Mapping String States into Partons:

Mapping String States into Partons: Form Factors, and the Hadron Spectrum in AdS/QCD Guy F. de Téramond Universidad de Costa Rica In Collaboration with Stan Brodsky Continuous Advances in QCD Minneapolis, May 11-14, 6 CAQCD, Minneapolis, May 11-14, 6 Page 1

Motivation: Holographic QCD Strings describe extended objects (no quarks), QCD degrees of freedom are pointlike confined particles: how can they be related? More precisely: how can we map string states into partons? Precise mapping of string amplitudes to light-front wavefuntions of hadrons for strongly coupled QCD in the conformal limit. Infinite tension limit of strings effective gravity description Holographic duality requires a higher dimensional warped space. Space with negative curvature and a 4-dim boundary: AdS 5 Eigenvalues of normalizable modes inside AdS give the hadronic spectrum. AdS modes represent also the probability amplitude for distribution of quarks at a given scale Non-normalizable modes are related to external currents: they probe the cavity interior CAQCD, Minneapolis, May 11-14, 6 Page

Outline Mapping String States to Partons Light-Front Wavefunctions in QCD The QCD Form Factor in the Light-Front Frame The Form Factor in AdS Space Holographic Mapping of AdS Modes to QCD LFWF Wave Equations in AdS and their Partonic Interpretation Scalar and Vector AdS Fields Meson Spectrum Spinor AdS Fields Baryon Spectrum Hadronic Form Factors in AdS/QCD Meson Form Factors Baryon Form Factors Outlook CAQCD, Minneapolis, May 11-14, 6 Page 3

Mapping States to Partons Light-Front Wave Functions in QCD Hadronic bound state expanded in n-particle Fock eigenstates ψ h = n ψ n/h ψ h of the LC Hamiltonian H LC = P = P + P P at fixed light-cone time x+ = x + x 3 Dirac 49; Brodsky, Pauli and Pinsky, Phys. Rept. 1988. Fock components ψ n/h (x i, k i ) = n; x i, k i, ψ h (P +, P ), frame independent and encode hadron properties and behavior in high momentum-transfer collisions. Momentum fraction x i = k + i /P + and k i are the relative coordinates of parton i in Fock-state n n x i = 1 i=1 n k i =. i=1 Define transverse position coordinates x i r i = x i R + b i n n b i =, x i r i = R. i=1 i=1 CAQCD, Minneapolis, May 11-14, 6 Page 4

The QCD Form Factor in the Light-Front Frame LFWF ψ n (x j, k j ) expanded in terms of n 1 independent coordinates b j, j = 1,,..., n 1 ψ n (x j, k j ) = (4π) n 1 n 1 j=1 d b j exp ( n 1 i j=1 b j k j ) ψn (x j, b j ). Normalization n n 1 j=1 dx j d b j ψ n (x j, b j ) = 1. The form factor has the exact representation (DYW) F (q ) = n n 1 j=1 dx j d b j exp ( iq n 1 j=1 x j b j ) ψn (x j, b j ), corresponding to a change of transverse momentum x j q for each of the n 1 spectators and elementary coupling to the struck parton. CAQCD, Minneapolis, May 11-14, 6 Page 5

Define effective single particle transverse density (Soper 77) 1 F (q ) = From DYW expression for FF in transverse position space: ρ(x, η ) = n n 1 j=1 dx dx j d b j δ ( 1 x d η e i η q ρ(x, η ). n 1 j=1 ) x j δ () ( n 1 j=1 ) x j b j η ψn (x j, b j ). Integration over the n 1 spectator partons, and x = x n is the coordinate of the active charged quark. η = n 1 j=1 x jb j is the x-weighted transverse position coordinate of the n 1 spectators. CAQCD, Minneapolis, May 11-14, 6 Page 6

The Form Factor in AdS Space Non-conformal metric dual to a confining gauge theory ds = R z ea(z) ( η µν dx µ dx ν dz ), where A(z) as z (Polchinski and Strassler, hep-th/19174). Hadronic matrix element for EM coupling with string mode Φ(x, z), x l = (x µ, z) ig 5 d 4 x dz g A l (x, z)φ P (x, z) l Φ P (x, z). Electromagnetic probe polarized along Minkowski coordinates, A µ = ɛ µ e iq x J(Q, z), A z =, with J(Q, z) = zqk 1 (zq), J(Q =, z) = J(Q, z = ) = 1 Hadronic modes are plane waves along the Poincaré coordinates with four-momentum P µ and invariant mass P µ P µ = M Φ(x, z) = e ip x f(z), f(z) z, z. CAQCD, Minneapolis, May 11-14, 6 Page 7

