Light-Front Quantization Approach to the Gauge/Gravity Correspondence and Higher Fock States

Light-Front Quantization Approach to the Gauge/Gravity Correspondence and Higher Fock States Guy F. de Téramond University of Costa Rica Southampton High Energy Physics Theory Group University of Southampton October 8, 2010 From Nick Evans SHEP, University of Southampton, October 8, 2010 Page 1

I. Introduction Lattice QCD Gravity Gauge/Gravity Correspondence and Light-Front QCD II. Higher Spin Modes in AdS Space III. Light Front Dynamics Light-Front Fock Representation Semiclassical Approximation to QCD in the Light Front IV. Light-Front Holographic Mapping Fermionic Modes V. Light-Front Holographic Mapping of Current Matrix Elements Electromagnetic Form Factors VI. Higher Fock Components Detailed Structure of Space-and Time Like Pion Form Factor SHEP, University of Southampton, October 8, 2010 Page 2

I. Introduction QCD fundamental theory of quarks and gluons QCD Lagrangian follows from the gauge invariance of the theory ψ(x) e iαa (x)t a ψ(x), [ T a, T b] = if abc T c Find QCD Lagrangian L QCD = 1 4g 2 Tr (Gµν G µν ) + iψd µ γ µ ψ + mψψ where D µ = µ igt a A a µ, Ga µν = µ A a ν ν A a µ + f abc A b µa c ν Quarks and gluons interactions from color charge, but... gluons also interact with each other: strongly coupled non-abelian gauge theory color confinement Most challenging problem of strong interaction dynamics: determine the composition of hadrons in terms of their fundamental QCD quark and gluon degrees of freedom SHEP, University of Southampton, October 8, 2010 Page 3

Lattice QCD Lattice numerical simulations at the teraflop/sec scale (resolution L/a) Sums over quark paths with billions of dimensions LQCD (2009) > 1 petaflop/sec a L Dynamical properties in Minkowski space-time not amenable to Euclidean lattice computations SHEP, University of Southampton, October 8, 2010 Page 4

Gravity Space curvature determined by the mass-energy present following Einstein s equations R µν 1 2 R g µν = κ T µν }{{}}{{} matter geometry R µν Ricci tensor, R space curvature g µν metric tensor ( ds 2 = g µν dx µ dx ν ) T µν energy-momentum tensor κ = 8πG/c 4, Matter curves space and space determines how matter moves! Annalen der Physik 49 (1916) p. 30 SHEP, University of Southampton, October 8, 2010 Page 5

Gauge Gravity Correspondence and Light-Front QCD The AdS/CFT correspondence [Maldacena (1998)] between gravity on AdS space and conformal field theories in physical spacetime has led to a semiclassical approximation for strongly-coupled QCD, which provides analytical insights into the confining dynamics of QCD Light-front (LF) quantization is the ideal framework to describe hadronic structure in terms of quarks and gluons: simple vacuum structure allows unambiguous definition of the partonic content of a hadron, exact formulae for form factors, physics of angular momentum of constituents... Light-front holography provides a remarkable connection between the equations of motion in AdS and the bound-state LF Hamiltonian equation in QCD [GdT and S. J. Brodsky, PRL 102, 081601 (2009)] Isomorphism of SO(4, 2) group of conformal transformations with generators P µ, M µν, K µ, D, with the group of isometries of AdS 5, a space of maximal symmetry, negative curvature and a four-dim boundary: Minkowski space Isometry group: most general group of transformations which leave invariant the distance between two points. Ej: S N O(N + 1) Dim: N(N + 1)/2 Dim isometry group of AdS d+1 is (d + 1)(d + 2)/2 SHEP, University of Southampton, October 8, 2010 Page 6

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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 = 0): 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 0 maps to UV conformal AdS 5 boundary z 0 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 SHEP, University of Southampton, October 8, 2010 Page 8

Nonconformal metric dual to a confining gauge theory ds 2 = R2 z 2 eϕ(z) ( η µν dx µ dx ν dz 2) where ϕ(z) 0 at small z for geometries which are asymptotically AdS 5 Gravitational potential energy for object of mass m V = mc 2 g 00 = mc 2 R eϕ(z)/2 z Consider warp factor exp(±κ 2 z 2 ) Plus solution: V (z) increases exponentially confining any object in modified AdS metrics to distances z 1/κ SHEP, University of Southampton, October 8, 2010 Page 9

