N N Transitions from Holographic QCD

N N Transitions from Holographic QCD Guy F. de Téramond Universidad de Costa Rica Nucleon Resonance Structure Electroproduction at High Photon Virtualities with the Class 12 Detector Workshop Jefferson Lab, May 16, 211 From Nick Evans N Electroproduction, JLab, May 16, 211 Page 1

1 Light Front Dynamics Light-Front Fock Representation Semiclassical Approximation to QCD in the Light Front 2 Light-Front Holographic Mapping Higher Spin Modes in AdS Space Dual QCD Light-Front Wave Equation AdS Bosonic Modes and Meson Spectrum AdS Fermionic Modes and Baryon Spectrum 3 Light-Front Holographic Mapping of Current Matrix Elements Electromagnetic Form Factors Nucleon Elastic Form Factors Nucleon Transition Form Factors N Electroproduction, JLab, May 16, 211 Page 2

1 Light Front Dynamics Different possibilities to parametrize space-time [Dirac (1949)] Parametrizations differ by the hypersurface on which the initial conditions are specified. Each evolve with different times and has its own Hamiltonian, but should give the same physical results Forms of Relativistic Dynamics: dynamical vs. kinematical generators [Dirac (1949)] ct Instant form: hypersurface defined by t =, the familiar one z H, K dynamical, L, P kinematical y 1-211 8811A1 ct Point form: hypersurface is an hyperboloid P µ dynamical, M µν kinematical y z Front form: hypersurface is tangent to the light cone at τ = t + z/c = ct 1-211 8811A2 P, L x, L y dynamical, P +, P, L z, K kinematical z y 1-211 8811A3 N Electroproduction, JLab, May 16, 211 Page 3

LF quantization is the ideal framework to describe hadronic structure in terms of quarks and gluons: simple vacuum structure allows unambiguous definition of partonic content of a hadron Calculation of a matrix elements P + q J P requires boosting the hadronic bound state from P to P + q : extremely complicated in the instant form, whereas K is trivial in the LF Invariant Hamiltonian equation for bound states similar structure of AdS equations of motion: direct connection of QCD and AdS/CFT possible LF coordinates x + = x + x 3 x = x x 3 light-front time longitudinal space variable k + = k + k 3 longitudinal momentum (k + > ) k = k k 3 light-front energy k x = 1 2 (k+ x + k x + ) k x On shell relation k 2 = m 2 leads to dispersion relation k = k2 +m2 k + N Electroproduction, JLab, May 16, 211 Page 4

Light-Front Fock Representation Construct LF Lorentz invariant Hamiltonian equation for the relativistic bound state P µ P µ ψ(p ) = ( P P + P 2 ) ψ(p ) = M 2 ψ(p ) State ψ(p ) is expanded in multi-particle Fock states n of the free LF Hamiltonian ψ = n ψ n n, n = { uud, uudg, uudqq, } with k 2 i = m2 i, k i = (k + i, k i, k i), for each constituent i in state n Fock components ψ n (x i, k i, λ z i ) independent of P + and P and depend only on relative partonic coordinates: momentum fraction x i = k + i /P +, transverse momentum k i and spin λ z i n x i = 1, i=1 n k i =. i=1 N Electroproduction, JLab, May 16, 211 Page 5

Semiclassical Approximation to QCD in the Light Front [GdT and S. J. Brodsky, PRL 12, 8161 (29)] Compute M 2 from hadronic matrix element ψ(p ) P µ P µ ψ(p ) =M 2 ψ(p ) ψ(p ) Relevant variable in the limit of zero quark masses ζ = x 1 x n 1 j=1 x j b j the x-weighted transverse impact coordinate of the spectator system (x active quark) For a two-parton system ζ 2 = x(1 x)b 2 To first approximation LF dynamics depend only on the invariant variable ζ, and hadronic properties are encoded in the hadronic mode φ(ζ) from ψ(x, ζ, ϕ) = e ilzϕ X(x) φ(ζ) 2πζ factoring angular ϕ, longitudinal X(x) and transverse mode φ(ζ) (P +, P and J z commute with P ) N Electroproduction, JLab, May 16, 211 Page 6

