QCD on the Light-Front -- A Systematic Approach to Hadron Physics

Abstract

Light-front Hamiltonian theory, derived from the quantization of the QCD Lagrangian at fixed light-front time x
+ = x
0 + x
3, provides a rigorous frame-independent framework for solving nonperturbative QCD. The eigenvalues of the light-front QCD Hamiltonian H
LF
predict the hadronic mass spectrum, and the corresponding eigensolutions provide the light-front wavefunctions which describe hadron structure, providing a direct connection to the QCD Lagrangian. In the semiclassical approximation the valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a single-variable relativistic equation of motion, analogous to the nonrelativistic radial Schrödinger equation, with an effective confining potential U which systematically incorporates the effects of higher quark and gluon Fock states. Remarkably, the potential U has a unique form of a harmonic oscillator potential if one requires that the chiral QCD action remains conformally invariant. A mass gap and the color confinement scale also arises when one extends the formalism of de Alfaro, Fubini and Furlan to light-front Hamiltonian theory. In the case of mesons, the valence Fock-state wavefunctions of H
LF
for zero quark mass satisfy a single-variable relativistic equation of motion in the invariant variable ζ2=b⊥2x(1−x){\zeta^2=b^2_\perp x(1-x)}, which is conjugate to the invariant mass squared Mqqˉ2{{M^2_{q\bar q}}}. The result is a nonperturbative relativistic light-front quantum mechanical wave equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories M2(n,L,S)=4κ2(n+L+S/2){M^2(n, L, S) = 4\kappa^2( n+L +S/2)} with the same slope in the radial quantum number n and orbital angular momentum L. Only one mass parameter κ{\kappa} appears. The corresponding light-front Dirac equation provides a dynamical and spectroscopic model of nucleons. The same light-front equations arise from the holographic mapping of the soft-wall model modification of AdS5 space with a unique dilaton profile to QCD (3 + 1) at fixed light-front time. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. We also discuss the implications of the underlying conformal template of QCD for renormalization scale-setting and the implications of light-front quantization for the value of the cosmological constant.

Full text

QCD on the Light-Front
A Systematic Approach to Hadron Physics
Stanley J. Brodsky,1Guy F. de T´eramond,2and Hans G¨unter Dosch3
1SLAC National Accelerator Laboratory
Stanford University, Stanford, California 94309, USA
2Universidad de Costa Rica, San Jos´e, Costa Rica
3Institut f¨ur Theoretische Physik, Philosophenweg 16, Heidelberg, Germany
Light-Front Hamiltonian theory, derived from the quantization of the QCD Lagrangian at fixed
light-front time x+=x0+x3, provides a rigorous frame-independent framework for solving non-
perturbative QCD. The eigenvalues of the light-front QCD Hamiltonian HLF predict the hadronic
mass spectrum, and the corresponding eigensolutions provide the light-front wavefunctions which
describe hadron structure, providing a direct connection to the QCD Lagrangian. In the semiclassi-
cal approximation the valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy
a single-variable relativistic equation of motion, analogous to the nonrelativistic radial Schr¨odinger
equation, with an effective confining potential Uwhich systematically incorporates the effects of
higher quark and gluon Fock states. Remarkably, the potential Uhas a unique form of a harmonic
oscillator potential if one requires that the chiral QCD action remains conformally invariant. A mass
gap and the color confinement scale also arises when one extends the formalism of de Alfaro, Fubini
and Furlan to light-front Hamiltonian theory. In the case of mesons, the valence Fock-state wave-
functions of HLF for zero quark mass satisfy a single-variable relativistic equation of motion in the
invariant variable ζ2=b2
⊥x(1−x), which is conjugate to the invariant mass squared M2
q¯q. The result
is a nonperturbative relativistic light-front quantum mechanical wave equation which incorporates
color confinement and other essential spectroscopic and dynamical features of hadron physics, includ-
ing a massless pion for zero quark mass and linear Regge trajectories M2(n, L, S ) = 4κ2(n+L+S/2)
with the same slope in the radial quantum number nand orbital angular momentum L. Only one
mass parameter κappears. The corresponding light-front Dirac equation provides a dynamical
and spectroscopic model of nucleons. The same light-front equations arise from the holographic
mapping of the soft-wall model modification of AdS5space with a unique dilaton profile to QCD
(3+1) at fixed light-front time. Light-front holography thus provides a precise relation between
the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front
wavefunctions describing the internal structure of hadrons in physical space-time. We also discuss
the implications of the underlying conformal template of QCD for renormalization scale-setting and
the implications of light-front quantization for the value of the cosmological constant.
I. INTRODUCTION
The remarkable advantages of using light-front time x+=x0+x3(the “front form”) to quantize a quantum field
theory instead of the standard time x0(the “instant form”) were first demonstrated by Dirac [1]. As Dirac showed,
the front form has the maximum number of kinematic generators of the Lorentz group, including the boost operator.
Thus the description of a hadron at fixed x+is independent of the observer’s Lorentz frame, making it the natural
formalism for addressing dynamical processes in quantum chromodynamics. In contrast, the boost of a wavefunction
of a hadron at fixed ‘instant’ time x0is a difficult nonperturbative problem [2]. An extensive review of light-front
quantization is given in Ref. [3].
The quantization of QCD at fixed light-front (LF) time – light-front quantization – provides a first-principles method
for solving nonperturbative QCD. Given the Lagrangian, one can compute the LF Hamiltonian HLF in terms of the
independent quark and gluon fields. The eigenvalues of HLF determine the mass-squared values of both the discrete
and continuum hadronic spectra. The eigensolutions |ΨHiprojected on the free n-parton Fock state hn|ΨHidetermine
the LF wavefunctions ψn/H (xi,~
k⊥i, λi) where the xi=k+
i
P+=k0
i+k3
i
P0+P3, with Pn
i=1 xi= 1, are the light-front moment
fractions. The eigenstates are defined at fixed x+within the causal horizon, so that causality is maintained without
normal-ordering. In fact, light-front physics is a fully relativistic field theory, but its structure is similar to non-
relativistic atomic physics, and the resulting bound-state equations can be formulated as relativistic Schr¨odinger-like
equations at equal light-front time.
Given the frame-independent light-front wavefunctions (LFWFs) ψn/H , one can compute a large range of hadronic
observables, starting with form factors, structure functions, generalized parton distributions, Wigner distributions,
etc., as illustrated in Fig. 1. For example, the “handbag” contribution [4] to the Eand Hgeneralized parton
distributions for deeply virtual Compton scattering can be computed from the overlap of LFWFs, automatically
satisfy the known sum rules. In the case of deep-inelastic lepton-proton scattering `p →`0Xthe lepton scatters at
arXiv:1310.8648v1 [hep-ph] 31 Oct 2013

