1 Introduction
###### Abstract

SLAC–PUB–13107

January 2008

Stanley J. Brodsky and Guy F. de Téramond

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA

Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau, France

Two lectures presented at the International School of Subnuclear Physics

Searching for the ‘Totally Unexpected’ in the LHC Era

Erice, Sicily, August 29 – September 7, 2007

## 1 Introduction

Quantum Chromodynamics, the Yang-Mills local gauge field theory of color symmetry provides a fundamental description of hadron and nuclear physics in terms of quark and gluon degrees of freedom. Yet, because of its strong coupling nature, it has been difficult to find analytic solutions to QCD or to make precise predictions outside of its perturbative domain. An important theoretical goal is thus to find an initial approximation to QCD which is both analytically tractable and which can be systematically improved. For example, in quantum electrodynamics, the Coulombic Schrödinger and Dirac equations provide quite accurate first approximations to atomic bound state problems, which can then be systematically improved using the Bethe-Salpeter formalism and correcting for quantum fluctuations, such as the Lamb Shift and vacuum polarization.

It was originally believed that the AdS/CFT mathematical correspondence could only be applied to strictly conformal theories, such as supersymmetric Yang-Mills gauge theory. Conformal symmetry is broken in physical QCD by quantum effects and quark masses. There are indications, however both from theory and phenomenology, that the QCD coupling is slowly varying at small momentum transfer. In these lectures we shall discuss how conformal symmetry, plus a simple ansatz for color confinement, provides a remarkably accurate first approximation for QCD.

The essential element for the application of AdS/CFT to hadron physics is the indication that the QCD coupling becomes large and constant in the low momentum domain GeV/c, thus providing a window where conformal symmetry can be applied. Solutions of the Dyson-Schwinger equations for the three-gluon and four-gluon couplings [1, 2, 3, 4, 5, 6, 7] and phenomenological studies [8, 9, 10] of QCD couplings based on physical observables such as decay [11] and the Bjorken sum rule [12], show that the QCD function vanishes and become constant at small virtuality; i.e., effective charges develop an “infrared fixed point.” Recent lattice simulations [13, 14] and nonperturbative analyses [15] have also indicated an infrared fixed point for QCD. One can understand this physically [16]: in a confining theory where gluons have an effective mass [17] or maximal wavelength, all vacuum polarization corrections to the gluon self-energy decouple at long wavelength; thus an infrared fixed point appears to be a natural consequence of confinement. Furthermore, if one considers a semi-classical approximation to QCD with massless quarks and without particle creation or absorption, then the resulting function is zero, the coupling is constant, and the approximate theory is scale and conformal invariant [18, 19], allowing the mathematical tools of conformal symmetry to be applied. One can use conformal symmetry as a template, systematically correcting for its nonzero function as well as higher-twist effects.

One of the key consequences of conformal invariance are the dimensional counting rules [20, 21]. The leading power fall-off of a hard exclusive process follows from the conformal scaling of the underlying hard-scattering amplitude: , where is the total number of fields (quarks, leptons, or gauge fields) participating in the hard scattering. Thus the reaction is dominated by subprocesses and Fock states involving the minimum number of interacting fields. In the case of scattering processes, this implies where and is the minimum number of constituents of . The near-constancy of the effective QCD coupling helps explain the empirical success of dimensional counting rules for the near-conformal power law fall-off of form factors and fixed angle scaling [22]. For example, one sees the onset of perturbative QCD scaling behavior even for exclusive nuclear amplitudes such as deuteron photodisintegration, here , constant at fixed CM angle.

In the case of hard exclusive reactions [23], the virtuality of the gluons exchanged in the underlying QCD process is typically much less than the momentum transfer scale , as several gluons share the total momentum transfer. Since the coupling is probed in the conformal window, this kinematic feature can explain why the measured proton Dirac form factor scales as up to GeV [24] with little sign of the logarithmic running of the QCD coupling. Thus conformal symmetry can be a useful first approximant even for physical QCD. The measured deuteron form factor also appears to follow the leading-twist QCD predictions at large momentum transfers in the few GeV region [25, 26, 27].

Recently the Hall A collaboration at Jefferson Laboratory [28] has reported a significant exception to the general empirical success of dimensional counting in fixed CM angle Compton scattering , instead of the predicted scaling. However, the hadron form factor , which multiplies the amplitude is found by Hall-A to scale as , in agreement with the PQCD and AdS/CFT prediction. In addition the timelike two-photon process appears to satisfy dimensional counting [29, 30].