II. Higher Spin Modes in AdS Space Description of higher spin modes in AdS space (Frondsal, Fradkin and Vasiliev) Lagrangian for scalar 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(z)( 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 = 0 with four-momentum P µ and invariant hadronic mass P µ P µ =M 2 Construct effective action in terms of spin-j modes Φ J with only physical degrees of freedom SHEP, University of Southampton, October 8, 2010 Page 10

Lagrangian for scalar field in AdS d+1 S = d d x dz g e ϕ(z) ( g MN M Φ N Φ µ 2 Φ Φ ) Factor out plane waves along 3+1: [ zd 1 e ϕ(z) z Φ P (x µ, z) = e ip x Φ(z) ( e ϕ ) ( ) ] (z) µr 2 z d 1 z + Φ(z) = M 2 Φ(z) z where P µ P µ = M 2 invariant mass of physical hadron with four-momentum P µ Spin-J mode Φ µ1 µ J with all indices along 3+1 and shifted dimensions Φ J (z) z J Φ(z) Find AdS wave equation [ zd 1 2J e ϕ(z) z ( e ϕ ) (z) z d 1 2J z + ( ) ] µr 2 Φ J (z) = M 2 Φ J (z) z SHEP, University of Southampton, October 8, 2010 Page 11

III. 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 Instant form: hypersurface defined by t = 0, the familiar one Front form: hypersurface is tangent to the light cone at τ = t + z/c = 0 x + = x 0 + x 3 x = x 0 x 3 light-front time longitudinal space variable k + = k 0 + k 3 longitudinal momentum (k + > 0) k = k 0 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 + SHEP, University of Southampton, October 8, 2010 Page 12

QCD Lagrangian L QCD = 1 4g 2 Tr (Gµν G µν ) + iψd µ γ µ ψ + mψψ LF Momentum Generators P = (P +, P, P ) in terms of dynamical fields ψ, A P = 1 dx d 2 x ψ γ + (i ) 2 + m 2 2 i + ψ + interactions P + = dx d 2 x ψ γ + i + ψ P = 1 dx d 2 x ψ γ + i ψ 2 LF Hamiltonian P generates LF time translations [ ψ(x), P ] = i and the generators P + and P are kinematical x + ψ(x) SHEP, University of Southampton, October 8, 2010 Page 13

Light-Front Fock Representation Dirac field ψ, expanded in terms of ladder operators on the initial surface x + = x 0 + x 3 P = dq + d 2 q ( q 2 + m 2 ) (2π) 3 q + b λ (q)b λ(q) + interactions λ 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 = 0. i=1 SHEP, University of Southampton, October 8, 2010 Page 14

Semiclassical Approximation to QCD in the Light Front [GdT and S. J. Brodsky, PRL 102, 081601 (2009)] Compute M 2 from hadronic matrix element ψ(p ) P µ P µ ψ(p ) =M 2 ψ(p ) ψ(p ) Find M 2 = n [dxi ][ d 2 ] k i l ( k 2 l + m 2 l x q ) ψ n (x i, k i ) 2 + interactions Semiclassical approximation to QCD: with k 2 i = m2 i ψ n (k 1, k 2,..., k n ) φ n ( (k1 + k 2 + + k n ) 2 }{{} for each constituent Functional dependence of Fock state n given by invariant mass M 2 n = ( n a=1 k µ a ) 2 = a M 2 n k 2 a + m2 a x a Key variable controlling bound state: off-energy shell E = M 2 M 2 n ) SHEP, University of Southampton, October 8, 2010 Page 15

In terms of n 1 independent transverse impact coordinates b j, j = 1, 2,..., n 1, M 2 = n 1 dx j d 2 b j ψn(x i, b i ) ( 2 + ) m2 b l l ψ n (x i, b i ) + interactions x n j=1 q l Relevant variable conjugate to invariant mass 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 imϕ X(x) φ(ζ) 2πζ factoring angular ϕ, longitudinal X(x) and transverse mode φ(ζ) SHEP, University of Southampton, October 8, 2010 Page 16

Ultra relativistic limit m q 0 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 is 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 but can be implemented in the LF Hamiltonian EOM or by applying the L-S formalism SHEP, University of Southampton, October 8, 2010 Page 17