2
fixed x+on any quark in any one of the proton’s Fock states. The higher Fock states of a proton such as |uudQ ¯
Qiare
the source of the sea-quark distributions, including contributions “intrinsic” to the proton structure [5]. One can also
analyze and prove factorization and evolution equations for exclusive processes [6–8]. Analogous light-front methods
can be applied to inclusive process where more than one parton from an initial- or final-state hadron are involved [9].
A measurement in the front form is analogous to taking a flash picture. The image in the resulting photograph
records the state of the object as the front of the light wave from the flash illuminates it; in effect, this is a measurement
within the spacelike causal horizon ∆xµ2≤0.Similarly, measurements such as deep inelastic lepton-hadron scattering
`H →`0X, determine the LF wavefunction and structure of the target hadron Hat fixed light-front time. For example,
the BFKL Regge behavior of structure functions can be demonstrated [10] from the behavior of LFWFs at small x.
The final-state interactions of the struck quark on the spectators quarks produce “lensing effects” such as the ‘Sivers’
effect [11], the pseudo-T-odd correlation ~
S·~q ×~pqbetween the proton polarization and the virtual photon-quark
scattering plane. Such lensing effects are leading twist; i.e. they are not power-law suppressed [12]. The Sivers
correlation from initial-state lensing in the Drell-Yan process has the opposite sign [13–15]. The lensing interactions
can be considered as properties of the LFWFs augmented by a Wilson line [16]. The existence of “lensing effects” at
leading twist, such as the T-odd “Sivers effect” in spin-dependent semi-inclusive deep-inelastic scattering, was first
demonstrated using LF methods [12].
The physics of diffractive deep inelastic scattering and other hard processes where the projectile hadron remains
intact, such as γ∗p→X+p0, is also most easily analysed using LF QCD [17]. LF quantization thus provides a
distinction [18] between static (the square of LFWFs) distributions versus non-universal dynamic structure functions,
such as the Sivers single-spin correlation and diffractive deep inelastic scattering which involve final state interactions.
The origin of nuclear shadowing and flavor-dependent anti-shadowing also becomes explicit [19]. QCD properties
such as “color transparency” [20], the “hidden color” of the deuteron LFWF [21], and the existence of intrinsic heavy
quarks in the LFWFs of light hadrons [5, 22] can be derived from the structure of hadronic LFWFs. It is also possible
to compute jet hadronization at the amplitude level from first principles from the LFWFs [23].
In principle, one can solve nonperturbative QCD by diagonalizing the light-front QCD Hamiltonian HLF directly
using the “discretized light-cone quantization” (DLCQ) method [3] which imposes periodic boundary conditions to
discretize the k+and k⊥momenta, or the Hamiltonian transverse lattice formulation introduced in refs. [24–26].
The hadronic spectra and light-front wavefunctions are then obtained from the eigenvalues and eigenfunctions of the
Heisenberg problem HLF |ψi=M2|ψi, an infinite set of coupled integral equations for the light-front components ψn=
hn|ψiin a Fock expansion [3]. The DLCQ method has been applied successfully in lower space-time dimensions [3],
such as QCD(1+1) [27]. For example, one can compute the complete spectrum of meson and baryon states in
QCD(1+ 1) and their LF wavefunctions, for any number of colors, flavors, and quark masses by matrix diagonalization
using DLCQ [28]. It has also been applied successfully to a range of 1+1 string theory problems by Hellerman and
Polchinski [29, 30].Unlike lattice gauge theory, the DLCQ method is relativistic, has no fermion-doubling, is formulated
in Minkowski space, and is independent of the hadron’s momentum P+and P⊥.
Solving the eigenvalue problem using DLCQ is a formidable computational task for a non-abelian quantum field
theory in four-dimensional space-time because of the large number of independent variables. Consequently, alternative
methods and approximations are necessary to better understand the nature of relativistic bound-states in the strong-
coupling regime. One of the most promising methods for solving nonperturbative (3+1) QCD is the “Basis Light-Front
Quantization” (BFLQ) method [31]. In the BLFQ method one constructs a complete orthonormal basis of eigenstates
based on the eigensolutions of the effective light-front Schr¨odinger equation derived from light-front holography, in
the spirit of the nuclear shell model. Matrix diagonalization for BLFQ should converge more rapidly than DLCQ
since the basis states have a mass spectrum close to the observed hadronic spectrum.
As we shall discuss here, light-front quantized field theory in physical 3 + 1 space-time has a holographic dual with
the dynamics of theories in five-dimensional anti-de Sitter (AdS) space [32], giving important insight into the nature of
color confinement in QCD. Light-front holography – the duality between the front form and classical gravity based on
the isometries of AdS5space, provides a new method for determining the eigenstates of the QCD LF Hamiltonian in
the strongly coupled regime. In the case of mesons, for example, the valence Fock-state wavefunctions of HLF for zero
quark mass satisfy a single-variable relativistic equation of motion in the invariant variable ζ2=b2
⊥x(1 −x), which
is conjugate to the invariant mass squared M2
q¯q. The effective confining potential U(ζ2) in this frame-independent
“light-front Schr¨odinger equation” systematically incorporates the effects of higher quark and gluon Fock states [33].
The hadron mass scale – its “mass gap” – is generated in a novel way. Remarkably, the potential U(ζ2) has a unique
form of a harmonic oscillator potential if one requires that the chiral QCD action remains conformally invariant [33].
The result is a nonperturbative relativistic light-front quantum mechanical wave equation which incorporates color
confinement and other essential spectroscopic and dynamical features of hadron physics.

3
PDFs FFs
TMDs
Charges
GTMDs
GPDs
TMSDs
TMFFs
Transverse density in
momentum space
Transverse density in position
space
Longitudinal
Transverse
Momentum space Position space
General remarks about orbital angular mo-
mentum
n(xi,
ki,i)
n
i=1(xi
R+
bi) =
R
xi
R+
bi
n
i
bi=
0
n
ixi= 1
•Light Front Wavefunctions:
Sivers, T-odd
observables
from lensing
FIG. 1: Examples of hadronic observables, including Wigner functions and T-odd observables which are based on overlaps of
light-front wavefunctions. Adopted from a figure by F. Lorce and B. Pasquini. [34]
II. WHAT IS THE ORIGIN OF THE QCD MASS SCALE?
If one sets the masses of the quarks to zero, no mass scale appears explicitly in the QCD Lagrangian. The classical
theory thus displays invariance under both scale (dilatation) and special conformal transformations [35]. Nevertheless,
the quantum theory built upon this conformal template displays color confinement, a mass gap, as well as asymptotic
freedom. A fundamental question is thus how does the mass scale which determines the masses of the light-quark
hadrons, the range of color confinement, and the running of the coupling appears in QCD?
A hint to the origin of the mass scale in nominally conformal theories was given in 1976 in a remarkable paper
by V. de Alfaro, S. Fubini and G. Furlan (dAFF) [36] in the context of one-dimensional quantum mechanics. They
showed that the mass scale which breaks dilatation invariance can appear in the equations of motion without violating
the conformal invariance of the action. In fact, this is only possible if the resulting potential has the form of a confining
harmonic oscillator, and the transformed time variable τthat appears in the confining theory has a limited range.
In this contribution we will review how the application of the dAFF procedure, together with light-front quantum
mechanics and light-front holographic mapping, leads to a new analytic approximation to QCD – a light-front Hamil-
tonian and corresponding one-dimensional light-front Schr¨odinger and Dirac equations which are frame-independent,
relativistic, and reproduce crucial features of the spectroscopy and dynamics of the light-quark hadrons. The predic-
tions of the LF equations of motion include a zero-mass pion in the chiral mq→0 limit, and linear Regge trajectories
M2(n, L)∝n+Lwith the same slope in the radial quantum number n(the number of nodes) and L= max |Lz|, the
internal orbital angular momentum. In fact, we will also show that the effective confinement potential which appears
in the LF equations of motion is unique if we require that the corresponding one-dimensional effective action which
encodes the chiral symmetry of QCD remains conformally invariant.
III. APPLICATIONS OF LIGHT-FRONT HAMILTONIAN THEORY
Computing hadronic matrix elements of currents is particularly simple in the light-front, since space-like current
matrix elements can be written as an overlap of frame-independent light-front wave functions as in the Drell-Yan-