Our main tool for implementing conformal symmetry will be the use of Anti-de-Sitter (AdS) space in five dimensions which provides a mathematical realization of the group , the group of Poincare’ plus conformal transformations. The AdS metric is

 ds2=R2z2(ημνdxμdxν−dz2), (1)

which is invariant under scale changes of the coordinate in the fifth dimension and . Thus one can match scale transformations of the theory in physical space-time to scale transformations in the fifth dimension . The isomorphism of the group of Poincare’ and conformal transformations to the group of isometries of Anti-de Sitter space underlies the AdS/CFT correspondence [31] between string states defined on the 5-dimensional Anti–de Sitter (AdS) space-time and conformal field theories in physical space-time [32, 33] . In particular, we shall show that there is an exact correspondence between the fifth-dimensional coordinate of AdS space and a specific impact variable which measures the separation of the quark and gluonic constituents within the hadron in ordinary space-time. This connection leads to AdS/CFT predictions for the analytic form of the frame-independent light-front wavefunctions (LFWFs) of mesons and baryons, the fundamental entities which encode hadron properties. The LFWFs in turn predict decay constants and spin correlations, as well as dynamical quantities such as form factors, structure functions, generalized parton distributions, and exclusive scattering amplitudes.

Scale-changes in the physical world can thus be represented by studying dynamics in a mathematical fifth dimension with the metric. Different values of the holographic variable determine the scale of the invariant separation between the partonic constituents. This is illustrated in Fig. 1. Hard scattering processes occur in the small- ultraviolet (UV) region of AdS space. In particular, the zero separation limit corresponds to the asymptotic boundary, where the QCD Lagrangian is defined.

As shown by Polchinski and Strassler [34], one can simulate confinement by imposing boundary conditions in the holographic variable . The infrared (IR) cut-off at breaks conformal invariance, allowing the introduction of the QCD mass scale and a spectrum of particle states. In the hard wall model [34] a cut-off is placed at a finite value and the spectrum of states is linear in the radial and angular momentum quantum numbers: . In the soft wall model a smooth infrared cutoff is chosen to model confinement and reproduce the usual Regge behavior  [35]. The resulting models, although ad hoc, provide a simple semi-classical approximation to QCD which has both constituent counting rule behavior at short distances and confinement at large distances.

Following the approach described above, a limited set of operators is introduced to construct phenomenological viable five-dimensional dual holographic models. This simple prescription, which has been described as a “bottom-up” approach, has been successful in obtaining general properties of scattering amplitudes of hadronic bound states at strong coupling  [34, 36, 37, 38, 39, 40], the low-lying hadron spectra [35, 41, 42, 43, 44, 45, 46, 47, 48, 49], hadron couplings and chiral symmetry breaking [41, 50, 51, 52, 53], quark potentials in confining backgrounds [54, 55], a description of weak hadron decays [56] and euclidean correlation functions [57]. Geometry back-reaction in AdS may also be relevant to the infrared physics [58] and wall dynamics [59]. The gauge theory/gravity duality also provides a convenient framework for the description of deep inelastic scattering structure functions at small  [60, 61, 62], a unified description of hard and soft pomeron physics [63] and gluon scattering amplitudes at strong coupling [64].

In the top-down approach, one introduces higher dimensional branes to the background [65] in order to have a theory of flavor. One can obtain models with massive quarks in the fundamental representation, compute the hadronic spectrum, and describe chiral symmetry breaking in the context of higher dimensional brane constructs  [65, 66, 67, 68, 69]. However, a theory dual to QCD is unknown, and this “top-down” approach is difficult to extend beyond theories exceedingly constrained by their symmetries [70].

As we shall discuss, there is a remarkable mapping between the AdS description of hadrons and the Hamiltonian formulation of QCD in physical space-time quantized on the light front. The light-front wavefunctions of bound states in QCD are relativistic and frame-independent generalizations of the familiar Schrödinger wavefunctions of atomic physics, but they are determined at fixed light-cone time —the “front form” advocated by Dirac [71]—rather than at fixed ordinary time . The light-front wavefunctions of a hadron are independent of the momentum of the hadron, and they are thus boost invariant; Wigner transformations and Melosh rotations are not required. The light-front formalism for gauge theories in light-cone gauge is particularly useful in that there are no ghosts and one has a direct physical interpretation of orbital angular momentum.