III. Light-Front Holographic Mapping Φ P (z) ψ(p ) LF Holographic mapping found originally matching expressions of EM and gravitational form factors of hadrons in AdS and LF QCD [Brodsky and GdT (2006, 2008)] 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 0 Scaling dimension τ of AdS mode Φ J is τ = 2 + L in agreement with twist scaling dimension of a two parton bound state in QCD SHEP, University of Southampton, October 8, 2010 Page 18

Positive dilaton background ϕ = κ 2 z 2 : 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 = 4κ 2 (n + L + S/2) 0.8 Φ Ζ a Φ Ζ b 0.5 0.6 0.4 0.0 0.2 0.0 0 2 4 6 8 10 Ζ 0.5 0 2 4 6 8 10 Ζ LFWFs φ n,l (ζ) in physical spacetime for dilaton exp(κ 2 z 2 ): a) orbital modes and b) radial modes SHEP, University of Southampton, October 8, 2010 Page 19

4κ 2 for n = 1 4κ 2 for L = 1 2κ 2 for S = 1 6 J PC 0-+ 1+- 2-+ 3+- 4-+ n=3 n=2 n=1 n=0 6 J PC 1-- 2++ 3-- 4++ n=3 n=2 n=1 n=0 f 2 (2300) M 2 4 2 0 6-2010 8796A5 π(1800) π(1300) π 0 π 2 (1880) π 2 (1670) b(1235) 1 2 3 4 L M 2 4 2 0 9-2009 8796A1 ρ(1700) ω(1650) ρ(1450) ω(1420) ρ(770) ω(782) 0 f 2 (1950) a 2 (1320) f 2 (1270) ρ 3 (1690) ω 3 (1670) a 4 (2040) f 4 (2050) 1 2 3 4 L Regge trajectories for the π (κ = 0.6 GeV) and the I =1 ρ-meson and I =0 ω-meson families (κ = 0.54 GeV) SHEP, University of Southampton, October 8, 2010 Page 20

Fermionic Modes From Nick Evans Soft wall model equivalent to Dirac equation in presence of a holographic linear confining potential [ i Solution (µr = ν + 1/2, ν = L + 1) ) ] (zη lm Γ l m + 2Γ z + µr + κ 2 z Ψ(x l ) = 0. Ψ + (z) z 5 2 +ν e κ2 z 2 /2 L ν n(κ 2 z 2 ) Ψ (z) z 7 2 +ν e κ2 z 2 /2 L ν+1 n (κ 2 z 2 ) Eigenvalues (how to fix the overall energy scale, see arxiv:1001.5193) M 2 = 4κ 2 (n + L + 1) Obtain spin-j mode Φ µ1 µ J 1/2, J > 1 2, with all indices along 3+1 from Ψ by shifting dimensions SHEP, University of Southampton, October 8, 2010 Page 21

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=0 (b) n=3 n=2 n=1 n=0 M 2 4 N(2200) Δ(2420) 2 N(1710) N(1440) N(1680) N(1720) Δ(1600) Δ(1950) Δ(1905) Δ(1920) Δ(1910) Δ(1232) 0 9-2009 8796A3 N(940) 0 1 2 3 4 L 0 1 2 3 4 L Parent and daughter 56 Regge trajectories for the N and baryon families for κ = 0.5 GeV spectrum identical to Forkel, Beyer and Frederico and Forkel and Klempt SHEP, University of Southampton, October 8, 2010 Page 22

V. Light-Front Holographic Mapping of Current Matrix Elements [S. J. Brodsky and GdT, PRL 96, 201601 (2006)] EM transition matrix element in QCD: local coupling to pointlike constituents where Q = P P and J µ = e q qγ µ q ψ(p ) J µ ψ(p ) = (P + P )F (Q 2 ) EM hadronic matrix element in AdS space from non-local coupling of external EM field propagating in AdS with extended mode Φ(x, z) d 4 x dz g A l (x, z)φ P (x, z) l Φ P (x, z) Are the transition amplitudes related? How to recover hard pointlike scattering at large Q out of soft collision of extended objects? [Polchinski and Strassler (2002)] Mapping at fixed light-front time: Φ P (z) ψ(p ) SHEP, University of Southampton, October 8, 2010 Page 23