4
West formula [37–39]. If the virtual space-like photon has q+= 0, only processes with the same number of initial and
final partons are allowed. One can also prove fundamental theorems for relativistic quantum field theories using the
front form, including: the cluster decomposition theorem [40], and the vanishing of the anomalous gravitomagnetic
moment for any Fock state of a hadron [41]. The contribution to the hadron’s anomalous gravitomagnetic moment
vanishes for each Fock state - a crucial test of the consistency of the front-form formalism. One also can show that a
nonzero anomalous magnetic moment of a bound state requires nonzero angular momentum of the constituents [39].
In contrast, if one uses ordinary fixed time x0the hadronic states must be boosted from the hadron’s rest frame to
a moving frame – an intractable dynamical problem. In fact, the boost of a composite system at fixed time x0is
only known at weak binding [2, 42]. Moreover, form factors at fixed instant time x0require computing off-diagonal
matrix elements as well as the contributions of currents arising from fluctuations of the vacuum in the initial state
which connect to the hadron wavefunction in the final state. Thus the knowledge of the wave functions of hadronic
eigenstates alone is not sufficient to compute covariant current matrix elements in the usual instant-form framework.
The gauge-invariant meson and baryon distribution amplitudes φH(xi) which control hard exclusive and direct
reactions are the valence LFWFs integrated over transverse momentum at fixed xi=k+/P +. The ERBL evolution of
distribution amplitudes and the factorization theorems for hard exclusive processes are derived using LF theory [7, 8].
Quantization in the light-front provides the rigorous field-theoretical realization of the intuitive ideas of the parton
model [43, 44] which is formulated at fixed tin the infinite-momentum frame (IMF) [45, 46]. The same results are
obtained in the front form for any frame; e.g., the structure functions and other probabilistic parton distributions
measured in deep inelastic scattering are obtained from the squares of the boost invariant LFWFs, the eigensolution
of the light-front Hamiltonian. The familiar kinematic variable xbj of deep inelastic scattering becomes identified with
the LF fraction at small x. The IMF prescription thus allows one to formally use the acausal instant form, but it is
awkward, since it is frame-dependent, requiring an unphysical “infinite” boost of a composite system.
Because of Wick’s theorem, light-front time-ordered perturbation theory is equivalent to covariant Feynman per-
turbation theory. The higher order calculation of the electron anomalous moment at order α3and the “alternating
denominator method” for renormalizing LF perturbation theory is given in Ref. [47]. LF Hamiltonian perturbation
theory provides a simple method for deriving analytic forms for the analog of Parke-Taylor amplitudes [48] where
each particle spin Szis quantized in the LF zdirection. The gluonic g6amplitude T(−1−1→+1+1+1+1+1+1)
requires ∆Lz= 8; it thus must vanish at tree level since each three-gluon vertex has ∆Lz=±1.However, the order g8
one-loop amplitude can be nonzero. Stasto and Cruz-Santiago [49] have shown that the cluster properties [50] of LF
time-ordered perturbation theory, together with Jzconservation, can also be used to elegantly derive the Parke-Taylor
rules for multi-gluon scattering amplitudes.
The counting-rule [51] behavior of structure functions at large xand Bloom-Gilman duality have also been derived
in light-front QCD as well as from holographic QCD [52]. The instantaneous fermion interaction in LF quantization
provides a simple derivation of the J= 0 fixed pole contribution to deeply virtual Compton scattering [53], i.e., the
e2
qs0F(t) contribution to the DVCS amplitude which is independent of photon energy and virtuality. LF quantization
also can provide a method to implement jet hadronization at the amplitude level [23].
Angular momentum Jz=Lz+Szis conserved at every interaction vertex and for every LF Fock State in the Front
Form. The basic physics of the angular momentum composition of hadrons is thus expressed most simply using the
front form. If one chooses light-cone gauge A+= 0, then gauge particles have positive metric and physical polarization
Sz=±1. It is also possible to quantize QCD in Feynman gauge in the light front [54].
In a LF Fock state with nconstituents there are n−1 independent relative orbital angular momentum components
Lz
iso that Pn−1
i=1 Lz
i=Lz. For example, the |e+e−iFock state of positronium has one relative angular momentum of
the electron relative to the positron. This agrees with the usual convention used in atomic physics. In fact, the total
sum Pn
i=1 Lz
ivanishes because of the vanishing of the anomalous gravitomagnetic moment [41].
Information on the spin composition of the proton is obtained from measurements of deep inelastic lepton scattering
on a polarized proton target. As we discuss below, the proton’s LF wavefunctions are predicted in detail from
AdS/QCD in the quark-diquark picture via the LF Dirac equation obtained from LF Holography using a unique
confinement potential. One predicts equal probability for the struck quark to have Sz=±1/2, and thus the proton
spin is carried on the average by quark orbital angular momentum < Lz>=Jz=±1/2, not quark spin. As Burkardt
has emphasized [55], the effects of lensing in deep inelastic lepton scattering also impacts the observed proton spin
composition.
Light-front quantization is thus the natural framework for interpreting measurements like deep inelastic scattering in
terms of the nonperturbative relativistic bound-state structure of hadrons in quantum chromodynamics. The LFWFs
of hadrons provide a direct connection between observables and the QCD Lagrangian. Moreover, the formalism is
rigorous, relativistic and frame-independent.

5
IV. THE LIGHT-FRONT VACUUM
It is conventional to define the vacuum in quantum field theory as the lowest energy eigenstate of the instant-form
Hamiltonian. Such an eigenstate is defined at a single time x0over all space ~x. It is thus acausal and frame-
dependent. The instant-form vacuum thus must be normal-ordered in order to avoid violations of causality when
computing correlators and other matrix elements. In contrast, in the front form, the vacuum state is defined as the
eigenstate of lowest invariant mass M. It is defined at fixed light-front time x+=x0+x3over all x−=x0−x3and
~x⊥, the extent of space that can be observed within the speed of light. It is frame-independent and only requires
information within the causal horizon.
Since all particles have positive k+=k0+kz>0 and + momentum is conserved in the front form, the usual
vacuum bubbles are kinematically forbidden in the front form. In fact the LF vacuum for QED, QCD, and even the
Higgs Standard Model is trivial up to possible zero modes – backgrounds with zero four-momentum. In this sense
it is already normal-ordered. In the case of the Higgs theory, the usual Higgs vacuum expectation value is replaced
by a classical k+= 0 background zero-mode field which is not sensed by the energy momentum tensor [56]. The
phenomenology of the Higgs theory is unchanged.
There are thus no quark or gluon vacuum condensates in the LF vacuum– as first noted by Casher and Susskind [57];
the corresponding physics is contained within the LFWFs themselves [58–62], thus eliminating a ma jor contribution
to the cosmological constant. In the light-front formulation of quantum field theory, phenomena such as the GMOR
relation – usually associated with condensates in the instant form vacuum – are properties of the the hadronic LF
wavefunctions themselves. An exact Bethe-Salpeter analysis [63] shows that the quantity that appears in the Gell-
Mann-Oakes-Renner (GMOR) relation [64] relation is the matrix element h0|¯
ψγ5ψ|πifor the pion to couple locally
to the vacuum via a pseudoscalar operator – not a vacuum expectation value h0|¯
ψψ|0i. In the front-form h0|¯
ψγ5ψ|πi
involves the pion LF Fock state with parallel qand ¯qspin and Lz=±1. This second pion Fock state automatically
appears when the quarks are massive.
The simple structure of the light-front vacuum thus allows an unambiguous definition of the partonic content of
a hadron in QCD. The frame-independent causal front-form vacuum is a good match to the “void” – the observed
universe without luminous matter. Thus it is natural in the front form to obtain zero cosmological constant from
quantum field theory. For a recent review see Ref. [65].
V. THE CONFORMAL SYMMETRY TEMPLATE
In the case of perturbative QCD (pQCD), the running coupling αs(Q2) becomes constant in the limit of zero
β-function and zero quark mass, and conformal symmetry becomes manifest. In fact, the renormalization scale uncer-
tainty in pQCD predictions can be eliminated by using the Principle of Maximum Conformality (PMC) [66]. Using
the PMC/BLM procedure [67], all non-conformal contributions in the perturbative expansion series are summed into
the running coupling by shifting the renormalization scale in αsfrom its initial value, and one obtains unique, scale-
fixed, scheme-independent predictions at any finite order. One can also introduce a generalization of conventional
dimensional regularization, the Rδschemes. For example, when one generalizes the MSBar scheme by subtracting
ln 4π−γE−δinstead of just ln 4π−γEthe new terms generated in the pQCD series proportional to δexpose the β
terms and thus the renormalization scheme dependence. Thus the Rδschemes uncover the renormalization scheme
and scale ambiguities of pQCD predictions, exposes the general pattern of nonconformal terms, and allows one to
systematically determine the argument of the running coupling order by order in pQCD in a form which can be readily
automatized [68, 69]. The resulting PMC scales and finite-order PMC predictions are to high accuracy independent
of the choice of initial renormalization scale. For example, PMC scale-setting leads to a scheme-independent pQCD
prediction [70] for the top-quark forward-backward asymmetry which is within one σof the Tevatron measurements.
The PMC procedure also provides scale-fixed, scheme-independent commensurate scale relations [71] between observ-
ables which are based on the underlying conformal behavior of QCD, such as the generalized Crewther relation [72].
The PMC satisfies all of the principles of the renormalization group: reflectivity, symmetry, and transitivity, and it
thus eliminates an unnecessary source of systematic error in pQCD predictions [73].
VI. LIGHT-FRONT HOLOGRAPHY
An important analysis tool for QCD is Anti-de Sitter space in five dimensions. In particular, AdS5provides a
remarkable geometric representation of the conformal group which underlies the conformal symmetry of classical QCD.
A simple way to obtain confinement and discrete normalizable modes is to truncate AdS space with the introduction
of a sharp cut-off in the infrared region of AdS space, as in the “hard-wall” model [74], where one considers a slice of