Ultra relativistic limit m q longitudinal modes X(x) decouple (L = L z ) M 2 = dζ φ (ζ) ζ ( d2 dζ 2 1 ) d ζ dζ + L2 φ(ζ) ζ 2 + dζ φ (ζ) U(ζ) φ(ζ) ζ where the confining forces from the interaction terms are summed up in the effective potential U(ζ) LF eigenvalue equation P µ P µ φ = M 2 φ is a LF wave equation for φ ( d2 dζ 2 1 4L2 ) 4ζ }{{ 2 + U(ζ) φ(ζ) = M 2 φ(ζ) }{{}} conf inement kinetic energy of partons Effective light-front Schrödinger equation: relativistic, frame-independent and analytically tractable Eigenmodes φ(ζ) determine the hadronic mass spectrum and represent the probability amplitude to find n-massless partons at transverse impact separation ζ within the hadron at equal light-front time Semiclassical approximation to light-front QCD does not account for particle creation and absorption N Electroproduction, JLab, May 16, 211 Page 7

2 Light-Front Holographic Mapping AdS 5 metric: ds 2 }{{} L AdS = R 2 ( ηµν z 2 dx µ dx ν 2 dz ) }{{} L Minkowski A distance L AdS shrinks by a warp factor z/r as observed in Minkowski space (dz = ): L Minkowski z R L AdS Since the AdS metric is invariant under a dilatation of all coordinates x µ λx µ, z λz, the variable z acts like a scaling variable in Minkowski space Short distances x µ x µ map to UV conformal AdS 5 boundary z Large confinement dimensions x µ x µ 1/Λ 2 QCD maps to large IR region of AdS 5, z 1/Λ QCD, thus there is a maximum separation of quarks and a maximum value of z Use the isometries of AdS to map the local interpolating operators at the UV boundary of AdS into the modes propagating inside AdS N Electroproduction, JLab, May 16, 211 Page 8

Higher Spin Modes in AdS Space Description of higher spin modes in AdS space (Frondsal, Fradkin and Vasiliev) Action for spin-j field in AdS d+1 in presence of dilaton background ϕ(z) ( x M = (x µ, z) ) S = 1 2 d d x dz g e ϕ(z2 ) ( g NN g M 1M 1 g M J M J DN Φ M1 M J D N Φ M 1 M J µ 2 g M 1M 1 g M J M J ΦM1 M J Φ M 1 M J + ) where D M is the covariant derivative which includes parallel transport [D N, D K ]Φ M1 M J = R L M 1 NKΦ L MJ R L M J NKΦ M1 L Physical hadron has plane-wave and polarization indices along 3+1 physical coordinates Φ P (x, z) µ1 µ J = e ip x Φ(z) µ1 µ J, Φ zµ2 µ J = = Φ µ1 µ 2 z = with four-momentum P µ and invariant hadronic mass P µ P µ =M 2 N Electroproduction, JLab, May 16, 211 Page 9

Construct effective action in terms of spin-j modes Φ J with only physical degrees of freedom [ H. G. Dosch, S. J. Brodsky and GdT (in preparation)] Introduce fields with tangent indices Find effective action ˆΦ A1 A 2 A J = e M 1 A 1 e M 2 A 2 e M J A J Φ M1 M 2 M J = ( z R) JΦA1 A 2 A J S = 1 2 d d x dz g e ϕ(z)( g NN η µ 1µ 1 η µ J µ J N ˆΦµ1 µ J N ˆΦµ 1 µ J ) µ 2 η µ 1µ 1 η µ J µ J ˆΦµ1 µ J ˆΦµ 1 µ J upon µ-rescaling Variation of the action gives AdS wave equation for spin-j mode Φ J = Φ µ1 µ J [ ( zd 1 2J e ϕ ) ( ) ] (z) µr 2 e ϕ(z) z z d 1 2J z + Φ J (z) = M 2 Φ J (z) z with ˆΦ J (z) = (z/r) J Φ J (z) and all indices along 3+1 N Electroproduction, JLab, May 16, 211 Page 1