An important feature of light-front quantization is the fact that it provides exact formulas to write down matrix elements as a sum of bilinear forms, which can be mapped into their AdS/CFT counterparts in the semi-classical approximation. One can thus obtain not only an accurate description of the hadron spectrum for light quarks, but also a remarkably simple but realistic model of the valence wavefunctions of mesons, baryons, and glueballs. In terms of light front coordinates the AdS metric is

 ds2=R2z2(dx+dx−−dx2⊥−dz2). (2)

At fixed light-front time , the metric depends only on the transverse and the holographic variable . Thus we can find an exact correspondence between the fifth-dimensional coordinate of anti-de Sitter space and a specific impact variable in the light-front formalism. The new variable measures the separation of the constituents within the hadron in ordinary space-time. The amplitude describing the hadronic state in can then be precisely mapped to the light-front wavefunctions of hadrons in physical space-time [45], thus providing a relativistic description of hadrons in QCD at the amplitude level. This connection allows one to compute the analytic form [45] of the light-front wavefunctions of mesons and baryons. AdS/CFT also provides a non-perturbative derivation of dimensional counting rules for the power-law fall-off of form factors and exclusive scattering amplitudes at large momentum transfer. The AdS/CFT approach thus leads to a model of hadrons which has both confinement at large distances and the conformal scaling properties which reproduce dimensional counting rules for hard exclusive reactions.

## 2 Gauge/Gravity Semiclassical Correspondence

The formal statement of the duality between a gravity theory on -dimensional Anti-de Sitter AdS space and the strong coupling limit of a conformal field theory (CFT) on the -dimensional asymptotic boundary of AdS at is expressed in terms of the partition function for a field propagating in the bulk

 Zgrav[Φ(x,z)]=eiSeff[Φ]=∫D[Φ]eiS[Φ], (3)

where is the effective action of the AdS theory, and the -dimensional generating functional of the conformal field theory in presence of an external source ,

 ZCFT[Φ0(x)]=eiWCFT[Φ0]=⟨exp(i∫ddxΦ0(x)O(x))⟩. (4)

The functional is the generator of connected Green’s functions of the boundary theory and is a QCD local interpolating operator. The precise relation of the gravity theory on AdS space to the conformal field theory at its boundary is [32, 33]

 Zgrav[Φ(x,z)|z=0=Φ0(x)]=ZCFT[Φ0], (5)

where the partition function (3) on AdS is integrated over all possible configurations in the bulk which approach its boundary value . If we neglect the contributions from the non-classical configurations to the gravity partition function, then the generator of connected Green’s functions of the four-dimensional gauge theory (4) is precisely equal to the classical (on-shell) gravity action (3)

 WCFT[ϕ0]=Seff[Φ(x,z)|z=0=Φ0(x)]on−shell, (6)

evaluated in terms of the classical solution to the bulk equation of motion. This defines the semiclassical approximation to the conformal field theory. In the limit , the independent solutions behave as

 Φ(z,x)→zΔΦ+(x)+zd−ΔΦ−(x), (7)

where is the conformal dimension. The non-normalizable solution is the boundary value of the bulk field which couples to a QCD gauge invariant operator in the asymptotic boundary, thus . The normalizable solution is the response function and corresponds to the physical states [72]. The interpolating operators of the boundary conformal theory are constructed from local gauge-invariant products of quark and gluon fields and their covariant derivatives, taken at the same point in four-dimensional space-time in the limit. Their conformal twist-dimensions are matched to the scaling behavior of the AdS fields in the limit and are thus encoded into the propagation of the modes inside AdS space.

AdS coordinates are the Minkowski coordinates and , the holographic coordinate, which we label . The metric of the full space-time is , where , and has diagonal components . Unless stated otherwise, 5-dimensional fields are represented by capital letters such as and . Holographic fields in 4-dimensional Minkowski space are represented by and and constituent quark and gluon fields by and . We begin by writing the action for scalar modes on AdS. We consider a quadratic action of a free field propagating in the AdS background

 S[Φ]=12∫dd+1x√g[gℓm∂ℓΦ∂mΦ−μ2Φ2], (8)

where in the conformal limit and is a fifth dimensional mass. Taking the variation of (8) we find the equation of motion

 1√g∂∂xℓ(√g gℓm∂∂xmΦ)+μ2Φ=0. (9)

Integrating by parts and using the equation of motion, the bulk contribution to the action vanishes, and one is left with a non-vanishing surface term in the ultraviolet boundary

 S=Rd−12limz→0∫ddx1zd−1Φ∂zΦ, (10)

which can be identified with the boundary functional . Substituting the leading dependence (7) of near in the ultraviolet surface action (10) and using the functional relation , it follows that is related to the expectation values of in the presence of the source  [72]: The exact relation depends on the normalization of the fields used [73]. The field thus acts as a classical field, and it is the boundary limit of the normalizable string solution which propagates in the bulk.