6
AdS space, 0 ≤z≤z0, and imposes boundary conditions on the fields at the IR border z0∼1/ΛQCD. As first shown
by Polchinski and Strassler [74], the modified AdS space, provides a derivation of dimensional counting rules [75, 76]
in QCD for the leading power-law fall-off of hard scattering beyond the perturbative regime. The modified theory
generates the point-like hard behavior expected from QCD, instead of the soft behavior characteristic of extended
objects [74]. The physical states in AdS space are represented by normalizable modes ΦP(x, z) = eiP ·xΦ(z), with
plane waves along Minkowski coordinates xµand a profile function Φ(z) along the holographic coordinate z. The
hadronic invariant mass PµPµ=M2is found by solving the eigenvalue problem for the AdS wave equation.
One can also modify AdS space by using a dilaton factor in the AdS action eϕ(z)to introduce the QCD confinement
scale. To a first semiclassical approximation, light-front QCD is formally equivalent to the equations of motion on a
fixed gravitational background asymptotic to AdS5. In fact, the introduction of a dilaton profile is equivalent to a
modification of the AdS metric, even for arbitrary spin [77], but it is left largely unspecified. However, we shall show
that if one imposes the requirement that the action of the corresponding one-dimensional effective theory remains
conformal invariant, then the dilaton profile ϕ(z)∝zsis constrained to have the specific power s= 2, a remarkable
result which follows from the dAFF construction of conformally invariant quantum mechanics [78]. A related argument
is given in Ref. [79]. The quadratic form ϕ(z) = ±κ2z2immediately leads to linear Regge trajectories [80] in the
hadron mass squared.
“Light-Front Holography” refers to the remarkable fact that dynamics in AdS space in five dimensions is dual to a
semiclassical approximation to Hamiltonian theory in physical 3+ 1 space-time quantized at fixed light-front time [32].
The correspondence between AdS and QCD, which was originally motivated by the AdS/CFT correspondence between
gravity on a higher-dimensional space and conformal field theories in physical space-time [81], has its most explicit
and simplest realization as a direct holographic mapping to light-front Hamiltonian theory [32]. For example, the
equation of motion for mesons on the light-front has exactly the same single-variable form as the AdS equation of
motion; one can then interpret the AdS fifth dimension variable zin terms of the physical variable ζ, representing the
invariant separation of the qand ¯qat fixed light-front time. There is a precise connection between the quantities that
enter the fifth dimensional AdS space and the physical variables of LF theory. The AdS mass parameter µR maps to
the LF orbital angular momentum. The formulae for electromagnetic [82] and gravitational [83] form factors in AdS
space map to the exact Drell-Yan-West formulae in light-front QCD [84–86].
The light-front holographic principle provides a precise relation between the bound-state amplitudes in AdS space
and the boost-invariant LF wavefunctions describing the internal structure of hadrons in physical space-time (See Fig.
2). The resulting valence Fock-state wavefunctions satisfy a single-variable relativistic equation of motion analogous
to the eigensolutions of the nonrelativistic radial Schr¨odinger equation. The quadratic dependence in the effective
quark-antiquark potential U(ζ2, J) = κ4ζ2+ 2κ2(J−1) is determined uniquely from conformal invariance. The
constant term 2κ2(J−1) = 2κ2(S+L−1) is fixed by the duality between AdS and LF quantization for spin-Jstates,
a correspondence which follows specifically from the separation of kinematics and dynamics on the light-front [77].
The LF potential thus has a specific power dependence–in effect, it is a light-front harmonic oscillator potential. It is
confining and reproduces the observed linear Regge behavior of the light-quark hadron spectrum in both the orbital
angular momentum Land the radial node number n. The pion is predicted to be massless in the chiral limit [87]
- the positive contributions to m2
πfrom the LF potential and kinetic energy are cancelled by the constant term in
U(ζ2, J ) for J= 0.This holds for the positive sign of the dilaton profile ϕ(z) = κ2z2. The LF dynamics retains
conformal invariance of the action despite the presence of a fundamental mass scale. The constant term in the LF
potential U(ζ2, J ) derived from LF Holography is essential; the masslessness of the pion and the separate dependence
on Jand Lare consequences of the potential derived from the holographic LF duality with AdS for general Jand
L[77, 78]. Thus the light-front holographic approach provides an analytic frame-independent first approximation to
the color-confining dynamics, spectroscopy, and excitation spectra of the relativistic light-quark bound states of QCD.
It is systematically improvable in full QCD using the basis light-front quantization (BLFQ) method [31] and other
methods.
We now give an example of light-front holographic mapping for the specific case of the elastic pion form factor. In
the higher-dimensional gravity theory, the hadronic transition amplitude corresponds to the coupling of an external
electromagnetic field AM(x, z), for a photon propagating in AdS space, with an extended field ΦP(x, z ) describing a
meson in AdS is [82]
Zd4x dz√g AM(x, z)Φ∗
P0(x, z)←→
∂MΦP(x, z)∼(2π)4δ4(P0−P−q)µ(P+P0)µFM(q2),(1)
where the coordinates of AdS5are the Minkowski coordinates xµand zlabeled xM= (xµ, z), with M, N = 1,··· 5, and
gis the determinant of the metric tensor. The expression on the right-hand side of (1) represents the space-like QCD
electromagnetic transition amplitude in physical space-time hP0|Jµ(0)|Pi= (P+P0)µFM(q2).It is the EM matrix
element of the quark current Jµ=Pqeq¯qγµq, and represents a local coupling to pointlike constituents. Although

7
LF(3 + 1) AdS5
φ(z)
z
ψ(x,b )
ζ = x(1 – x) b2
φ(ζ)
ψ(x, ζ, ϕ) = eiLϕ x(1 – x) 2πζ
(1 – x)
x
b
8-2013
8839A1
FIG. 2: Light-Front Holography: Mapping between the hadronic wavefunctions of the Anti-de Sitter approach and eigensolutions
of the light-front Hamiltonian theory derived from the equality of LF and AdS formula for EM and gravitational current matrix
elements and their identical equations of motion.
the expressions for the transition amplitudes look very different, one can show that a precise mapping of the matrix
elements can be carried out at fixed light-front time [84, 85].
The form factor is computed in the light front formalism from the matrix elements of the plus current J+in order
to avoid coupling to Fock states with different numbers of constituents and is given by the Drell-Yan-West expression.
The form factor can be conveniently written in impact space as a sum of overlap of LFWFs of the j= 1,2,··· , n −1
spectator constituents [88]
FM(q2) = X
n
n−1
Y
j=1 Zdxjd2b⊥jexpiq⊥·
n−1
X
j=1
xjb⊥jψn/M (xj,b⊥j)2,(2)
corresponding to a change of transverse momentum xjq⊥for each of the n−1 spectators with Pn
i=1 b⊥i= 0. The
formula is exact if the sum is over all Fock states n.
For simplicity, consider a two-parton bound-state. The q¯qLF Fock state wavefunction for a meson can be written
as
ψ(x, ζ, ϕ) = eiLϕ X(x)φ(ζ)
√2πζ ,(3)
thus factoring the longitudinal, X(x), transverse φ(ζ) and angular dependence ϕ. If both expressions for the form
factor are to be identical for arbitrary values of Q, we obtain φ(ζ)=(ζ/R)3/2Φ(ζ) and X(x) = px(1 −x) [84], where
we identify the transverse impact LF variable ζwith the holographic variable z,z→ζ=px(1 −x)|b⊥|, where x
is the longitudinal momentum fraction and b⊥is the transverse-impact distance between the quark and antiquark.
Extension of the results to arbitrary nfollows from the x-weighted definition of the transverse impact variable of
the n−1 spectator system given in Ref. [84]. Identical results follow from mapping the matrix elements of the
energy-momentum tensor [86].
VII. THE LIGHT-FRONT SCHR ¨
ODINGER EQUATION: A SEMICLASSICAL APPROXIMATION TO
QCD
It is advantageous to reduce the full multiparticle eigenvalue problem of the LF Hamiltonian to an effective light-
front Schr¨odinger equation which acts on the valence sector LF wavefunction and determines each eigensolution
separately [89]. In contrast, diagonalizing the LF Hamiltonian yields all eigensolutions simultaneously, a complex
task. The central problem then becomes the derivation of the effective interaction Uwhich acts only on the valence
sector of the theory and has, by definition, the same eigenvalue spectrum as the initial Hamiltonian problem. In order