Dual QCD Light-Front Wave Equation Φ P (z) ψ(p ) [GdT and S. J. Brodsky, PRL 12, 8161 (29)] Upon substitution z ζ and φ J (ζ) ζ 3/2+J e ϕ(z)/2 Φ J (ζ) in AdS WE [ ( zd 1 2J e ϕ ) ( ) ] (z) µr 2 e ϕ(z) z z d 1 2J z + Φ J (z) = M 2 Φ J (z) z find LFWE (d = 4) ( d2 dζ 2 1 4L2 4ζ 2 ) + U(ζ) φ J (ζ) = M 2 φ J (ζ) with U(ζ) = 1 2 ϕ (z) + 1 4 ϕ (z) 2 + 2J 3 ϕ (z) 2z and (µr) 2 = (2 J) 2 + L 2 AdS Breitenlohner-Freedman bound (µr) 2 4 equivalent to LF QM stability condition L 2 Scaling dimension τ of AdS mode ˆΦ J is τ = 2 + L in agreement with twist scaling dimension of a two parton bound state in QCD and determined by QM stability condition N Electroproduction, JLab, May 16, 211 Page 11

Bosonic Modes and Meson Spectrum Soft wall model: linear Regge trajectories [Karch, Katz, Son and Stephanov (26)] Dilaton ϕ(z) = +κ 2 z 2 (Minkowski metrics), ϕ(z) = κ 2 z 2 (Euclidean metrics) Effective potential: U(z) = κ 4 ζ 2 + 2κ 2 (L + S 1) Normalized eigenfunctions φ φ = dζ φ(z) 2 = 1 φ nl (ζ) = κ 1+L 2n! (n+l)! ζ1/2+l e κ2 ζ 2 /2 L L n(κ 2 ζ 2 ) Eigenvalues M 2 n,l,s = 4κ2 (n + L + S/2).8 Φ Ζ a Φ Ζ b.5.6.4..2. 2 4 6 8 1 Ζ.5 2 4 6 8 1 Ζ LFWFs φ n,l (ζ) in physical space time for dilaton exp(κ 2 z 2 ): a) orbital modes and b) radial modes N Electroproduction, JLab, May 16, 211 Page 12

4κ 2 for n = 1 4κ 2 for L = 1 2κ 2 for S = 1 6 J PC -+ 1+- 2-+ 3+- 4-+ n=3 n=2 n=1 n= 6 J PC 1-- 2++ 3-- 4++ n=3 n=2 n=1 n= f 2 (23) M 2 4 2 2-211 8796A5 π(18) π(13) π π 2 (188) π 2 (167) b(1235) 1 2 3 4 L M 2 4 2 9-29 8796A1 ρ(17) ω(165) ρ(145) ω(142) ρ(77) ω(782) f 2 (195) a 2 (132) f 2 (127) ρ 3 (169) ω 3 (167) a 4 (24) f 4 (25) 1 2 3 4 L Regge trajectories for the π (κ =.6 GeV) and the I =1 ρ-meson and I = ω-meson families (κ =.54 GeV) N Electroproduction, JLab, May 16, 211 Page 13

Fermionic Modes and Baryon Spectrum Same multiplicity of states for mesons and baryons! 4κ 2 for n = 1 4κ 2 for L = 1 2κ 2 for S = 1 6 (a) n=3 n=2 n=1 n= (b) n=3 n=2 n=1 n= M 2 4 N(22) Δ(242) 2 N(171) N(144) N(168) N(172) Δ(16) Δ(195) Δ(195) Δ(192) Δ(191) Δ(1232) 9-29 8796A3 N(94) 1 2 3 4 L 1 2 3 4 L Regge trajectories for positive parity N and baryon families (κ =.5 GeV) N Electroproduction, JLab, May 16, 211 Page 14

