± 0.00 b 0.7 ± 0.0 b Lattice π 0.69 ± c K 0.07 ± c 0.60 ± c M. A. Donnellan et al., Lattice Results for Vector Meson Couplings and Parton Distribution Amplitudes, arxiv: [hep-lat]. March 8, V. M. Braun et al., Moments of pseudoscalar meson distribution amplitudes from the lattice, Phys. Rev. D 74, (006) [arxiv:hep-lat/060601]. W. I. Weisberger, Partons, electromagnetic mass shifts, b: Lattice c: Lattice 9

8 AdS/CFT and Integrability L. Infeld, On a new treatment of some eigenvalue problems, Phys. Rev. 59, 737 (1941). Generate eigenvalues and eigenfunctions using Ladder Operators Apply to Covariant Light-Front Radial Dirac and Schrodinger Equations March 8,

9 Algebraic Structure, Integrability and Stability Conditions (HW Model) If L > 0 the LF Hamiltonian, H LF, can be written as a bilinear form H L LF (ζ) = Π L (ζ)π L(ζ) in terms of the operator Π L (ζ) = i ( d dζ L + 1 ζ ), and its adjoint ( Π L (ζ) = i d dζ + L + 1 ζ ), with commutation relations [Π L (ζ), Π L (ζ) ] = L + 1 ζ. For L 0 the Hamiltonian is positive definite φ H L LF φ = dζ Π L φ(z) 0 and thus M 0. March 8,

10 Ladder Construction of Orbital States Orbital excitations constructed by the L-th application of the raising operator a L = iπ L on the ground state: a L = c L L + 1. In the light-front ζ-representation φ L (ζ) = ζ L = C L ζ ( ζ) L = C L ζjl (ζm). ( 1 ζ ) d L J 0 (ζm) dζ The solutions φ L are solutions of the light-front equation (L = 0, ±1, ±, ) [ d dζ 1 ] L 4ζ φ(ζ) = M φ(ζ), Mode spectrum from boundary conditions : φ (ζ = 1/Λ QCD ) = 0. March 8,

11 Non-Conformal Extension of Algebraic Integrability (SW Model) Soft-wall model [Karch, Katz, Son and Stephanov (006)] retain conformal AdS metrics but introduce smooth cutoff which depends on the profile of a dilaton background field ϕ(z). Consider the generator (short-distance Coulombic and long-distance linear potential) Π L (ζ) = i ( d dζ L + 1 ζ κ ζ ), and its adjoint ( Π L (ζ) = i d dζ + L + 1 ζ + κ ζ ), with commutation relations The LF Hamiltonian [Π L (ζ), Π L (ζ) ] = L + 1 ζ κ. H LF = Π L Π L + C Integrable! is positive definite φ H LF φ 0 for L 0, and C 4κ. Orbital and radial excited states are constructed from the ladder operators from the L = 0 state. March 8,

12 Baryons in Ads/CFT Baryons in bottom-up approach to holographic QCD: { GdT and SJB (004)}. Top-down Sakai-Sugimoto model: { Hong, Rho, Yee and Yi (007); Hata, Sakai,Sugimoto, Yamato (007)}. March 8,

13 Baryons Spectrum in bottom-up holographic QCD GdT and Brodsky: hep-th/ , hep-th/ Baryons in Ads/CFT Action for massive fermionic modes on AdS d+1 : S[Ψ, Ψ] = d d+1 x g Ψ(x, z) ( ) iγ l D l µ Ψ(x, z). Equation of motion: ( iγ l D l µ ) Ψ(x, z) = 0 [ ( i zη lm Γ l m + d ) Γ z ] + µr Ψ(x l ) = 0. March 8,

14 Baryons Holographic Light-Front Integrable Form and Spectrum In the conformal limit fermionic spin- 1 modes ψ(ζ) and spin- 3 modes ψ µ(ζ) are two-component spinor solutions of the Dirac light-front equation where H LF = απ and the operator απ(ζ)ψ(ζ) = Mψ(ζ), Π L (ζ) = i ( d dζ L + ) 1 γ 5, ζ and its adjoint Π L (ζ) satisfy the commutation relations ] [Π L (ζ), Π L (ζ) = L + 1 ζ γ 5. Supersymmetric QM between bosonic and fermionic modes in AdS? March 8,