8
to carry out this program one must systematically express the higher Fock components as functionals of the lower
ones. This method has the advantage that the Fock space is not truncated, and the symmetries of the Lagrangian
are preserved [89].
A hadron has four-momentum P= (P−, P +,P⊥), P±=P0±P3and invariant mass P2=M2. The generators
P= (P−, P +,~
P⊥) are constructed canonically from the QCD Lagrangian by quantizing the system on the light-front
at fixed LF time x+,x±=x0±x3[3]. The LF Hamiltonian P−generates the LF time evolution with respect to the
LF time x+, whereas the LF longitudinal P+and transverse momentum ~
P⊥are kinematical generators.
In the limit of zero quark masses the longitudinal modes decouple from the invariant LF Hamiltonian equation
HLF |φi=M2|φi, with HLF =PµPµ=P−P+−P2
⊥. The result is a relativistic and frame-independent light-front
wave equation for φ[32] (See Fig. 3)
−d2
dζ2−1−4L2
4ζ2+Uζ2, J φn,J,L (ζ2) = M2φn,J,L(ζ2).(4)
This equation describes the spectrum of mesons as a function of n, the number of nodes in ζ, the total angular
momentum J, which represent the maximum value of |Jz|,J= max |Jz|, and the internal orbital angular momentum
of the constituents L= max |Lz|. The variable zof AdS space is identified with the LF boost-invariant transverse-
impact variable ζ[84], thus giving the holographic variable a precise definition in LF QCD [32, 84]. For a two-parton
bound state ζ2=x(1 −x)b2
⊥.
Coupled Fock States
Eliminate Higher Fock States
(includes retarded interactions)
Effective Two-particle Equation
Azimuthal Basis ζ,ϕ
Confining AdS/QCD Potentials!
(1 – x)
x
b
8-2013
8839A2
HLF
QCD
HLF
0+HLF
Iψ > = M2 ψ >
U(ζ) = κ4ζ2 + 2κ2(L + S – 1)
U(ζ,S,L) ψLF(ζ) = M2 ψLF(ζ)
x(1 – x)
k2 + m2
ψLF(x, k ) = M2 ψLF (
+VLF
eff x,k )
d2
AdS/QCD:
ζ2 = x(1 – x) b2
dζ24ζ2
–++
–1 + 4L2
(m = o)
FIG. 3: Reduction of the QCD light-front Hamiltonian to an effective q¯qbound state equation. The potential is determined
from spin-Jrepresentations on AdS5space. The harmonic oscillator form of U(ζ2) is determined by the requirement that the
action remain conformally invariant.
In the exact QCD theory the potential in the the Light-Front Schr¨odinger equation (4) is determined from the
two-particle irreducible (2PI) q¯q→q¯qGreens’ function. In particular, the reduction from higher Fock states in the
intermediate states leads to an effective interaction Uζ2, J for the valence |q¯qiFock state [89]. A related approach
for determining the valence light-front wavefunction and studying the effects of higher Fock states without truncation
has been given in Ref. [90].
Unlike ordinary instant-time quantization, the light-front Hamiltonian equations of motion are frame independent;
remarkably, they have a structure which matches exactly the eigenmode equations in AdS space. This makes a direct
connection of QCD with AdS methods possible. In fact, one can derive the light-front holographic duality of AdS
by starting from the light-front Hamiltonian equations of motion for a relativistic bound-state system in physical
space-time [32].

9
VIII. EFFECTIVE CONFINEMENT FROM THE GAUGE/GRAVITY CORRESPONDENCE
Recently we have derived wave equations for hadrons with arbitrary spin Jstarting from an effective action in AdS
space [77]. An essential element is the mapping of the higher-dimensional equations to the LF Hamiltonian equation
found in Ref. [32]. This procedure allows a clear distinction between the kinematical and dynamical aspects of the
LF holographic approach to hadron physics. Accordingly, the non-trivial geometry of pure AdS space encodes the
kinematics, and the additional deformations of AdS encode the dynamics, including confinement [77].
A spin-Jfield in AdSd+1 is represented by a rank Jtensor field ΦM1···MJ, which is totally symmetric in all its
indices. In presence of a dilaton background field ϕ(z) the effective action is [77]
Seff =Zddx dz p|g|eϕ(z)gN1N0
1·· ·gNJN0
JgMM 0DMΦ∗
N1...NJDM0ΦN0
1...N0
J−µ2
eff (z) Φ∗
N1...NJΦN0
1...N0
J,(5)
where DMis the covariant derivative which includes parallel transport. The effective mass µeff (z), which encodes
kinematical aspects of the problem, is an a priori unknown function, but the additional symmetry breaking due to its
z-dependence allows a clear separation of kinematical and dynamical effects [77]. The dilaton background field ϕ(z)
in (5) introduces an energy scale in the five-dimensional AdS action, thus breaking conformal invariance. It vanishes
in the conformal ultraviolet limit z→0.
A physical hadron has plane-wave solutions and polarization indices along the 3 + 1 physical coordinates
ΦP(x, z)ν1···νJ=eiP ·xΦJ(z)ν1···νJ(P), with four-momentum Pµand invariant hadronic mass PµPµ=M2. All other
components vanish identically. The wave equations for hadronic modes follow from the Euler-Lagrange equation for
tensors orthogonal to the holographic coordinate z, ΦzN2···NJ= 0. Terms in the action which are linear in tensor
fields, with one or more indices along the holographic direction, ΦzN2···NJ, give us the kinematical constraints required
to eliminate the lower-spin states [77]. Upon variation with respect to ˆ
Φ∗
ν1...νJ, we find the equation of motion [77]
−zd−1−2J
eϕ(z)∂zeϕ(z)
zd−1−2J∂z+(m R)2
z2ΦJ=M2ΦJ,(6)
with (m R)2= (µeff (z)R)2−Jz ϕ0(z) + J(d−J+ 1), which is the result found in Refs. [32, 91] by rescaling the wave
equation for a scalar field. Similar results were found in Ref. [92]. Upon variation with respect to ˆ
Φ∗
N1···z···NJwe find
the kinematical constraints which eliminate lower spin states from the symmetric field tensor [77]
ηµν Pµνν2···νJ(P)=0, ηµν µνν3···νJ(P) = 0.(7)
Upon the substitution of the holographic variable zby the LF invariant variable ζand replacing ΦJ(z) =
(R/z)J−(d−1)/2e−ϕ(z)/2φJ(z) in (6), we find for d= 4 the LF wave equation (4) with effective potential [93]
U(ζ2, J ) = 1
2ϕ00(ζ2) + 1
4ϕ0(ζ2)2+2J−3
2ζϕ0(ζ2),(8)
provided that the AdS mass min (6) is related to the internal orbital angular momentum L=max|Lz|and the total
angular momentum Jz=Lz+Szaccording to (m R)2=−(2 −J)2+L2. The critical value L= 0 corresponds to
the lowest possible stable solution, the ground state of the LF Hamiltonian. For J= 0 the five dimensional mass m
is related to the orbital momentum of the hadronic bound state by (m R)2=−4 + L2and thus (m R)2≥ −4. The
quantum mechanical stability condition L2≥0 is thus equivalent to the Breitenlohner-Freedman stability bound in
AdS [94].
The effective interaction U(ζ2, J) is instantaneous in LF time and acts on the lowest state of the LF Hamiltonian.
This equation describes the spectrum of mesons as a function of n, the number of nodes in ζ2, the internal orbital
angular momentum L=Lz, and the total angular momentum J=Jz, with Jz=Lz+Szthe sum of the orbital angular
momentum of the constituents and their internal spins. The SO(2) Casimir L2corresponds to the group of rotations
in the transverse LF plane. The LF wave equation is the relativistic frame-independent front-form analog of the
non-relativistic radial Schr¨odinger equation for muonium and other hydrogenic atoms in presence of an instantaneous
Coulomb potential. The LF harmonic oscillator potential could in fact emerge from the exact QCD formulation when
one includes contributions from the effective potential Uwhich are due to the exchange of two connected gluons; i.e.,
“H” diagrams [95]. We notice that Ubecomes complex for an excited state since a denominator can vanish; this
gives a complex eigenvalue and the decay width. The multi gluon exchange diagrams also could be connected to the
Isgur-Paton flux-tube model of confinement; the collision of flux tubes could give rise to the ridge phenomena recently
observed in high energy pp collisions at RHIC [96].