Form factor in AdS is the overlap of normalizable modes dual to the incoming and outgoing hadrons Φ P and Φ P with the non-normalizable mode J(Q, z) dual to the external source F (Q ) = R 3 Polchinski and Strassler, hep-th/911 dz z 3 e3a(z) Φ P (z)j(q, z)φ P (z). Integrate Soper formula over angles: F (q ) = π 1 (1 x) dx x ζdζj (ζq ) 1 x x ρ(x, ζ). Transversality variable ζ = x 1 x n 1 j=1 x j b j. Compare AdS and QCD expressions of FFs for arbitrary Q using identity: 1 ) 1 x dxj (ζq = ζqk 1 (ζq), x the solution for J(Q, ζ)! CAQCD, Minneapolis, May 11-14, 6 Page 8

Hadronic QCD transverse density ρ is identified with the string mode density Φ in AdS space! ρ(x, ζ) = R3 π x 1 x e3a(ζ) Φ(ζ) ζ 4 The variable ζ represents the invariant separation between point-like constituents and it is also the holographic variable: ζ = z. For two-partons ρ(x, ζ) = 1 (1 x) ψ (x, ζ). Two-parton bound state LFWF ψ(x, ζ) = R3 π x(1 x) e3a(ζ) Φ(ζ) ζ 4. Brodsky and de Teramond, arxiv:hep-ph/65 Short distance behavior of LFWF: ψ(x, b ) (b ). CAQCD, Minneapolis, May 11-14, 6 Page 9

Example: Two parton LFWF bound state: ψ qq/π (x, ζ) = B L,k x(1 x)jl (ζβ L,k Λ QCD ) θ ( z Λ 1 QCD), (a) x (b) (c) x.5 1.5 1 1 x.5....1 ψ(x,ζ).1.1.1.1.1 1 1 1 ζ(gev 1 ) ζ(gev 1 ) ζ(gev 1 ) -6 871A14 3 3 3 (a) ground state L =, k = 1, (b) first orbital L = 1, k = 1, (c) first radial L =, k =. CAQCD, Minneapolis, May 11-14, 6 Page 1

Summary Effective Schrödinger equation in terms of the transverse impact variable ζ ] [ d dζ + V (ζ) φ(ζ) = M φ(ζ), with effective potential in the conformal limit. V (ζ) 1 4L 4ζ, Conformal Analytical AdS machinery to extend the hadron into the fifth dimensions and back! 3 + 1 AdS 5 (z) 3 + 1 (z ζ) A different approach: Match the AdS results with Migdal 77 regularization approach to UV correlators (Padé approx) to recover zeros of Bessel functions! Two point function: Erlich, Kribs and Low (hep-th/611). Three point function: Radyushkin (hep-ph/65116). CAQCD, Minneapolis, May 11-14, 6 Page 11

Wave Equations in AdS and their Partonic Interpretation Isomorphism of SO(4, ) of conformal QCD with the group of isometries of AdS space SO(1, 5) ds = R z (η µνdx µ dx ν dz ), x µ λx µ, z λz, maps scale transformations into the holographic coordinate z. Different values of z correspond to different scales at which the hadron is examined. The AdS boundary at z correspond to the Q, UV zero separation limit. There is a maximum separation of quarks and a maximum value of z at the IR boundary. Truncated AdS/CFT model: cut-off at z = 1/Λ QCD breaks conformal invariance and allows the introduction of the QCD scale. Conformal behavior at short distances and color confinement at large interquark separation. CAQCD, Minneapolis, May 11-14, 6 Page 1