15 Note: in the Weyl representation (iα = γ 5 β) iα = 0 I, β = I 0 0 I, γ 5 = I 0 I 0. 0 I Baryon: twist-dimension 3 + L (ν = L + 1) O 3+L = ψd {l1... D lq ψd lq+1... D lm }ψ, L = m l i. i=1 Solution to Dirac eigenvalue equation with UV matching boundary conditions ψ(ζ) = C ζ [J L+1 (ζm)u + + J L+ (ζm)u ]. Baryonic modes propagating in AdS space have two components: orbital L and L + 1. Hadronic mass spectrum determined from IR boundary conditions ψ ± (ζ = 1/Λ QCD ) = 0, given by M + ν,k = β ν,kλ QCD, M ν,k = β ν+1,kλ QCD, with a scale independent mass ratio. March 8,

16 8 (a) N (600) I = 1/ (b) I = 3/! (40) (GeV ) 6 4 N (1700) N (1675) N (1650) N (1535) N (150) N (50) N (190) N (0)! (13)! (1950)! (190)! (1910)! (1905)! (1930) A14 N (939) N (170) N (1680) 0 L 4 6! (1700)! (160) 0 L Fig: Light baryon orbital spectrum for Λ QCD = 0.5 GeV in the HW model. The 56 trajectory corresponds to L even P = + states, and the 70 to L odd P = states. March 8,

17 SU(6) S L Baryon State N 1 + (939) (13) 1 N 1 1 N N 3 1 (1535) N 3 (150) (1650) N 3 (1700) N 5 (1675) + (1910) 3 3 N 5 3 N 3 N N N 9 5 N 7 (160) 3 (1700) + (170) N 5 + (1680) + (190) 5 + (1905) 7 + (1950) N 7 N 7 (190) N 9 (50) (1930) 7 N 9 N 9 + (0) 9 + N 11 N (600) + (40) N 13 March 8,

18 Non-Conformal Extension of Algebraic Structure (Soft Wall Model) We write the Dirac equation (απ(ζ) M) ψ(ζ) = 0, in terms of the matrix-valued operator Π Π ν (ζ) = i ( d dζ ν + ) 1 γ 5 κ ζγ 5, ζ and its adjoint Π, with commutation relations [ ] Π ν (ζ), Π ν(ζ) = ( ) ν + 1 ζ κ γ 5. Solutions to the Dirac equation ψ + (ζ) z 1 +ν e κ ζ / L ν n(κ ζ ), ψ (ζ) z 3 +ν e κ ζ / L ν+1 n (κ ζ ). Eigenvalues M = 4κ (n + ν + 1). March 8,

19 Baryon: twist-dimension 3 + L (ν = L + 1) O 3+L = ψd {l1... D lq ψd lq+1... D lm }ψ, L = m l i. i=1 Define the zero point energy (identical as in the meson case) M M 4κ : M = 4κ (n + L + 1). Proton Regge Trajectory March 8, κ = 0.49GeV 104

20 Space-Like Dirac Proton Form Factor Consider the spin non-flip form factors F + (Q ) = g + F (Q ) = g dζ J(Q, ζ) ψ + (ζ), dζ J(Q, ζ) ψ (ζ), where the effective charges g + and g are determined from the spin-flavor structure of the theory. Choose the struck quark to have S z = +1/. The two AdS solutions ψ + (ζ) and ψ (ζ) correspond to nucleons with J z = +1/ and 1/. For SU(6) spin-flavor symmetry F p 1 (Q ) = dζ J(Q, ζ) ψ + (ζ), F1 n (Q ) = 1 dζ J(Q, ζ) [ ψ + (ζ) ψ (ζ) ], 3 where F p 1 (0) = 1, F n 1 (0) = 0. March 8,

21 Scaling behavior for large Q : Q 4 F p 1 (Q ) constant Proton τ = 3 Q 4 F p 1 (Q ) (GeV 4 ) A Q (GeV ) SW model predictions for κ = 0.44 GeV. Data analysis from: M. Diehl et al. Eur. Phys. J. C 39, 1 (005). March 8,

