S = 0 (b) S = 0. AdS/QCD. Stan Brodsky, SLAC INT. March 28, Fig: Orbital and radial AdS modes in the soft wall model for κ = 0.6 GeV.

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1 6 5 4 Φ(z) Φ(z) A2 4 8 z A z Fig: Orbital and radial AdS modes in the soft wall model for κ =.6 GeV. (a) S = (b) S = (GeV 2 ) 4 π 2 (167) π (18) 2 b 1 (1235) π (13) π (14) π (14) A L 2 4 n Light meson orbital (a) and radial (b) spectrum for κ =.6 GeV. March 28,

2 Higher Spin Bosonic Modes SW Effective LF Schrödinger wave equation [ d2 [ d2 dz 2 1 4L2 4z 2 with eigenvalues ] + κ 4 z κ 4 2κ ζ (L+ 2κ 2 (L+ S 1) φ (ζ) = M 2 S (z) = Mφ S (ζ) 2 φ S (z) dζ 2 1 4L2 4ζ 2 M 2 = 2κ 2 (2n + 2L + S). Compare with Nambu string result (rotating flu tube): M 2 n(l) = 2πσ (n + L + 1/2). (a) S = 1 f 4 (25) a 4 (24) (b) S = 1 (GeV 2 ) 4 a 2 (132) ρ 3 (169) ρ (145) ρ (17) 2 ω (782) ρ (77) ω 3 (167) ρ (77) f 2 (127) A2 2 4 L 2 4 Vector mesons orbital (a) and radial (b) spectrum for κ =.54 GeV. Glueballs in the bottom-up approach: (HW) Boschi-Filho, Braga and Carrion (25); (SW) Colangelo, De Facio, Jugeau and Nicotri( 27). n March 28,

3 α(t) t listed in the data tab Soft Wall Model -- Reproduces Linear Regge Trajectories 58

4 Propagation of eternal perturbation suppressed inside AdS. At large enough Q r/r 2, the interaction occurs in the large-r conformal region. Importa Consider a specific AdS mode Φ (n) dual to an n partonic Fock state n. At small z, Φ ( scales as Φ (n) z n. Thus: Hadron Form Factors from AdS/CFT J(Q, z) = zqk 1 (zq) contribution to the FF integral from the boundary near z 1/Q. F (Q 2 ) I F = dz z 3Φ F (z)j(q, z)φ I (z) J(Q, z), Φ(z) 1 J(Q, z) High Q 2 Φ(z).8 Polchinski, Strassler from.6 de Teramond, sjb small z ~ 1/Q.4 F (Q 2 ) z [ ] 1 τ 1 Q 2, Dimensional Quark Counting Rule General result from AdS/CFT where τ = n σ n, σ n = n i=1 σ i. The twist is equal to the number of partons, τ = n. March 28,

5 Current Matri Elements in AdS Space (HW) Hadronic matri element for EM coupling with string mode Φ( l ), l = ( µ, z) ig 5 d 4 dz g A l (, z)φ P (, z) l Φ P (, z). Electromagnetic probe polarized along Minkowski coordinates (Q 2 = q 2 > ) A(, z) µ = ɛ µ e iq J(Q, z), A z =. Propagation of eternal current inside AdS space described by the AdS wave equation [ z 2 2 z z z z 2 Q 2] J(Q, z) =, subject to boundary conditions J(Q =, z) = J(Q, z = ) = 1. Solution J(Q, z) = zqk 1 (zq). Substitute hadronic modes Φ(, z) in the AdS EM matri element Φ P (, z) = e ip Φ(z), Φ(z) z, z. March 28,