10
0
2
4
0 2
L
M2
(GeV2)
2-2012
8820A20
π(140)
π(1300)
π(1800)
b1(1235)
n=2 n=1 n=0
π2(1670)
π2(1880)
0
2
4
0 2
M2
(GeV2)
2-2012
8820A24 L
ρ(770)
ρ(1450)
ρ(1700)
a2(1320)
a4(2040)
ρ3(1690)
n=2 n=1 n=0
FIG. 4: I= 1 parent and daughter Regge trajectories for the π-meson family (left) with κ= 0.59 GeV; and the ρ-meson family
(right) with κ= 0.54 GeV.
0.2
0.4
0.6
02 4 60
Q2 (GeV2)
Q2
Fπ (Q2) (GeV2)
2-2012
8820A21
FIG. 5: Light-front holographic prediction for the space-like pion form factor.
The correspondence between the LF and AdS equations thus determines the effective confining interaction Uin
terms of the infrared behavior of AdS space and gives the holographic variable za kinematical interpretation. The
identification of the orbital angular momentum is also a key element of our description of the internal structure of
hadrons using holographic principles.
The dilaton profile exp ±κ2z2leads to linear Regge trajectories [80]. For the confining solution ϕ= exp κ2z2
the effective potential is U(ζ2, J) = κ4ζ2+ 2κ2(J−1) leads to eigenvalues M2
n,J,L = 4κ2n+J+L
2, with a string
Regge form M2∼n+L. A detailed discussion of the light meson and baryon spectrum, as well as the elastic and
transition form factors of the light hadrons using LF holographic methods, is given in Ref. [91]. As an example, the
spectral predictions for the J=L+Slight pseudoscalar and vector meson states are compared with experimental
data in Fig. 4 for the positive sign dilaton model.
The predictions of the resulting LF Schr¨odinger and Dirac equations for hadron light-quark spectroscopy and form
factors for mq= 0 and κ'0.5 GeV are shown in Figs. 4-7 for a dilaton profile ϕ(z) = κ2z2. A detailed discussion of
the computations is given in Ref. [91].
IX. UNIQUENESS OF THE CONFINING POTENTIAL
If one starts with a dilaton profile eϕ(z)with ϕ∝zs,the existence of a massless pion in the limit of massless
quarks determines uniquely the value s= 2. To show this, one can use the stationarity of bound-state energies with

11
0 2 4
6
4
2
0
L
M2 (GeV2)
2-2012
8820A12
n=3 n=2 n=1 n=0
N(2220)
N(1720)
N(1710)
N(1440)
N(940)
N(1900)
N(1680)
0 2 4
0
2
4
6
L
M2 (GeV2)
2-2012
8820A3
Δ(2420)
n=0n=1n=2n=3
Δ(1950)
Δ(1920)
Δ(1600)
Δ(1232)
Δ(1910)
Δ(1905)
FIG. 6: Light front holographic predictions of the light-front Dirac equation for the nucleon spectrum. Orbital and radial
excitations for the positive-parity sector are shown for the N(left) and ∆ (right) for κ= 0.49 GeV and κ= 0.51 GeV
respectively. All confirmed positive and negative-parity resonances from PDG 2012 are well accounted using the procedure
described in [91].
0
0.4
0.8
1.2
10 20 300
Q2 (GeV2)
Q4
Fp
1 (Q2) (GeV4)
2-2012
8820A18
0
-0.2
10 20 300
Q2 (GeV2)
Q4
Fn
1 (Q2) (GeV4)
2-2012
8820A17
0
1
2
0246
Q2 (GeV2)
Fnp
2 (Q2)
2-2012
8820A8
-2
-1
0
0246
Q2 (GeV2)
Fn
2 (Q2)
2-2012
8820A7
FIG. 7: Light-front holographic predictions for the nucleon form factors normalized to their static values.
respect to variation of parameters. More generally, the effective theory should incorporate the fundamental conformal
symmetry of the four-dimensional classical QCD Lagrangian in the limit of massless quarks. To this end we study the
invariance properties of a one-dimensional field theory under the full conformal group following the dAFF construction
of Hamiltonian operators described in Ref. [36].

12
One starts with the one-dimensional action
S=1
2Zdt ˙
Q2−g
Q2,(9)
which is invariant under conformal transformations in the variable t. In addition to the Hamiltonian Ht(Q, ˙
Q) =
1
2˙
Q2+g
Q2there are two more invariants of motion for this field theory, namely the dilation operator Dand K,
corresponding to the special conformal transformations in t. Specifically, if one introduces the the new variable τ
defined through dτ =dt/(u+v t +w t2) and the rescaled fields q(τ) = Q(t)/(u+v t +w t2)1/2, it then follows that
the the operator G=u Ht+v D +w K generates the quantum mechanical evolution in τ[36]. The Hamiltonian
corresponding to the operator G, which introduces the mass scale, is a linear combination of the old Hamiltonian Ht,
D, the generator of dilations, and K, the generator of special conformal transformations.
One can show explicitly [36, 78] that a confinement length scale appears in the action when one expresses the
action (9) in terms of the new time variable τand the new fields q(t), without affecting its conformal invariance.
Furthermore, for g≥ −1/4 and 4 uw −v2>0 the corresponding Hamiltonian Hτ(q, ˙q) = 1
2˙q2+g
q2+4uw−v2
4q2is a
compact operator. Finally, we can transform back to the original field operator Q(t) in (9). We find
Hτ(Q, ˙
Q) = 1
2u˙
Q2+g
Q2−1
4vQ˙
Q+˙
QQ+1
2wQ2(10)
=uHt+vD +wK,
at t= 0. We thus recover the evolution operator G=uHt+vD +wK which describes the evolution in the variable
τ, but expressed in terms of the original field Q.
The Shr¨odinger picture follows by identifying Q→xand ˙
Q→ −id
dx . Then the evolution operator in the new time
variable τis
Hτ=1
2u−d2
dx2+g
x2+i
4vxd
dx +d
dxx+1
2wx2(11)
If we now compare the Hamiltonian (11) with the light-front wave equation (4) and identifying the variable x
with the light-front invariant variable ζ, we have to choose u= 2, v = 0 and relate the dimensionless constant g
to the LF orbital angular momentum, g=L2−1/4, in order to reproduce the light-front kinematics. Furthermore
w= 2λ2fixes the confining light-front potential to a quadratic λ2ζ2dependence. The mass scale brought in via
w= 2κ2then generates the confining mass scale κ. The dilaton is also unique: eφ(z)=eκ2z2,where z2is matched to
ζ2=b2
⊥x(1 −x) via LF holography. The spin-Jrepresentations in AdS5[77] then leads uniquely to the LF confining
potential U(ζ2) = κ4ζ2−2κ2(J−1) .
The new time variable τis related to the variable tfor the case uw > 0, v = 0 by
τ=1
√u w arctan rw
ut,(12)
i.e.,τhas only a limited range. The finite range of invariant LF time τ=x+/P +can be interpreted as a feature of the
internal frame-independent LF time difference between the confined constituents in a bound state. For example, in the
collision of two mesons, it would allow one to compute the LF time difference between the two possible quark-quark
collisions [78].
X. SUMMARY
The triple complementary connection of (a) AdS space, (b) its LF holographic dual, and (c) the relation to the
algebra of the conformal group in one dimension, is characterized by a quadratic confinement LF potential, and thus a
dilaton profile with the power zs, with the unique power s= 2. In fact, for s= 2 the mass of the J=L=n= 0 pion
is automatically zero in the chiral limit. The separate dependence on Jand Lleads to a mass ratio of the ρand the
a1mesons which coincides with the result of the Weinberg sum rules [97]. One predicts linear Regge trajectories with
the same slope in the relative orbital angular momentum Land the LF radial quantum humber n. The AdS approach,
however, goes beyond the purely group theoretical considerations of dAFF, since features such as the masslessness of
the pion and the separate dependence on Jand Lare a consequence of the potential (8) derived from the duality with
AdS for general high-spin representations. The pion distribution amplitude predicted in the nonperturbative domain
is φ(x) = 4
√3πfπpx(1 −x), and the pion decay constant is fπ=√3
8κ.