Electromagnetic probe polarized along Minkowski coordinates, (Q 2 = q 2 > 0) A(x, z) µ = ɛ µ e iq x V (Q, z), A z = 0 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) = 0 Solution V (Q, z) = zqk 1 (zq) 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 0 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 (2002)]. F (Q 2 ) = R 3 dz z 3 V (Q, z) Φ2 J(z) ( 1 Q 2 ) τ 1 J(Q,z), Φ(z) 1.2 0.8 0.4 0 5-2006 0 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 (2001)] SHEP, University of Southampton, October 8, 2010 Page 24

Electromagnetic Form-Factor [S. J. Brodsky and GdT, PRL 96, 201601 (2006); PRD 77, 056007 (2008)] 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 =0) = 1 0 2 Integrating over angle F π +(q 2 ) = 2π 1 0 dx x(1 x) ) 1 x ζdζj 0 (ζq x ψud/π (x, ζ) 2 where ζ 2 = x(1 x)b 2 SHEP, University of Southampton, October 8, 2010 Page 25

Compare with electromagnetic FF in AdS space where V (Q, z) = zqk 1 (zq) F (Q 2 ) = R 3 dz z 3 V (Q, z)φ2 π + (z) Use the integral representation V (Q, z) = 1 0 dx J 0 (ζq ) 1 x x Find F (Q 2 ) = R 3 1 0 ( dz 1 x dx z 3 J 0 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 Φ(ζ), ζ z R Same results from mapping of gravitational form factor [S. J. Brodsky and GdT, PRD 78, 025032 (2008)] SHEP, University of Southampton, October 8, 2010 Page 26

Expand X(x) in Gegenbauer polynomials (DA evolution equation [Lepage and Brodsky (1980)]) X(x) = x(1 x) = x(1 x) n=0 a n C 3/2 n (2x 1) Normalization 1 0 dx x(1 x) X2 (x) = n P n = 1 C λ n C λ m = 1 0 dx x λ 1/2 (1 x) λ 1/2 C λ n(2x 1)C λ m(2x 1) = 21 4λ πγ(n + 2λ) n!(n + λ)γ 2 (λ) Compute asymptotic probability P n=0 (Q ) P n=0 = π2 32 0.3 ( a 0 = 3π 4 ) SHEP, University of Southampton, October 8, 2010 Page 27

VI. Higher Fock Components [GdT and S. J Brodsky, arxiv:1010.1204 [hep-ph]] LF Lorentz invariant Hamiltonian equation for the relativistic bound state system P µ P µ ψ(p ) = M 2 ψ(p ), where P µ P µ = P P + P 2 H LF H LF sum of kinetic energy of partons H 0 LF plus an interaction H I, H LF = H 0 LF + H I Expand in Fock eigenstates of HLF 0 : ( n M 2 i=1 an infinite number of coupled integral equations ψ = n ψ n n, ) k 2 i + m2 ψ n = x i m n V m ψ m Only interaction in AdS/QCD is the confinement potential In QFT the resulting LF interaction is the 4-point effective interaction H I = ψψv (ζ 2 )ψψ wich leads to qq qq, qq qq, q qqq and q qqq Create Fock states with extra quark-antiquark pairs. No mixing with qqg Fock states (g s ψγ Aψ) Explain the dominance of quark interchange in large angle elastic scattering SHEP, University of Southampton, October 8, 2010 Page 28

Detailed Structure of Space-and Time Like Pion Form Factor Holographic variable for n-parton hadronic bound state 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) 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 (2008)] F (Q 2 ) = 1 ( )( ) ( ) 1 + Q2 1 + Q2 1 + Q2 M 2 ρ M 2 ρ M 2 ρ N 2 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 = 0.54 GeV, Γ ρ = 130, Γ ρ = 400, Γ ρ = 300 MeV, P qqqq = 13% SHEP, University of Southampton, October 8, 2010 Page 29

0.6 Q 2 F Π Q 2 2 M Ρ 2 log F Π Q 2 0.5 0.4 1 2 M Ρ 0.3 0 2 M Ρ 0.2 1 0.1 Q 2 GeV 2 0.0 0 1 2 3 4 5 6 7 2 Q 2 GeV 2 0 1 2 3 4 5 SHEP, University of Southampton, October 8, 2010 Page 30