6 Current Matri Elements in AdS Space (SW) sjb and GdT Grigoryan and Radyushkin Propagation of eternal current inside AdS space described by the AdS wave equation [ z 2 2 z z ( 1 + 2κ 2 z 2) z Q 2 z 2] J κ (Q, z) =. Solution bulk-to-boundary propagator ) J κ (Q, z) = Γ (1 + Q2 4κ 2 where U(a, b, c) is the confluent hypergeometric function Γ(a)U(a, b, z) = Form factor in presence of the dilaton background ϕ = κ 2 z 2 For large Q 2 4κ 2 ( ) Q 2 U 4κ 2,, κ2 z 2, e zt t a 1 (1 + t) b a 1 dt. F (Q 2 ) = R 3 dz z 3 e κ2 z 2 Φ(z)J κ (Q, z)φ(z). J κ (Q, z) zqk 1 (zq) = J(Q, z), the eternal current decouples from the dilaton field. March 28,

7 Space and Time-Like Pion Form Factor Hadronic string modes Φ π (z) z 2 as z (twist τ = 2) Φ HW π (z) = Φ SW π (z) = 2ΛQCD R 3/2 J 1 (β,1 ) z2 J (zβ,1 Λ QCD ), 2κ R 3/2 z2. F π has analytical solution in the SW model F π (Q 2 ) = 4κ2 4κ 2 +Q F π (q 2 ) q 2 (GeV 2 ) Fig: F π (q 2 ) for κ =.375 GeV and Λ QCD =.22 GeV. Continuous line: SW, dashed line: HW. March 28, A3 62

8 Note: Analytical Form of Hadronic Form Factor for Arbitrary Twist Form factor for a string mode with scaling dimension τ, Φ τ in the SW model ) Γ (1+ Q2 F (Q 2 4κ ) = Γ(τ) ( 2 ). Γ τ + Q2 4κ 2 For τ = N, Γ(N + z) = (N 1 + z)(n 2 + z)... (1 + z)γ(1 + z). Form factor epressed as N 1 product of poles F (Q 2 ) = F (Q 2 ) = F (Q 2 ) = 1, N = 2, 1 + Q2 4κ 2 2 ( )( ), N = 3, 1 + Q2 4κ 2 + Q2 2 4κ 2 ( 1 + Q2 4κ 2 )( 2 + Q2 4κ 2 ) (N 1)! ( ), N. N 1+ Q2 4κ 2 For large Q 2 : [ ] 4κ F (Q 2 2 (N 1) ) (N 1)!. March 28, Q 2 63

9 Analytical continuation to time-like region q 2 q 2 M ρ = 2κ = 75 MeV (M ρ = 4κ 2 = 75 MeV) Strongly coupled semiclassical gauge/gravity limit hadrons have zero widths (stable). 2 1 log IF π (q 2 )I q 2 (GeV 2 ) March 28, A4 Space and time-like pion form factor for κ =.375 GeV in the SW model. Vector Mesons: Hong, Yoon and Strassler (24); Grigoryan and Radyushkin (27). 64

10 HolographicLight-Front Model for QCDRepresentation Light-Front Wavefunctions SJB and GdT in preparation of Two-Body Meson Form Factor Drell-Yan-West form factor in the light-cone (two-parton state) F (q 2 ) = q e q 1 d d 2 k 16π 3 ψ P (, k q ) ψ P (, k ). Fourrier transform to impact parameter space b ψ(, k ) = 4π d 2 b e i b k ψ(, b ) Find (b = b ) : F (q 2 ) = 1 = 2π d d 2 b e i b q ψ(, b) 2 1 d b db J (bq) ψ(, b) 2, Soper March 28,

11 Holographic Mapping of AdS Modes to QCD LFWFs Integrate Soper formula over angles: F (q 2 ) = 2π 1 d (1 ) ζdζj (ζq ) 1 ρ(, ζ), with ρ(, ζ) QCD effective transverse charge density. Transversality variable ζ = 1 n 1 j=1 j b j. Compare AdS and QCD epressions of FFs for arbitrary Q using identity: 1 ) 1 dj (ζq = ζqk 1 (ζq), the solution for J(Q, ζ) = ζqk 1 (ζq)! March 28,