3 Light-Front Holographic Mapping of Current Matrix Elements [S. J. Brodsky and GdT, PRL 96, 2161 (26)], PRD 77, 567 (28)] EM transition matrix element in QCD: local coupling to pointlike constituents ψ(p ) J µ ψ(p ) = (P + P ) µ F (Q 2 ) where Q = P P and J µ = e q qγ µ q EM hadronic matrix element in AdS space from coupling of external EM field propagating in AdS with extended mode Φ(x, z) d 4 x dz g e ϕ(z) A M (x, z)φ P (x, z) M Φ P (x, z) (2π) 4 δ 4 ( P P ) ( ɛ µ P + P ) µ F (Q 2 ) How to recover hard pointlike scattering at large Q out of soft collision of extended objects? [Polchinski and Strassler (22)] Mapping of J + elements at fixed light-front time: Φ P (z) ψ(p ) N Electroproduction, JLab, May 16, 211 Page 15

Electromagnetic probe polarized along Minkowski coordinates, (Q 2 = q 2 > ) A(x, z) µ = ɛ µ e iq x V (Q, z), A z = Propagation of external current inside AdS space described by the AdS wave equation [ z 2 2 z z z z 2 Q 2] V (Q, z) = Solution V (Q, z) = zqk 1 (zq) for free current Substitute hadronic modes Φ(x, z) in the AdS EM matrix element Φ P (x, z) = e ip x Φ(z), Φ(z) z τ, z Find form factor in AdS as overlap of normalizable modes dual to the in and out hadrons Φ P and Φ P, with the non-normalizable mode V (Q, z) dual to external source [Polchinski and Strassler (22)]. F (Q 2 ) = R 3 dz z 3 eϕ(z) V (Q, z) Φ 2 J(z) ( 1 Q 2 ) τ 1 J(Q,z), Φ(z) 1.2.8.4 5-26 1 8721A16 2 z 3 4 5 At large Q important contribution to the integral from z 1/Q where Φ z τ and power-law point-like scaling is recovered [Polchinski and Susskind (21)] N Electroproduction, JLab, May 16, 211 Page 16

Electromagnetic Form-Factor Drell-Yan-West electromagnetic FF in impact space [Soper (1977)] F (q 2 ) = n n 1 j=1 ( n 1 ) dx j d 2 b j e q exp iq x k b k ψ n (x j, b j ) 2 q k=1 Consider a two-quark π + Fock state ud with e u = 2 3 and e d = 1 3 1 F π +(q 2 ) = dx d 2 b e iq b (1 x) ψ ud/π (x, b ) with normalization F + π (q =) = 1 2 Integrating over angle F π +(q 2 ) = 2π 1 dx x(1 x) ) 1 x ζdζj (ζq x ψud/π (x, ζ) 2 where ζ 2 = x(1 x)b 2 N Electroproduction, JLab, May 16, 211 Page 17

Compare with electromagnetic FF in AdS space [Polchinski and Strassler (22)] where V (Q, z) = zqk 1 (zq) F (Q 2 ) = R 3 dz z 3 eϕ(z) V (Q, z)φ 2 π + (z) Use the integral representation V (Q, z) = 1 dx J (ζq ) 1 x x Find F (Q 2 ) = R 3 1 ( dz 1 x dx z 3 J zq x ) Φ 2 π + (z) Compare with electromagnetic FF in LF QCD for arbitrary Q. Expressions can be matched only if LFWF is factorized ψ(x, ζ, ϕ) = e imϕ X(x) φ(ζ) 2πζ Find X(x) = x(1 x), φ(ζ) = ( ) ζ 3/2 e ϕ(z)/2 Φ(ζ), z ζ R N Electroproduction, JLab, May 16, 211 Page 18