Factoring out the dependence of the hadronic modes along the Poincaré coordinates , in (9), we find the effective AdS wave equation for the scalar string mode

 [z2∂2z−(d−1)z∂z+z2M2−(μR)2]Φ(z)=0. (11)

The eigenvalues of (11) are the hadronic invariant mass states and the fifth-dimensional mass is related to the conformal dimension . Stable solutions satisfy the condition , according to the Breitenlohner-Freedman bound [74].

Higher spin- bosonic modes in AdS are described by a set of coupled differential equations [75]. Each hadronic state of integer spin , , is dual to a normalizable string mode , with four-momentum and spin polarization indices along the 3+1 physical coordinates. For string modes with all the polarization indices along the Poncaré coordinates, the coupled differential wave equations for a spin- bosonic mode reduce to the homogeneous equation [75]

 [z2∂2z−(d−1−2S)z∂z+z2M2−(μR)2]ΦS(z)=0, (12)

with . We expect to avoid large anomalous dimensions associated with since modes with do not couple to stringy excitations.

## 3 The Holographic Light-Front Hamiltonian and Schrödinger Equation

We shall show in Sect. 5 how the string amplitude can be mapped to the light-front wave functions of hadrons in physical space-time [45]. In fact, we find an exact correspondence between the holographic variable and an impact variable which measures the transverse separation of the constituents within a hadron, we can identify . The mapping of from AdS space to in the LF space allows the equations of motion in AdS space to be recast in the form of a light-front Hamiltonian equation [76]

 HLF|ϕ⟩=M2|ϕ⟩, (13)

a remarkable result which maps AdS/CFT solutions to light-front equations in physical 3+1 space-time. By substituting , in the AdS scalar wave equation (11) for , we find an effective Schrödinger equation as a function of the weighted impact variable

 [−d2dζ2+V(ζ)]ϕ(ζ)=M2ϕ(ζ), (14)

with the conformal potential , an effective two-particle light-front radial equation for mesons [16, 45]. Its eigenmodes determine the hadronic mass spectrum. We have written above . The holographic hadronic light-front wave functions are normalized according to

 ⟨ϕ|ϕ⟩=∫dζ|⟨ζ|ϕ⟩|2=1, (15)

and represent the probability amplitude to find -partons at transverse impact separation . Its eigenvalues are set by the boundary conditions at and are given in terms of the roots of Bessel functions: . The normalizable modes are

 ϕL,k(ζ)=√2ΛQCDJ1+L(βL,k)√ζJL(ζβL,kΛQCD)θ(ζ≤Λ−1QCD). (16)

The lowest stable state is determined by the Breitenlohner-Freedman bound [74]. Higher excitations are matched to the small asymptotic behavior of each string mode to the corresponding conformal dimension of the boundary operators of each hadronic state. The effective wave equation (14) is a relativistic light-front equation defined at . The AdS metric (2) is invariant if and at equal light-front time . The Casimir operator for the rotation group in the transverse light-front plane is . This shows the natural holographic connection to the light front.

For higher spin bosonic modes we can also recast the wave equation AdS (12) into its light-front form (13). Using the substitution , , we find a LF Schrödinger equation identical to (14) with , provided that . Stable solutions satisfy a generalized Breitenlohner-Freedman bound , and thus the lowest stable state has scaling dimensions , independent of . The fundamental LF equation of AdS/CFT has the appearance of a Schrödinger equation, but it is relativistic, covariant, and analytically tractable.

The pseudoscalar meson interpolating operator , written in terms of the symmetrized product of covariant derivatives with total internal space-time orbital momentum , is a twist-two, dimension operator with scaling behavior determined by its twist-dimension . Likewise the vector-meson operator has scaling dimension . The scaling behavior of the scalar and vector AdS modes is precisely the scaling required to match the scaling dimension of the local pseudoscalar and vector-meson interpolating operators. The light meson spectrum is compared in Figure 2 with the experimental values.