12 LF(3+1) AdS 5 ψ(, b ) φ(z) ζ = ψ(, b ) (1 ) b 2 z b (1 ) Holography: Unique mapping derived from equality of LF and AdS formula for current matri elements March 28, 28 ψ(, ζ) = (1 )ζ 1/2 φ(ζ) 67 67

13 Consider the AdS 5 metric: ds 2 = R2 z 2 (η µν d µ d ν dz 2 ). ds 2 invariant if µ λ µ, z λz, Maps scale transformations to scale changes of the the holographic coordinate z. We define light-front coordinates ± = ± 3. Then η µν d µ d ν = d 2 d 3 2 d 2 = d + d d 2 and ds 2 = R2 z 2 (d 2 + dz 2 ) for + =. Light-Front AdS 5 Duality ds 2 is invariant if d 2 λ[ 2 d 2, and z λz, ] at equal LF time. Maps scale transformations in transverse LF space to scale changes of the holographic coordinate z. Holographic connection of AdS 5 to the light-front. The Casimir effective for the wave rotation equation groupinso(2). the two-dim transverse LF plane has the Casimir representation L 2 corresponding to the SO(2) rotation group [The Casimir for SO(N) S N 1 is L(L + N 2) ]. March 28,

14 Light-front Hamiltonian equation H LF φ = M 2 φ, leads to effective LF Schrödinger wave equation (KKSS) [ d2 dζ 2 1 4L2 4ζ 2 ] + κ 4 ζ 2 + 2κ 2 (L 1) φ(ζ) = M 2 φ(ζ) with eigenvalues M 2 = 4κ 2 (n + L) and eigenfunctions φ L (ζ) = κ 1+L 2n! (n + L)! ζ1/2+l e κ2 ζ 2 /2 L L n ( κ 2 ζ 2). Transverse oscillator in the LF plane with SO(2) rotation subgroup has Casimir L 2 representing rotations for the transverse coordinates b in the LF. SW model is a remarkable eample of integrability to a non-conformal etension of AdS/CFT [Chim and Zamolodchikov (1992) - Potts Model.] March 28,

15 Eample: Pion LFWF Two parton LFWF bound state: ψ qq/π HW (, b ) = Λ ( QCD (1 ) ( ) πj1+l (β L,k ) J L (1 ) b β L,k Λ QCD θ b 2 ) Λ 2 QCD, (1 ) ψ SW qq/π (, b ) = κ L+1 2n! (n + L)! [(1 )] 1 2 +L b L e 1 2 κ ( 2 (1 )b 2 L L n κ 2 (1 )b 2 ). (a) b 1 2 (b) b 1 2 ψ(,b) A1 Fig: Ground state pion LFWF in impact space. (a) HW model Λ QCD =.32 GeV, (b) SW model κ =.375 GeV. March 28,

16 Prediction from AdS/CFT: Meson LFWF (, de Teramond, sjb ψ M (, k 2 ) Soft Wall model 1.3 κ =.375 GeV 1.4 k (GeV) 1.5 massless quarks ψ M (, k ) = 4π κ (1 e k 2 2κ 2 (1 ) φ M (, Q ) (1 ) March 28,

17 Eample: Evaluation of QCD Matri Elements Pion decay constant f π defined by the matri element of EW current J + W : with ψ u γ (1 γ 5)ψ d π = i P + f π 2 π = du = 1 1 N C 2 NC c=1 Find light-front epression (Lepage and Brodsky 8): ( b c d d c u b c d d c u ). f π = 2 N C 1 d d 2 k 16π 3 ψ qq/π(, k ). Using relation between AdS modes and QCD LFWF in the ζ limit f π = 1 3 Φ(ζ) 8 2 R3/2 lim ζ ζ 2. Holographic result (Λ QCD =.22 GeV and κ=.375 GeV from pion FF data): Ep: f π =92.4 MeV f HW π = 3 8J 1 (β,k ) Λ QCD = 91.7 MeV, f SW π = March 28, κ = 81.2 MeV, 72