Scalar and Vector AdS Fields Consider a scalar wave equation on AdS d+1 [ z z (d 1)z z + z M (µr) ] Φ(z) =, with solution Φ(z) z d/ J d (zm), (µr) = ( d). For d = 4, lowest stable solution determined by the Breitenlohner-Freedman (B-F) stability bound, (µr) 4, on the fifth dimensional mass. Orbital excitations correspond to higher values of µ. Allowed values determined by the condition that conformal dimensions are spaced by integers, according to the spectral relation (µr) = ( d) = κ(κ + d). If lowest stable state corresponds to the lowest orbital, L =, then κ = L, = + L and (µr) = 4 + L. CAQCD, Minneapolis, May 11-14, 6 Page 13

Wave equation in AdS for bound state of two scalar partons with conformal dimension = + L [ z z 3z z + z M L + 4 ] Φ(z) =, with solution Φ(z) = Ce ip x z J L (zm). For spin-carrying constituents: τ = σ, σ = n i=1 σ i. The twist τ is equal to the number of partons τ = n. Same form of equation for vector wave equation in AdS with lowest stable solution (µr) 1 and (µr) = ( 1)( d 1) = κ(κ + d ). Two-quark vector meson described by wave equation [ z z 3z z + z M L + 4 ] Φ µ (z) =, with solution Φ µ (x, z) = Ce ip x z J L (zm) ɛ µ. CAQCD, Minneapolis, May 11-14, 6 Page 14

Meson Spectrum 4-d mass spectrum from boundary conditions on the normalizable string modes at z = 1/Λ QCD, Φ(x, z o ) =, given by the zeros of Bessel functions β α,k : M α,k = β α,k Λ QCD. Normalizable AdS modes Φ(z) 5 4 4 Φ(z) 3 Φ(z) 1-6 871A7 1 3 4 z - 3-6 871A13 1 3 4 z Fig: Meson orbital and radial AdS modes for Λ QCD =.3 GeV. CAQCD, Minneapolis, May 11-14, 6 Page 15

(a) f 4 (5) a 4 (4) (b) (GeV ) 4 ρ (17) ρ 3 (169) a (145) a (13) f 1 (185) b 1 (135) ω ω (78) 3 (167) ω (165) ρ (77) π (14) π (167) 1-6 8694A1 f (17) a 1 (16) 4 L 4 L Fig: Light meson orbital spectrum Λ QCD =.3 GeV CAQCD, Minneapolis, May 11-14, 6 Page 16

Spinor AdS Fields Baryon: twist-three, dimension = 9 + L O 9 +L = ψd {l 1... D lq ψd lq+1... D lm }ψ, L = m l i. i=1 Solve full 1-dim Dirac Eq., /D ˆΨ =, since baryons are charged under SU(4) SO(6). Baryon number conservation? ˆΨ is expanded in terms of eigenfunctions η κ (y) of the Dirac operator on compact space X with eigenvalues λ κ : ˆΨ(x, z, y) = Ψ κ (x, z)η κ (y). κ From the 1-dim Dirac equation, /D ˆΨ = : [ z z d z z + z M (λ κ + µ) R + d i /D X η(y) = λ η(y), ( ) ] d + 1 + (λ κ + µ)r ˆΓ f(z) =, where Ψ(x, z) = e ip x f(z), P µ P µ = M and ˆΓu ± = ±u ± ( For d = 4, ˆΓ = γ 5 ). CAQCD, Minneapolis, May 11-14, 6 Page 17

Spinor field in AdS: Ψ(x, z) = Ce ip x z d+1 [ ] J (µ+λκ )R 1 (zm) u + (P ) + J (µ+λκ )R+ 1 (zm) u (P ), with = d + (µ + λ κ)r and spinors u ± (P ) defined along 4-dim coordinates. See: Muck and Viswanathan, hep-ph/985945. µ determined asymptotically by spectral comparison with orbital excitations in the boundary: µ = k/r and λ κ are the eigenvalues of the Dirac equation on S d+1 : ( λ κ R = ± κ + d + 1 ), κ =, 1,... See: Camporesi and Higuchi: gr-gc/9559. Spin- 3 Rarita-Schwinger eq. in AdS similar to spin- 1 in the Ψ z = gauge for polarization along Minkowski coordinates, Ψ µ. See: Volovich, hep-th/9899. CAQCD, Minneapolis, May 11-14, 6 Page 18