22 Dirac Neutron Form Factor (Valence Approximation) Truncated Space Confinement Q 4 F n 1 (Q ) [GeV 4 ] Q [GeV ] Prediction for Q 4 F n 1 (Q ) for Λ QCD = 0.1 GeV in the hard wall approximation. Data analysis from Diehl (005). March 8,

23 Scaling behavior for large Q : Q 4 F n 1 (Q ) constant Neutron τ = 3 0 Q 4 F n 1 (Q ) (GeV 4 ) A1 Q (GeV ) SW model predictions for κ = 0.44 GeV. Data analysis from M. Diehl et al. Eur. Phys. J. C 39, 1 (005). March 8,

24 1.5 F p (Q ) 1 Spacelike Pauli Form Factor From overlap of L = 1 and L = 0 LFWFs Harmonic Oscillator Confinement Normalized to anomalous moment κ = 0.49 GeV Preliminary Q (GeV ) March 8, G. de Teramond, sjb 109

25 Holographic Connection between LF and AdS/CFT Predictions for hadronic spectra, light-front wavefunctions, interactions Deduce meson and baryon wavefunctions, distribution amplitude, structure function from holographic constraint Identification of Orbital Angular Momentum Casimir for SO(): LF Rotations Extension to massive quarks March 8,

26 Diffractive Dissociation of Pion into Quark Jets E791 Ashery et al. M k ψ π (x, k ) Measure Light-Front Wavefunction of Pion Minimal momentum transfer to nucleus Nucleus left Intact! March 8,

27 E791 FNAL Diffractive DiJet Gunion, Frankfurt, Mueller, Strikman, sjb Frankfurt, Miller, Strikman Two-gluon exchange measures the second derivative of the pion light-front wavefunction π q q M k ψ π (x, k ) N N March 8,

28 Key Ingredients in E791 Experiment Brodsky Mueller Frankfurt Miller Strikman Small color-dipole moment pion not absorbed; interacts with each nucleon coherently QCD COLOR Transparency π q q M A = A M N dσ dt (πa q qa ) = A dσ dt (πn q qn ) FA (t) A N A Target left intact Diffraction, Rapidity gap March 8,

29 Color Transparency Bertsch, Gunion, Goldhaber, sjb A. H. Mueller, sjb Fundamental test of gauge theory in hadron physics Small color dipole moments interact weakly in nuclei Complete coherence at high energies Clear Demonstration of CT from Diffractive Di-Jets March 8,

30 ± ± ± Nuclear coherence Nuclear coherence F A (q ) e 1 3 R A q March 8,

31 Mueller, sjb; Bertsch et al; Frankfurt, Miller, Strikman Measure pion LFWF in diffractive dijet production Confirmation of color transparency A-Dependence results: σ A α k t range (GeV/c) α α (CT) 1.5 < k t < < k t < ± < k t < ± Ashery E791 α (Incoh.) = 0.70 ± 0.1 Conventional Glauber Theory Ruled Out! March 8, Factor of 7 116

32 ( ) E791 Diffractive Di-Jet transverse momentum distribution dσ dk T k 6.5 T Gaussian Two Components High Transverse momentum dependence consistent with PQCD, ERBL Evolution k 6.5 T Gaussian component similar to AdS/CFT HO LFWF k T (GeV) March 8,

33 asympt CZ x x Ashery E791 Narrowing of x distribution at higher jet transverse momentum x Fig.. The u distribution of diffractive dijets from the platinum target for 1.5 k t 1.5 GeV/c (left) and for 1.5 k t.5 GeV/c (right). The solid line is a fit to a combination of the asymptotic and CZ distribution amplitudes. The dashed line shows the contribution from the asymptotic function and the dotted line that of the CZ function. March 8, 008 Possibly two components: Nonperturbative (AdS/CFT) and φ(x) x(1 x) Perturbative (ERBL) Evolution to asymptotic distribution

34 1.5 Linear potential(m=0. GeV,β= GeV) HO potential(m=0.5 GeV,β= GeV) φ as (x)~x(1-x) φ AdS/CFT (x)~[x(1-x)] 1/ φ asympt x(1 x) AdS/CFT : φ(x, Q 0 ) x(1 x) φ π (x) Increases PQCD leading twist predictio F π (Q ) by factor 16/ x March 8,