### 3.1 Integrability of AdS/CFT Equations

The integrability methods of [78] find a remarkable application in the AdS/CFT correspondence. Integrability follows if the equations describing a physical model can be factorized in terms of linear operators. These ladder operators generate all the eigenfunctions once the lowest mass eigenfunction is known. In holographic QCD, the conformally invariant 3 + 1 light-front differential equations can be expressed in terms of ladder operators and their solutions can then be expressed in terms of analytical functions. In the conformal limit the ladder algebra for bosonic () or fermionic () modes is given in terms of the operator ()

 ΠB,Fν(ζ)=−i⎛⎝ddζ−ν+12ζΓB,F⎞⎠, (17)

 ΠB,Fν(ζ)†=−i⎛⎝ddζ+ν+12ζΓB,F⎞⎠, (18)

with commutation relations

 [ΠB,Fν(ζ),ΠB,Fν(ζ)†]=2ν+1ζ2ΓB,F. (19)

For bosonic modes the Hamiltonian is written as a bilinear form: . For the Hamiltonian is positive definite

 ⟨ϕ∣∣HνLC∣∣ϕ⟩=∫dζ|Πνϕ(z)|2≥0, (20)

and its eigenvalues are positive: . For the Hamiltonian is not bounded from below. The critical value of the potential corresponds to with potential . LF quantum-mechanical stability conditions are thus equivalent to the stability conditions which follows from the Breitenlohner-Freedman stability bound [74]. Higher orbital states are constructed from the -th application of the raising operator on the ground state . In the light-front coordinate representation

 ⟨ζ|L⟩∼√ζ(−ζ)L(1ζddζ)LJ0(ζM)∼√ζJL(ζM). (21)

In the fermionic case the eigenmodes also satisfy a first order LF Dirac equation as will be shown in Sect. 4.

### 3.2 Soft-Wall Holographic Model

The predicted mass spectrum in the truncated space hard-wall (HW) model is linear at high orbital angular momentum , in contrast to the quadratic dependence in the usual Regge parameterization. It has been shown recently that by choosing a specific profile for a non-constant dilaton, the usual Regge dependence can be obtained [35]. This procedure retains conformal AdS metrics (1) while introducing a smooth cutoff which depends on the profile of a dilaton background field

 S=∫d4xdz√ge−φ(z)L, (22)

where is a function of the holographic coordinate which vanishes in the ultraviolet limit . The IR hard-wall or truncated space holographic model corresponds to a constant dilaton field in the confining region, , and to very large values elsewhere: for . The introduction of a soft cutoff avoids the ambiguities in the choice of boundary conditions at the infrared wall. A convenient choice [35] for the background field with usual Regge behavior is . The resulting wave equations are equivalent to the radial equation of a two-dimensional oscillator, previously found in the context of mode propagation on AdS, in the light-cone formulation of Type II supergravity [79]. Also, equivalent results follow from the introduction of a gaussian warp factor in the AdS metric for the particular case of massless vector modes propagating in the distorted metric [80]. A different approach to the soft-wall (SW) consists in the non-conformal extension of the algebraic expressions found in the previous section to obtain directly the corresponding holographic LF wave equations. This method is particularly useful to extend the non-conformal results to the fermionic sector where the corresponding linear wave equations become exactly solvable. The extended generators are given in terms of the matrix-valued operator and its adjoint ()

 ΠB,Fν(ζ) = −i⎛⎝ddζ−ν+12ζΓB,F−κ2ζΓB,F⎞⎠, (23) ΠB,Fν(ζ)† = −i⎛⎝ddζ+ν+12ζΓB,F+κ2ζΓB,F⎞⎠, (24)

with commutation relations

 [ΠB,Fν(ζ),ΠB,Fν(ζ)†]=(2ν+1ζ2−2κ2)ΓB,F. (25)

An account of the extended algebraic holographic model and a possible supersymmetric connection between the bosonic and fermionic operators used in the holographic construction will be described elsewhere.

## 4 Baryonic Spectra in AdS/QCD

The holographic model based on truncated AdS space can be used to obtain the hadronic spectrum of light quark and bound states. Specific hadrons are identified by the correspondence of the AdS amplitude with the twist dimension of the interpolating operator for the hadron’s valence Fock state, including its orbital angular momentum excitations. Bosonic modes with conformal dimension are dual to the interpolating operator with . For fermionic modes .