18 Hadron Distribution Amplitudes φ H ( i, Q) i = 1 i Fundamental gauge invariant non-perturbative input to hard eclusive processes, heavy hadron decays. Defined for mesons, baryons Evolution Equations from PQCD, OPE, Conformal Invariance Fied τ = t + z/c Compute from valence light-front wavefunction in Q light-cone gauge φ M (, Q) = d 2 k ψq q(, k ) 1 Lepage, sjb k 2 < Q 2 Lepage, sjb F rishman, Lepage, Sachrajda, sjb P eskin Braun Ef remov, Radyushkin Chernyak etal March 28,

19 Second Moment of Pion Distribution Amplitude < ξ 2 >= 1 1 dξ ξ 2 φ(ξ) ξ = 1 2 < ξ 2 > π = 1/5 =.2 < ξ 2 > π = 1/4 =.25 φ asympt (1 ) φ (1 ) Lattice (I) < ξ 2 > π =.28 ±.3 Lattice (II) < ξ 2 > π =.269 ±.39 March 28, Donnellan et al. Braun et al. 74

20 1.8 Spacelike pion form factor from AdS/CFT F π (q 2 ) 1.8 F π (q 2 ).6.4 q 2 (GeV 2 ).6.4 q 2 (GeV 2 ).2.2 However J/ψ ρπ F π However (q 2 ) F π (q 2 ) J/ψ ρπ is largest two-body hadron -1-8 decay is largest two-body q 2 (GeV 2 ) However J/ψ ρπ is largest two-body hadron decay Small value One parameter for ψ - set ρπ by pion decay constant Small value for ψ ρπ de Teramond, sjb ρ March 28, q 2 (GeV 2 ) However J/ψ ρπ is largest two-body hadr Small value SW: for Harmonic ψ Oscillator ρπ Confinement Small value for ψ ρ Data Compilation from Baldini, Kloe and Volmer HW: Truncated Space Confinement ρ ρ 75

21 Note: Contributions to Mesons Form Factors at Large Q in Write form factor in terms of an effective partonic transverse density in impact space b 1 F π (q 2 ) = d db 2 ρ(, b, Q), with ρ(, b, Q) = πj [b Q(1 )] ψ(, b) 2 and b = b. Contribution from ρ(, b, Q) is shifted towards small b and large 1 as Q increases. (a) 2 b 1 (b) 2 b ρ(,b) A5 Fig: LF partonic density ρ(, b, Q): (a) Q = 1 GeV/c, (b) very large Q. March 28,

22 Energy-momentum tensor in QCD Θ µν = 2 δ δg µν () d 4 L QCD. = 1 2 ψi(γ µ D ν + γ ν D µ ) ψ g µν ψ ( iγ λ D λ m ) ψ G a µλ G a ν λ gµν G a µνg a µν. dq Θ ++ + d 2 q dq + d 2 q f () = (2π) 3 (2π) 3 q + { b (q)b (q ) + b (q)b (q ) + d (q)d (q ) + d (q)d (q ) } for each quark field f Etra compared to J + () March 28,

23 Gravitational Form Factor on the LF A(q 2 ) = f 1 j=1 d d 2 η e i η q ρ(, η ), where d 2 q ρ(, η ) = (2π) 2 e i η q ρ(, q ) f f = n 1 ( n 1 ) d j d 2 b j δ 1 j n j=1 j=1 δ (2)( n 1 ) j b j η ψn ( j, b j ) 2. Integrate over angle 1 ) 1 A(q 2 ) = 2π d (1 ) ζdζ J (ζq ρ(, ζ) f f n 1 ζ = j b j 1 j=1 f Etra factor of relative to charge form factor For each quark and f March 28,

24 Anomalous gravitomagnetic moment B() Okun, Kobzarev, Teryaev: B() Must vanish because of Equivalence Theorem q 2 = q 2 graviton sum over constituents - Hwang, Schmidt, sjb; Holstein et al B() = Each Fock State March 28,