Baryon Spectrum For spin-carrying constituents: τ = σ, σ = n i=1 σ i. For a three quark state 3/. Change compensated in µ by the shift k L 1 and Ψ(z) z 1 Ψ(z). Three-quark baryon described by wave equation (d = 4, κ = ) [ z z 3z z + z M L ± + 4 ] f ± (z) = with L + = L + 1, L = L +, and solution Ψ(x, z) = Ce ip x z [ ] J 1+L (zm) u + (P ) + J +L (zm) u (P ). 4-d mass spectrum Ψ(x, z o ) ± = = parallel Regge trajectories for baryons! M + α,k = β α,kλ QCD, M α,k = β α+1,kλ QCD. Ratio of eigenvalues determined by the ratio of zeros of Bessel functions! CAQCD, Minneapolis, May 11-14, 6 Page 19

SU(6) multiplet structure for N and orbital states, including internal spin S and L. SU(6) S L Baryon State 56 1 N 1 + (939) 3 3 + (13) 7 1 1 N 1 (1535) N 3 (15) 3 1 N 1 (165) N 3 (17) N 5 (1675) 1 1 1 (16) 3 (17) 56 1 N 3 + (17) N 5 + (168) 3 1 + (191) 3 + (19) 5 + (195) 7 + (195) 7 1 3 N 5 N 7 3 3 N 3 N 5 N 7 (19) N 9 (5) 1 3 5 (193) 7 56 1 4 N 7 + 3 4 5 + 7 + 7 1 5 N 9 3 5 N 7 N 9 + () 9 + 11 + (4) N 11 (6) N 9 N 11 N 13 CAQCD, Minneapolis, May 11-14, 6 Page

8 (a) N (6) I = 1/ (b) I = 3/ (4) (GeV ) 6 4 N (17) N (1675) N (165) N (1535) N (5) N (19) N () (195) (19) (191) (195) N (15) (13) (193) 1-6 8694A14 N (939) N (17) N (168) L 4 6 (17) (16) L 56 7 4 6 Fig: Predictions for the light baryon orbital spectrum for Λ QCD =.5 GeV. The 56 trajectory corresponds to L even P = + states, and the 7 to L odd P = states. CAQCD, Minneapolis, May 11-14, 6 Page 1

Example: Gaussian Confining Background Karch, Katz, Son and Stephanov : Non-constant dilaton V (z) z (arxiv:hep-ph/69) Two-dim harmonic oscillator in terms of transverse variable ζ = z [ z z 3z z + z M (µr) κ 4 z 4] Φ(z) =. Normalizable solutions exist if (µr) = 4 + L (obtained in truncated model by B-F bound) Φ(z) = κl+1 n! R 3/ (n + L)! z+l e κ z / L L ( n κ z ), with eigenvalues M = κ (n + L + 1). Equivalent problem: Z(z) = e κ z / Φ(z) and metric e A(z) = e κ ζ /3 with no potential! Andreev: Gaussian metric term (arxiv:hep-th/6317) CAQCD, Minneapolis, May 11-14, 6 Page

Φ(z) 4 3 4 1 - -4 1 3 4 5 6 7 z -6 1 3 4 5 6 7 z Orbital and radial modes in a confining Gaussian metric asymptotic to AdS 5. CAQCD, Minneapolis, May 11-14, 6 Page 3

Hadronic Form Factors in AdS/QCD Form factor in AdS (Polchinski and Strassler, hep-th/911). F (Q ) A B R 3 Λ 1 QCD dz z 3 Φ B(z) J(Q, z) Φ A (z). At large Q, important contribution is from the boundary conformal region z 1/Q where Φ z : F (Q ) [ 1 Q ] 1 Constituent counting rule for hard scattering! (Brodsky and Farrar 73; Matveev et al. 73) Spin carrying constituents τ. 1. J(Q,z), Φ(z).8.4 5-6 871A16 1 3 4 5 z Fig: Suppression of external perturbations for large Q inside AdS. CAQCD, Minneapolis, May 11-14, 6 Page 4

Meson Form Factor (Valence Approximation) 1.8 F π (Q ).6.4. -1-8 -6-4 - Q Space-like pion form factor in holographic model for Λ QCD =. GeV. CAQCD, Minneapolis, May 11-14, 6 Page 5