As an example, we will outline here the analysis of the baryon spectrum in AdS/CFT. The action for massive fermionic modes on AdS is

 S[¯¯¯¯Ψ,Ψ]=∫dd+1x√g¯¯¯¯Ψ(iΓℓDℓ−μ)Ψ, (26)

with the equation of motion

 [i(zηℓmΓℓ∂m+d2Γz)+μR]Ψ(xℓ)=0. (27)

Upon the substitution  ,  , we find the light-front Dirac equation

 (αΠF(ζ)−M)ψ(ζ)=0, (28)

where the generator is given by (17) and in the Weyl representation. The solution is

 ψ(ζ)=C√ζ[JL+1(ζM)u++JL+2(zM)u−], (29)

with . A discrete four-dimensional spectrum follows when we impose the boundary condition : , with a scale-independent mass ratio [44].

Figure 3(a) shows the predicted orbital spectrum of the nucleon states and Fig. 3(b) the orbital resonances. The spin-3/2 trajectories are determined from the corresponding Rarita-Schwinger equation. The solution of the spin-3/2 for polarization along Minkowski coordinates, , is identical to the spin-1/2 solution. The data for the baryon spectra are from [77]. The internal parity of states is determined from the SU(6) spin-flavor symmetry.

Since only one parameter, the QCD mass scale is introduced, the agreement with the pattern of physical states is remarkable. In particular, the ratio of to nucleon trajectories is determined by the ratio of zeros of Bessel functions.

We can solve the LF Dirac equation (28) with the non-conformal extended generator given by (23). The solutions to the Dirac equation are

 ψ+(ζ) ∼ z12+νe−κ2ζ2/2Lνn(κ2ζ2), (30) ψ−(ζ) ∼ z32+νe−κ2ζ2/2Lν+1n(κ2ζ2). (31)

with eigenvalues . Comparing with usual Dirac equation in AdS space we find

 [i(zηℓmΓℓ∂m+d2Γz)+μR+V(z)]Ψ(xℓ)=0. (32)

with Thus for fermions the “soft-wall” corresponds to fermion modes propagating in AdS conformal metrics in presence of a linear confining potential.

The AdS/QCD correspondence is particularly relevant for the description of hadronic form factors, since it incorporates the connection between the twist of the hadron to the fall-off of its current matrix elements, as well as essential aspects of vector meson dominance. It also provides a convenient framework for analytically continuing the space-like results to the time-like region. Recent applications to the electromagnetic [81, 82, 83, 84, 85, 86, 87, 88] and gravitational [89] form factors of hadrons have followed from the original work described in [60, 90].

### 5.1 Meson Form Factors

In AdS/CFT, the hadronic matrix element for the electromagnetic current has the form of a convolution of the string modes for the initial and final hadrons with the external electromagnetic source which propagates inside AdS. We discuss first the truncated space or hard wall [34] holographic model, where quark and gluons as well as the external electromagnetic current propagate freely into the AdS interior according to the AdS metric. Assuming minimal coupling the form factor has the form [60, 90]

 ig5∫d4xdz√gAℓ(x,z)Φ∗P′(x,z)↔∂ℓΦP(x,z), (33)

where is a five-dimensional effective coupling constant and is a normalizable mode representing a hadronic state, , with hadronic invariant mass given by . We consider the propagation inside AdS space of an electromagnetic probe polarized along Minkowski coordinates () , where has the value 1 at zero momentum transfer, since we are normalizing the bulk solutions to the total charge operator, and as boundary limit the external current . Thus .

The propagation of the external current inside AdS space is described by the wave equation

 [z2∂2z−z∂z−z2Q2]J(Q2,z)=0, (34)

with the solution . Substituting the normalizable mode in (33) and extracting an overall delta function from momentum conservation at the vertex, we find the matrix element , with

 F(Q2)=R3∫dzz3Φ(z)J(Q2,z)Φ(z). (35)

The form factor in AdS is thus represented as the overlap of the normalizable modes dual to the incoming and outgoing hadrons, and , with the non-normalizable mode, , dual to the external source [60]. Since for large , it follows that the external electromagnetic field is suppressed inside the AdS cavity for large . At small the string modes scale as . At large enough , the important contribution to (35) is from the region near : , and the ultraviolet point-like behavior [91] responsible for the power law scaling [20, 21] is recovered. This is a remarkable consequence of truncating AdS space since we are describing the coupling of an electromagnetic current to an extended mode, and instead of soft collision amplitudes characteristic of strings, hard point-like ultraviolet behavior is found [34].