25 Gravity in AdS/CFT ds 2 = g lm d l d m = R2 z 2 ( ηµν d µ d ν dz 2) g lm = ( R 2 /z 2) η lm g = (R/z) 5 The action for gravity coupled to a scalar field in AdS5 space: S = 1 κ 2 d 5 g (R + 2Λ + 12 glm l Φ m Φ 12 ) µ2 Φ 2 eqn. of motion from variation of action R lm 1 2 g lmr Λg lm = solution: R ilkm = 1 R 2 [g ilg km g ik g lm ] R = g il R il = g il g km R ilkm = 2 R 2 Λ = 6 R 2 (µr) 2 = ( 4) March 28,

26 The AdS Gravitational Form Factor ds 2 = R2 z 2 ( (ηµν + h µν )d µ d ν dz 2). Abidin & Carlson linearized metric h zz = h zµ = gauge choice d 4 dz g h lm (, z) l Φ P (, z) m Φ P (, z) ( ) 1 z 3 z z 3 zhµ ν ρ ρ hµ ν =. gravitational coupling eqn. of motion from action propagation of graviton into AdS from eternal source h ν µ (, z) = η ν µ e iq H(q 2, z) H(q 2 =, z) = H(q 2, z = ) = 1. H(Q 2, z) = 1 2 Q2 z 2 K 2 (zq). solution! A(Q 2 ) = R 3 dz z 3 Φ(z)H(Q2, z)φ(z). March 28, Gravitational Form Factor 81

27 [ ] d2 Holography: d2 Map AdS/CFT d 2 ζ + V (ζ) to 3+1 LF Theory Relativistic LF radial equation [ [ d2 dζ 2 + V (ζ) ] = M 2 φ(ζ) d 2 ζ + V (ζ) φ(ζ) = M2 φ(ζ) d2 dζ 2 + V (ζ) ] Frame Independent = M 2 φ(ζ) φ(ζ) = M 2 φ(ζ) ] ζ 2 = (1 )b 2. G. de Teramond, sjb ζ 2 = (1 )b 2. L = P R Effective conformal potential: March 28, 28 J z = S z p = n i=1 S z i + n 1 (1 ) b ψ(, b ) = ψ(ζ) each Fock State φ(z) V (ζ) = 1 4L2 4ζ 2. ζ = Jp z = Sq z + Sg z + L z q + z L z g = 1 ζ = 2 quation has a stable range z 82 z (1 ) b 2 (1 ) b i=1 lz i = 1 2 ψ(, b ) = ψ(ζ) (1 ) b φ(z) ψ(, b ) = ψ(ζ) ζ = φ(z) z z (1 ) b 2 +κ 4 ζ 2 confining potential: (1 ) b 2 82

28 H(Q 2, z) = 2 A(Q 2 ) = 2 R 3 1 d J zq 1 ( dz 1 d z 3 J zq ). Φ(z) 2. AdS Compare with gravitational form factor from LF A(Q 2 ) = 2π 1 d (1 ) ρ(, ζ) = 2 R3 2π 1 March 28, ) 1 ζdζ J (ζq ρ(, ζ) Holography: identify AdS and LF density for all Q with ζ z ζ = 1 Φ(ζ) 2 ζ 4. n 1 j=1 j b j LF 83

29 LF(3+1) AdS 5 ψ(, b ) φ(z) ζ = ψ(, b ) (1 ) b 2 z b (1 ) Holography: Unique mapping derived from equality of LF and AdS formula for current matri elements: em and gravitational March 28, 28 ψ(, ζ) = (1 )ζ 1/2 φ(ζ) 84 84

30 Holographic result for LFWF identical for electroweak and gravity couplings! Highly nontrivial consistency test can predict Momentum fractions for each quark flavor and the gluons A f () =< f >, f Orbital Angular Momentum for each quark flavor and the gluons B f () =< L 3 f >, f A f () = A() = 1 B f () = B() = Vanishing Anomalous Gravitomagnetic Moment Shape and Asymptotic Behavior of A f (Q 2 ), B f (Q 2 ) March 28,