Free current V (Q, z) = zqk 1 (zq) infinite radius (mauve), no pole structure in time-like region Dressed current non-perturbative sum of an infinite number of terms finite radius (blue) Form factor in soft-wall model expressed as N 1 product of poles along vector radial trajectory ( 2 Mρ 4κ 2 (n + 1/2) ) [Brodsky and GdT (28)] F (Q 2 ) = 1 ( )( ) ( ) 1 + Q2 M 1 + Q2 1 + Q2 2 ρ M 2 ρ M 2 ρ N 2 1. F Π q 2.8.6.4.2 2.5 2. 1.5 1..5. Pion form factor (lowest mode) N Electroproduction, JLab, May 16, 211 Page 19

Higher Fock components in pion LFWF π = ψ qq/π qq τ=2 + ψ qqqq/π qqqq τ=4 + Expansion of LFWF up to twist 4 (monopole + tripole) κ =.54 GeV, Γ ρ = 13, Γ ρ = 4, Γ ρ = 3 MeV, P qqqq = 13%.6 Q 2 F Π Q 2 2 M Ρ 2 log F Π Q 2.5.4 1 2 M Ρ.3 2 M Ρ.2 1.1 Q 2 GeV 2. 1 2 3 4 5 6 7 2 Q 2 GeV 2 1 2 3 4 5 Only interaction in LF holographic semiclassical approx is the confinement potential: create Fock states with extra quark-antiquark pairs, no dynamical gluons N Electroproduction, JLab, May 16, 211 Page 2

Nucleon Form Factors Light Front Holographic Approach [Brodsky and GdT] EM hadronic matrix element in AdS space from non-local coupling of external EM field in AdS with fermionic mode Ψ P (x, z) d 4 x dz g e ϕ(z) Ψ P (x, z) e M A Γ A A M (x, z)ψ P (x, z) (2π) 4 δ 4 ( P P ) ɛ µ ψ(p ), σ J µ ψ(p ), σ Effective AdS/QCD model [Abidin and Carlson, Phys. Rev. D79, 1153 (29) ] Additional term in the 5-dim action: η d 4 x dz g e ϕ(z) Ψ e M A e N B [ Γ A, Γ B] F MN Ψ Couplings η determined by static quantities N Electroproduction, JLab, May 16, 211 Page 21

Compute Dirac form factor using SU(6) flavor symmetry Nucleon AdS wave function in soft-wall model Ψ + (z) = κ2+l R 2 F p 1 (Q2 ) = R 4 dz z 4 V (Q, z)ψ2 +(z) Ψ (z) = κ3+l R 2 1 2n! (n + L)! z7/2+l L L+1 n 2n! n + L + 2 ( κ 2 z 2) (n + L)! z9/2+l L L+2 ( n κ 2 z 2) Bulk-to-boundary propagator [Grigoryan and Radyushkin (27)] Find 1 V (Q, z) = κ 2 z 2 dx (1 x) 2 x Q 2 4κ 2 e κ2 z 2 x/(1 x) F p 1 (Q2 ) = ( 1 + Q2 M 2 ρ 1 )( ) 1 + Q2 M 2 ρ N Electroproduction, JLab, May 16, 211 Page 22

Q 4 F p 1 (Q2 ) (GeV 4 ) 1.2.8.4 9-27 8757A2 1 2 3 Q 2 (GeV 2 ) Data compilation from M. Diehl (26) N Electroproduction, JLab, May 16, 211 Page 23

Nucleon Transition Form Factors Compute spin non-flip EM transition N(94) N (144) ( n =, L =, sz = 1 2 n = 1, L =, s z = 1 2) Ψ n=,l= + Ψ n=1,l= + F p 1 (Q2 ) = R 4 dz z 4 Ψn=1,L= + (z)v (Q, z)ψ n=,l= + (z) Find F p 1 N N (Q 2 ) = 2 2 3 Q 2 M 2 P ( )( )( ) 1 + Q2 1 + Q2 1 + Q2 M 2 ρ M 2 ρ M 2 ρ N Electroproduction, JLab, May 16, 211 Page 24

Data from I. Aznauryan, et al. CLAS (29) N Electroproduction, JLab, May 16, 211 Page 25