The form factor in AdS space in presence of the dilaton background has the additional term in the metric

 F(Q2)=R3∫dzz3e−κ2z2Φ(z)Jκ(Q2,z)Φ(z). (36)

Since the non-normalizable modes also couple to the dilaton field, we must study the solutions of the modified wave equation describing the propagation in AdS space of an electromagnetic probe. The solution is [84, 85]

 Jκ(Q2,z)=Γ(1+Q24κ2)U(Q24κ2,0,κ2z2), (37)

where is the confluent hypergeometric function with the integral representation In the large limit, we find that . Thus, for large transverse momentum the current decouples from the dilaton background.

We can compute the pion form factor from the AdS expressions (35) and (36) for the hadronic string modes in the hard-wall (HW)

 ΦHWπ(z)=√2ΛQCDR3/2J1(β0,1)z2J0(zβ0,1ΛQCD), (38)

and soft-wall (SW) model

 ΦSWπ(z)=√2κR3/2z2, (39)

respectively. For the soft wall model the results for form factors can be expressed analytically. For integer twist the form factor is expressed as a product of poles, corresponding to the first states along the vector meson trajectory [85]. Since the pion mode couples to a twist-two boundary interpolating operator which creates a two-component hadronic bound state, the form factor is given in the SW model by a simple monopole form. In Fig. 4, we plot the product for the soft and hard-wall holographic models.

When the results for the pion form factor are analytically continued to the time-like region, we obtain the results shown in Figure 5 for in the SW model. The monopole form of the SW model exhibits a pole at the mass and reproduces well the peak with MeV. In the strongly coupled semiclassical gauge/gravity limit hadrons have zero widths and are stable. The form factor accounts for the scaling behavior in the space-like region, but it does not give rise to the additional structure found in the time-like region since the pole saturates 100% of the monopole form.

### 5.2 The Nucleon Dirac Form Factors

As an example of a twist fall-off we compute the spin non-flip nucleon form factor in the soft wall model. Consider the spin non-flip form factors

 F+(Q2) = g+R4∫dzz4e−κ2z2Jκ(Q,z)|Ψ+(z)|2, (40) F−(Q2) = g−R4∫dzz4e−κ2z2Jκ(Q,z)|Ψ−(z)|2, (41)

where the effective charges and are determined from the spin-flavor structure of the theory. We choose the struck quark to have . The two AdS solutions and correspond to nucleons with total angular momentum and . For the spin-flavor symmetry

 Fp1(Q2) = R4∫dzz4e−κ2z2Jκ(Q,z)|Ψ+(ζ)|2, (42) Fn1(Q2) = −13R4∫dzz4e−κ2z2Jκ(Q,z)[|Ψ+(z)|2−|Ψ−(z)|2], (43)

where . The bulk-to-boundary propagator is the solution (37) of the AdS wave equation for the external electromagnetic current, and the plus and minus components of the twist 3 nucleon mode in the SW model are

 Ψ+(z)=√2κ2R2z7/2,      Ψ−(z)=κ3R2z9/2. (44)

For the SW model the results for and follow from the analytic form for the form factors for any given in Appendix D of reference [85] and are shown in Figure 6.

## 6 The Light-Front Fock Representation

The light-front expansion of any hadronic system is constructed by quantizing quantum chromodynamics at fixed light-cone time [71] . In terms of the hadron four-momentum , , the light-cone Lorentz invariant Hamiltonian for the composite system, , has eigenvalues given in terms of the eigenmass squared corresponding to the mass spectrum of the color-singlet states in QCD [76].

The hadron wavefunction is an eigenstate of the total momentum and and the longitudinal spin projection , and is normalized according to

 ⟨ψh(P+,P⊥,Sz)∣∣ψh(P′+,P′⊥,S′z)⟩=2P+(2π)3δSz,S′zδ(P+−P′+)δ(2)(P⊥−P′⊥). (45)

The momentum generators and are kinematical; i.e., they are independent of the interactions. The LF time evolution operator can be derived directly from the QCD Lagrangian in the light-cone gauge . In principle, the complete set of bound states and scattering eigensolutions of can be obtained by solving the light-front Heisenberg equation , where is an expansion in multi-particle Fock eigenstates of the free LF Hamiltonian: . The LF Heisenberg equation has in fact been solved for QCD and a number of other theories using the discretized light-cone quantization method [95]. The light-cone gauge has the advantage that all gluon degrees of freedom have physical polarization and positive metric. In addition, orbital angular momentum has a simple physical interpretation in this representation. The light-front wavefunctions (LFWFs) provide a frame-independent representation of a hadron which relates its quark and gluon degrees of freedom to their asymptotic hadronic state.