13
0
0.2
0.4
0.6
0.8
1
10 -1 1 10
Q (GeV)
!s(Q)/"
!g1/" (pQCD)
!g1/" world data
!#/" OPAL
AdS/QCD
LF Holography
AdS/QCD with
!g1 extrapolation
Lattice QCD
!g1/" Hall A/CLAS
!g1/" JLab CLAS
!F3/"
GDH limit
FIG. 8: Light-front holographic results for the QCD running coupling from Ref. [98] normalized to αs(0)/π = 1.The result is
analytic, defined at all scales and exhibits an infrared fixed point.
The QCD mass scale κin units of GeV has to be determined by one measurement; e.g., the pion decay constant
fπ.All other masses and size parameters are then predicted. The running of the QCD coupling is predicted in the
infrared region for Q2<4κ2to have the form αs(Q2)∝exp −Q2
4κ2. As shown in Fig. 8, the result agrees with
the shape of the effective charge defined from the Bjorken sum rule [98], displaying an infrared fixed point. In the
nonperturbative domain soft gluons are in effect sublimated into the effective confining potential. Above this region,
hard-gluon exchange becomes important, leading to asymptotic freedom. The scheme-dependent scale ΛQCD that
appears in the QCD running coupling in any given renormalization scheme such as ΛM S could be determined in terms
of κ.
In our previous papers we have applied LF holography to baryon spectroscopy, space-like and time-like form
factors, as well as transition amplitudes such as γ∗γ→π0,γ∗N→N∗, all based on essentially the single mass
scale parameter κ. Many other applications have been presented in the literature, including recent results by Forshaw
and Sandapen [99] for diffractive ρelectroproduction which are based on the light-front holographic prediction for the
longitudinal ρLFWF. Other recent applications include predictions for generalized parton distributions (GPDs) [100],
and a model for nucleon and flavor form factors [101].
The treatment of the chiral limit in the LF holographic approach to strongly coupled QCD is substantially different
from the standard approach based on chiral perturbation theory. In the conventional approach, spontaneous symmetry
breaking by a non-vanishing chiral quark condensate h¯
ψψiplays the crucial role. In QCD sum rules [102] h¯
ψψibrings
in non-perturbative elements into the perturbatively calculated spectral sum rules. It should be noted, however, that
the definition of the condensate, even in lattice QCD necessitates a renormalization procedure for the operator product,
and it is not a directly observable quantity. In contrast, in Bethe-Salpeter [103] and light-front analyses [65], the Gell
Mann-Oakes-Renner relation [64] for m2
π/mqinvolves the decay matrix element h0|¯
ψγ5ψ|πiinstead of h0|¯
ψψ|0i.
In the color-confining light-front holographic model discussed here, the vanishing of the pion mass in the chiral limit,
a phenomenon usually ascribed to spontaneous symmetry breaking of the chiral symmetry, is obtained specifically
from the precise cancellation of the LF kinetic energy and LF potential energy terms for the quadratic confinement
potential. This mechanism provides a viable alternative to the conventional description of nonperturbative QCD
based on vacuum condensates, and it eliminates a major conflict of hadron physics with the empirical value for the
cosmological constant [58, 59].

14
Acknowledgments
Invited talk, presented by SJB at LightCone 2013+ May 20- May 24, 2013, Skiathos, Greece This work was
supported by the Department of Energy contract DE–AC02–76SF00515. SLAC-PUB-15818
[1] P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949).
[2] S. J. Brodsky and J. R. Primack, Annals Phys. 52, 315 (1969).
[3] S. J. Brodsky, H. -C. Pauli and S. S. Pinsky, Phys. Rept. 301, 299 (1998) [hep-ph/9705477].
[4] S. J. Brodsky, M. Diehl and D. S. Hwang, Nucl. Phys. B 596, 99 (2001) [hep-ph/0009254].
[5] S. J. Brodsky, P. Hoyer, C. Peterson and N. Sakai, Phys. Lett. B 93, 451 (1980).
[6] G. P. Lepage and S. J. Brodsky, Phys. Lett. B 87, 359 (1979).
[7] G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
[8] A. V. Efremov and A. V. Radyushkin, Phys. Lett. B 94, 245 (1980).
[9] E. L. Berger and S. J. Brodsky, Phys. Rev. Lett. 42, 940 (1979).
[10] A. H. Mueller, Nucl. Phys. B 415, 373 (1994).
[11] D. W. Sivers, Phys. Rev. D 41, 83 (1990).
[12] S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B 530, 99 (2002) [hep-ph/0201296].
[13] J. C. Collins, Phys. Lett. B 536, 43 (2002) [hep-ph/0204004].
[14] S. J. Brodsky, D. S. Hwang and I. Schmidt, Nucl. Phys. B 642, 344 (2002) [hep-ph/0206259].
[15] S. J. Brodsky, D. S. Hwang, Y. V. Kovchegov, I. Schmidt and M. D. Sievert, Phys. Rev. D 88, 014032 (2013)
[arXiv:1304.5237 [hep-ph]].
[16] S. J. Brodsky, B. Pasquini, B. -W. Xiao and F. Yuan, Phys. Lett. B 687, 327 (2010) [arXiv:1001.1163 [hep-ph]].
[17] S. J. Brodsky, P. Hoyer, N. Marchal, S. Peigne and F. Sannino, Phys. Rev. D 65, 114025 (2002) [hep-ph/0104291].
[18] S. J. Brodsky, Nucl. Phys. A 827, 327C (2009) [arXiv:0901.0781 [hep-ph]].
[19] S. J. Brodsky, I. Schmidt and J. -J. Yang, Phys. Rev. D 70, 116003 (2004) [hep-ph/0409279].
[20] S. J. Brodsky and A. H. Mueller, Phys. Lett. B 206, 685 (1988).
[21] S. J. Brodsky, C. -R. Ji and G. P. Lepage, Phys. Rev. Lett. 51, 83 (1983).
[22] M. Franz, M. V. Polyakov and K. Goeke, Phys. Rev. D 62, 074024 (2000) [hep-ph/0002240].
[23] S. J. Brodsky, G. F. de Teramond and R. Shrock, AIP Conf. Proc. 1056, 3 (2008) [arXiv:0807.2484 [hep-ph]].
[24] W. A. Bardeen and R. B. Pearson, Phys. Rev. D 14, 547 (1976).
[25] M. Burkardt and S. K. Seal, Phys. Rev. D 65, 034501 (2002) [hep-ph/0102245].
[26] J. Bratt, S. Dalley, B. van de Sande and E. M. Watson, Phys. Rev. D 70, 114502 (2004) [hep-ph/0410188].
[27] H. C. Pauli and S. J. Brodsky, Phys. Rev. D 32, 1993 (1985).
[28] K. Hornbostel, S. J. Brodsky and H. C. Pauli, Phys. Rev. D 41, 3814 (1990).
[29] S. Hellerman and J. Polchinski, In *Shifman, M.A. (ed.): The many faces of the superworld* 142-155 [hep-th/9908202].
[30] J. Polchinski, Prog. Theor. Phys. Suppl. 134, 158 (1999) [hep-th/9903165].
[31] J. P. Vary, H. Honkanen, J. Li, P. Maris, S. J. Brodsky, A. Harindranath, G. F. de Teramond and P. Sternberg et al.,
Phys. Rev. C 81, 035205 (2010) [arXiv:0905.1411 [nucl-th]].
[32] G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. 102, 081601 (2009) [arXiv:0809.4899 [hep-ph]].
[33] S. J. Brodsky, G. F. de Teramond and H. G. Dosch, arXiv:1302.4105 [hep-th].
[34] C. Lorce and B. Pasquini, Int. J. Mod. Phys. Conf. Ser. 20, 84 (2012) [arXiv:1208.3065 [hep-ph]].
[35] G. Parisi, Phys. Lett. B 39, 643 (1972).
[36] V. de Alfaro, S. Fubini and G. Furlan, Nuovo Cim. A 34, 569 (1976).
[37] S. D. Drell and T. -M. Yan, Phys. Rev. Lett. 24, 181 (1970).
[38] G. B. West, Phys. Rev. Lett. 24, 1206 (1970).
[39] S. J. Brodsky and S. D. Drell, Phys. Rev. D 22, 2236 (1980).
[40] S. J. Brodsky and C. -R. Ji, Phys. Rev. D 33, 2653 (1986).
[41] S. J. Brodsky, D. S. Hwang, B. -Q. Ma and I. Schmidt, Nucl. Phys. B 593, 311 (2001) [hep-th/0003082].
[42] S. J. Brodsky and J. R. Primack, Phys. Rev. 174, 2071 (1968).
[43] R. P. Feynman, Phys. Rev. Lett. 23, 1415 (1969).
[44] R. P. Feynman, Reading 1972, 282p
[45] S. Fubini and G. Furlan, Physics 1, 229 (1965).
[46] S. Weinberg, Phys. Rev. 150, 1313 (1966).
[47] S. J. Brodsky, R. Roskies and R. Suaya, Phys. Rev. D 8, 4574 (1973).
[48] L. Motyka and A. M. Stasto, Phys. Rev. D 79, 085016 (2009) [arXiv:0901.4949 [hep-ph]].
[49] C. A. Cruz-Santiago and A. M. Stasto, Nucl. Phys. B 875, 368 (2013) [arXiv:1308.1062 [hep-ph]].
[50] F. Antonuccio, S. J. Brodsky and S. Dalley, Phys. Lett. B 412, 104 (1997) [hep-ph/9705413].
[51] S. J. Brodsky, M. Burkardt and I. Schmidt, Nucl. Phys. B 441, 197 (1995) [hep-ph/9401328].
[52] T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, arXiv:1306.0366 [hep-ph].
[53] S. J. Brodsky, F. J. Llanes-Estrada, J. T. Londergan and A. P. Szczepaniak, arXiv:0906.5515 [hep-ph].