Baryon Form Factors Coupling of the extended AdS mode with an external gauge field A µ (x, z) ig 5 d 4 x dz g A µ (x, z) Ψ(x, z)γ µ Ψ(x, z), where Ψ(x, z) = e ip x [ψ + (z)u + (P ) + ψ (z)u (P )], ψ + (z) = Cz J 1 (zm), ψ (z) = Cz J (zm), and In the large P + limit u(p ) ± = 1 ± γ 5 ψ + (z) ψ (z), u(p ). ψ (z) ψ (z), the LC ± spin projection along ẑ. Constant C determined by charge normalization: C = ΛQCD R 3/ [ J (β 1,1 )J (β 1,1 )] 1/. CAQCD, Minneapolis, May 11-14, 6 Page 6

Consider the spin non-flip form factors in the infinite wall approximation F + (Q ) = g + R 3 dz z 3 J(Q, z) ψ +(z), F (Q ) = g R 3 dz z 3 J(Q, z) ψ (z), 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 + (z) and ψ (z) correspond to nucleons with J z = +1/ and 1/. For SU(6) spin-flavor symmetry (proton up) N u = 5 3, N u = 1 3, N d = 1 3, N d = 3. Final result F p 1 (Q ) = R 3 dz z 3 J(Q, z) ψ +(z), F n 1 (Q ) = 1 3 R3 dz z 3 J(Q, z) [ ψ + (z) ψ (z) ], where F p 1 () = 1, F n 1 () =. CAQCD, Minneapolis, May 11-14, 6 Page 7

Dirac Proton Form Factor (Valence Approximation) Q 4 F p 1 (Q ) [GeV 4 ] 1.75 1.5 1.5 1.75.5.5 5 1 15 5 3 35 Q [GeV ] Prediction for Q 4 F p 1 (Q ) for Λ QCD =.1 GeV in the hard wall approximation. Analysis of the data is from Diehl (5). Red points are from Sill (1993). Superimposed Green points are from Kirk (1973). CAQCD, Minneapolis, May 11-14, 6 Page 8

Dirac Neutron Form Factor (Valence Approximation) Q 4 F n 1 (Q ) [GeV 4 ] -.5 -.1 -.15 -. -.5 -.3 -.35 1 3 4 5 6 Q [GeV ] Prediction for Q 4 F n 1 (Q ) for Λ QCD =.1 GeV in the hard wall approximation. Data analysis from Diehl (5). CAQCD, Minneapolis, May 11-14, 6 Page 9

Example: Evaluation of QCD Matrix Elements Pion decay constant f π defined by the matrix element of EW current J + W : ψu γ + (1 γ 5 )ψ d π = i P + f π, with π = du = 1 NC 1 N C c=1 ( b c d d c u b c d d c u ). Use light-cone expression: Lepage and Brodsky 8 f π = N C 1 dx d k 16π 3 ψ qq/π(x, k ). Find: f π = 3ΛQCD 8J 1 (β,1 ) = 83.4 Mev, for Λ QCD =. GeV. Experiment: f π = 9.4 Mev. CAQCD, Minneapolis, May 11-14, 6 Page 3

Outlook Only one scale Λ QCD determines hadronic spectrum (slightly different for mesons and baryons). Ratio of Nucleon to Delta trajectories determined by zeros of Bessel functions. Non-zero orbital angular momentum and higher Fock-states require introduction of quantum fluctuations. Initial good approximation for description of the structure of hadronic form factors and other observables. Covariant version of the bag model with confinement and conformal symmetry. Light-cone frame is the natural frame to establish the AdS/QCD holographic duality. Precise mapping of string modes to partonic states. String modes inside AdS represent the probability amplitude for the distribution of quarks at a given scale. Write eigenvalue problem in terms of 3+1 QCD degrees of freedom. CAQCD, Minneapolis, May 11-14, 6 Page 31

... basic features of QCD can be understood in terms of a higher dimensional dual gravity theory which holographically encodes multi parton boundary states into string modes and allows the computation of physical observables at strong coupling... 5-Dimensional Anti-de Sitter Spacetime Ψ (x, z = z ) = Ψ (x, z) z Black Hole z = 1/Λ QCD Ψ~z 4-Dimensional Flat Spacetime (hologram) 1-6 8685A5 CAQCD, Minneapolis, May 11-14, 6 Page 3