Each hadronic eigenstate is expanded in a Fock-state complete basis of non-interacting -particle states with an infinite number of components

 (46)

where the sum begins with the valence state; e.g., for mesons. The coefficients of the Fock expansion

 ψn/h(xi,k⊥i,λi)=⟨n:xi,k⊥i,λi∣∣ψh⟩, (47)

are independent of the total momentum and of the hadron and depend only on the relative partonic coordinates, the longitudinal momentum fraction , the relative transverse momentum , and , the projection of the constituent’s spin along the direction. Thus, given the Fock-projection (47), the wavefunction of a hadron is determined in any frame. The amplitudes represent the probability amplitudes to find on-mass-shell constituents with longitudinal momentum , transverse momentum , helicity and invariant mass

 M2n=n∑i=1kμikiμ=n∑i=1k2⊥i+m2ixi, (48)

in the hadron . Momentum conservation requires and . In addition, each light front wavefunction obeys the angular momentum sum rule [96] , where and the orbital angular momenta have the operator form . It should be emphasized that the assignment of quark and gluon spin and orbital angular momentum of a hadron is a gauge-dependent concept. The LF framework in light-cone gauge provides a physical definition since there are no gauge field ghosts and the gluon has spin-projection ; moreover, it is frame-independent.

The LFWFs are normalized according to

 ∑n∫[dxi][d2k⊥i]∣∣ψn/h(xi,k⊥i)∣∣2=1, (49)

where the measure of the constituents phase-space momentum integration is

 ∫[dxi]≡n∏i=1∫dxiδ(1−n∑j=1xj), (50)
 ∫[d2k⊥i]≡n∏i=1∫d2k⊥i2(2π)3(16π3)δ(2)(n∑j=1k⊥j), (51)

for the normalization given by (45). The spin indices have been suppressed.

Given the light-front wavefunctions one can compute a large range of hadron observables. For example, the valence and sea quark and gluon distributions which are measured in deep inelastic lepton scattering are defined from the squares of the LFWFs summed over all Fock states . Form factors, exclusive weak transition amplitudes [97] such as , and the generalized parton distributions [98] measured in deeply virtual Compton scattering are (assuming the “handbag” approximation) overlaps of the initial and final LFWFs with and . In the case of deeply virtual meson production such as and , the meson enters the amplitude directly through its LFWF. In inclusive reactions such as electron-positron annihilation to jets, the hadronic light-front wavefunctions are the amplitudes which control the coalescence of comoving quarks and gluons into hadrons. Thus one can study hadronization at the amplitude level. Light-front wavefunctions also control higher-twist contributions to inclusive and semi-inclusive reactions [99, 100].

The gauge-invariant distribution amplitude defined from the integral over the transverse momenta of the valence (smallest ) Fock state provides a fundamental measure of the hadron at the amplitude level [101, 102]; they are the nonperturbative input to the factorized form of hard exclusive amplitudes and exclusive heavy hadron decays in perturbative QCD. The resulting distributions obey the DGLAP and ERBL evolution equations as a function of the maximal invariant mass, thus providing a physical factorization scheme [23]. In each case, the derived quantities satisfy the appropriate operator product expansions, sum rules, and evolution equations. However, at large where the struck quark is far-off shell, DGLAP evolution is quenched [103], so that the fall-off of the DIS cross sections in satisfies inclusive-exclusive duality at fixed

The holographic mapping of hadronic LFWFs to AdS string modes is most transparent when one uses the impact parameter space representation. The total position coordinate of a hadron or its transverse center of momentum , is defined in terms of the energy momentum tensor

 R⊥=1P+∫dx−∫d2x⊥T++x⊥. (52)

In terms of partonic transverse coordinates , where the are the physical transverse position coordinates and frame independent internal coordinates, conjugate to the relative coordinates . Thus, and . The LFWFs can be expanded in terms of the independent transverse coordinates ,

 ψn(xj,k⊥j)=(4π)(n−1)/2exp(in−1∑j=1b⊥j⋅k⊥j)˜ψn(xj,b⊥j). (53)

The normalization is defined by

 ∑nn−1∏j=1∫dxjd2b⊥j∣∣˜ψn(xj,b⊥j