15
[54] P. P. Srivastava and S. J. Brodsky, Phys. Rev. D 61, 025013 (2000) [hep-ph/9906423].
[55] M. Burkardt, Acta Phys. Polon. Supp. 6, 125 (2013).
[56] P. P. Srivastava and S. J. Brodsky, Phys. Rev. D 66, 045019 (2002) [hep-ph/0202141].
[57] A. Casher and L. Susskind, Phys. Rev. D 9, 436 (1974).
[58] S. J. Brodsky and R. Shrock, Proc. Nat. Acad. Sci. 108, 45 (2011) [arXiv:0905.1151 [hep-th]].
[59] S. J. Brodsky, C. D. Roberts, R. Shrock and P. C. Tandy, Phys. Rev. C 82, 022201 (2010) [arXiv:1005.4610 [nucl-th]].
[60] L. Chang, I. C. Cloet, J. J. Cobos-Martinez, C. D. Roberts, S. M. Schmidt and P. C. Tandy, Phys. Rev. Lett. 110, 132001
(2013) [arXiv:1301.0324 [nucl-th]].
[61] L. Chang, C. D. Roberts and S. M. Schmidt, arXiv:1308.4708 [nucl-th].
[62] S. D. Glazek, Acta Phys. Polon. B 42, 1933 (2011) [arXiv:1106.6100 [hep-th]].
[63] P. Maris and C. D. Roberts, Phys. Rev. C 56, 3369 (1997) [nucl-th/9708029].
[64] M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. 175, 2195 (1968).
[65] S. J. Brodsky, C. D. Roberts, R. Shrock and P. C. Tandy, Phys. Rev. C 85, 065202 (2012) [arXiv:1202.2376 [nucl-th]].
[66] S. J. Brodsky and L. Di Giustino, Phys. Rev. D 86, 085026 (2012) [arXiv:1107.0338 [hep-ph]].
[67] S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 28, 228 (1983).
[68] M. Mojaza, S. J. Brodsky and X. -G. Wu, Phys. Rev. Lett. 110, 192001 (2013) [arXiv:1212.0049 [hep-ph]].
[69] S. J. Brodsky, M. Mojaza and X. -G. Wu, arXiv:1304.4631 [hep-ph].
[70] S. J. Brodsky and X. -G. Wu, Phys. Rev. D 85, 114040 (2012) [arXiv:1205.1232 [hep-ph]].
[71] S. J. Brodsky and H. J. Lu, Phys. Rev. D 51, 3652 (1995) [hep-ph/9405218].
[72] S. J. Brodsky, G. T. Gabadadze, A. L. Kataev and H. J. Lu, Phys. Lett. B 372, 133 (1996) [hep-ph/9512367].
[73] X. -G. Wu, S. J. Brodsky and M. Mojaza, Prog. Part. Nucl. Phys. 72, 44 (2013) [arXiv:1302.0599 [hep-ph]].
[74] J. Polchinski and M. J. Strassler, Phys. Rev. Lett. 88, 031601 (2002) [hep-th/0109174].
[75] S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973).
[76] V. A. Matveev, R. M. Muradian and A. N. Tavkhelidze, Lett. Nuovo Cim. 7, 719 (1973).
[77] G. F. de Teramond, H. G. Dosch and S. J. Brodsky, Phys. Rev. D 87, 075005 (2013) [arXiv:1301.1651 [hep-ph]].
[78] S. J. Brodsky, G. F. de Teramond and H. G. Dosch, arXiv:1302.4105 [hep-th].
[79] S. D. Glazek and A. P. Trawi´nski, arXiv:1307.2059 [hep-ph].
[80] A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D 74, 015005 (2006) [hep-ph/0602229].
[81] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [hep-th/9711200].
[82] J. Polchinski and M. J. Strassler, JHEP 0305, 012 (2003) [hep-th/0209211].
[83] Z. Abidin and C. E. Carlson, Phys. Rev. D 77, 095007 (2008) [arXiv:0801.3839 [hep-ph]].
[84] S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett. 96, 201601 (2006) [hep-ph/0602252].
[85] S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 77, 056007 (2008) [arXiv:0707.3859 [hep-ph]].
[86] S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 78, 025032 (2008) [arXiv:0804.0452 [hep-ph]].
[87] G. F. de Teramond and S. J. Brodsky, Nucl. Phys. Proc. Suppl. 199, 89 (2010) [arXiv:0909.3900 [hep-ph]].
[88] D. E. Soper, Phys. Rev. D 15, 1141 (1977).
[89] H. C. Pauli, Eur. Phys. J. C 7, 289 (1999) [hep-th/9809005].
[90] S. S. Chabysheva and J. R. Hiller, Phys. Lett. B 711, 417 (2012) [arXiv:1103.0037 [hep-ph]].
[91] G. F. de Teramond and S. J. Brodsky, arXiv:1203.4025 [hep-ph].
[92] T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 85, 076003 (2012) [arXiv:1108.0346 [hep-ph]].
[93] G. F. de Teramond and S. J. Brodsky, AIP Conf. Proc. 1296, 128 (2010) [arXiv:1006.2431 [hep-ph]].
[94] P. Breitenlohner and D. Z. Freedman, Annals Phys. 144, 249 (1982).
[95] T. Appelquist, M. Dine and I. J. Muzinich, Phys. Lett. B 69, 231 (1977).
[96] J. D. Bjorken, S. J. Brodsky and A. Scharff Goldhaber, Phys. Lett. B 726, 344 (2013) [arXiv:1308.1435 [hep-ph]].
[97] S. Weinberg, Phys. Rev. Lett. 18, 507 (1967).
[98] S. J. Brodsky, G. F. de Teramond and A. Deur, Phys. Rev. D 81, 096010 (2010) [arXiv:1002.3948 [hep-ph]].
[99] J. R. Forshaw and R. Sandapen, Phys. Rev. Lett. 109, 081601 (2012) [arXiv:1203.6088 [hep-ph]].
[100] A. Vega, I. Schmidt, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 83, 036001 (2011) [arXiv:1010.2815 [hep-ph]].
[101] D. Chakrabarti and C. Mondal, arXiv:1307.7995 [hep-ph].
[102] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979).
[103] P. Maris, C. D. Roberts and P. C. Tandy, Phys. Lett. B 420, 267 (1998) [nucl-th/9707003].