^{1}

^{3}.

The QCD light-front Hamiltonian equation

A profound question in hadron physics is how the proton mass and other hadronic mass scales can be determined by QCD since there is no explicit parameter with mass dimensions in the QCD Lagrangian for vanishing quark mass. This dilemma is compounded by the fact that the chiral QCD Lagrangian has no knowledge of the conventions used for units of mass such as

A remarkable principle, first demonstrated by de Alfaro, Fubini, and Furlan (dAFF) [

Brodsky et al. [

In this contribution, I will review a number of recent advances in holographic QCD, extending earlier reviews given in [

Light-Front quantization is the natural formalism for relativistic quantum field theory. Measurements of hadron structure, such as deep inelastic lepton-proton scattering, are made at fixed light-front time

The eigenfunctions of the light-front Hamiltonian

The LFWFs are Poincaré invariant: they are independent of

Light-front wavefunctions thus provide a direct link between the QCD Lagrangian and hadron structure. Since they are defined at a fixed

The meson LFWF connects the intermediate

One of the most elegant features of quantum field theory is supersymmetry, where fermionic and bosonic eigensolutions have the same mass. The conformal group has an elegant

As shown by Guy de Téramond, Günter Dosch and myself, the bound-state equations of superconformal algebra are, in fact, Lorentz invariant, frame-independent, relativistic light-front Schrödinger equations, and the resulting eigensolutions are the eigenstates of a light-front Hamiltonian obtained from

Superconformal algebra leads to effective QCD light-front bound-state equations for both mesons and baryons [

The LF Schrödinger equations for baryons and mesons for zero quark mass derived from the Pauli

The comparison between the meson and baryon masses of the

Comparison of the

As illustrated in Figure

The eigenstates of superconformal algebra have a

Note that the same slope controls the Regge trajectories of both mesons and baryons in both the orbital angular momentum

The LF Schrödinger equations for baryons and mesons derived from superconformal algebra are shown in Figure

Superconformal algebra also predicts that the LFWFs of the superpartners are related, and thus the corresponding elastic and transition form factors are identical. The resulting predictions for meson and baryon timelike form factors can be tested in

One can generalize these results to heavy-light

The LFWFs thus play the same role in hadron physics as the Schrödinger wavefunctions which encode the structure of atoms in QED. The elastic and transition form factors of hadrons, weak-decay amplitudes, and distribution amplitudes are overlaps of LFWFs; structure functions, transverse momentum distributions, and other inclusive observables are constructed from the squares of the LFWFs. In contrast one cannot compute form factors of hadrons or other current matrix elements of hadrons from overlap of the usual “instant" form wavefunctions since one must also include contributions where the photon interacts with connected but acausal vacuum-induced currents. The calculation of deeply virtual Compton scattering using LFWFs is given in [

The hadronic LFWFs predicted by light-front holography and superconformal algebra are functions of the LF kinetic energy

Prediction from AdS/QCD and Light-Front Holography for meson LFWFs

Five-dimensional

The holographic dictionary which maps the fifth dimension variable

Comparison of the AdS/QCD prediction

Light-Front Holography predicts not only meson and baryon spectroscopy successfully, but also hadron dynamics, including vector meson electroproduction, hadronic light-front wavefunctions, distribution amplitudes, form factors, and valence structure functions. The application to the deuteron elastic form factors and structure functions is given in [

Remarkably, the light-front potential using the dAFF procedure has the unique form of a harmonic oscillator

It is interesting to note that the contribution of the

It should be emphasized that the value of the mass scale

Doubly virtual Compton scattering on a proton (or nucleus) can be measured for two

One can derive the exact form of the light-front Hamiltonian

PQCD factorization theorems and the DGLAP [

The LF Heisenberg equation can in principle be solved numerically by matrix diagonalization using the “Discretized Light-Cone Quantization” (DLCQ) [

One can in fact measure the LFWFs of QED atoms using diffractive dissociation.

For example, suppose one creates a relativistic positronium beam. It will dissociate by Coulomb exchange in a thin target:

Positronium dissociation is analogous to the Ashery measurements of the pion LFWF:

LF-time-ordered perturbation theory can be advantageous for perturbative QCD calculations. An excellent example of LF-time-ordered perturbation theory is the computation of multigluon scattering amplitudes by Cruz-Santiago et al. [

A remarkable advantage of LF time-ordered perturbation theory (LFPth) is that the calculation of a subgraph of any order in pQCD only needs to be done once; the result can be stored in a “history" file. This is due to the fact that in LFPth the numerator algebra is independent of the process; the denominator changes, but only by a simple shift of the initial

The new insights into color confinement given by AdS/QCD suggest that one could compute “hadronization at the amplitude level” [

The invariant mass of a color-singlet cluster

A model for the two stages of hadronization and evolution is illustrated in Figure

(a) A model for evolution starting with a nonperturbative hadronic LFWF. (b) Hadronization and evolution ending with a hadronic LFWF. The intermediate quark and gluon states are off the

Thus quarks and gluons can appear in intermediate off-shell states, but only hadrons are produced asymptotically. Thus the AdS/QCD Light-Front Holographic model suggests how one can implement the transition between perturbative and nonperturbative QCD. For a QED analog, see [

A central, unique property of light-front quantization is

Particles in the front form move with positive

By definition, spin and helicity can be used interchangeably in the front form. LF chirality is conserved by the vector current in electrodynamics and the axial current of electroweak interactions. Each coupling conserves quark chirality when the quark mass is set to zero. A compilation of LF spinor matrix elements is given in [

Light-front spin is not the same as the usual “Wick helicity," where spin is defined as the projection of the particle’s three-momentum

The twist of a hadronic interpolating operator corresponds to the number of fields plus

One can use LF

Illustration of spin flow in

Similarly one can utilize the behavior of the amplitude

It is important to distinguish the LF vacuum from the conventional instant-form vacuum. The eigenstates of the instant-form Hamiltonian describe a state defined at a single instant of time

In contrast, the vacuum in LF Hamiltonian theory is defined as the eigenstate of

The physics associated with quark and gluon QCD vacuum condensates of the instant form are replaced by physical effects contained within the hadronic LFWFs. This is referred to as “in-hadron” condensates [

The universe is observed within the causal horizon, not at a single instant of time. The causal, frame-independent light-front vacuum can thus provide a viable match to the empty visible universe [

The QCD running coupling

The dilaton

(a) Comparison of the predicted nonperturbative coupling, based on the dilaton

Sum rules for deep inelastic scattering are usually analyzed using the operator product expansion of the forward virtual Compton amplitude, assuming it depends in the limit

The “handbag” approximation to deeply virtual Compton scattering also defines the “static" contribution [

The Glauber propagation of the vector system

Diffractive deep inelastic scattering is leading-twist, and it is an essential component of the two-step amplitude which causes shadowing and antishadowing of the nuclear PDF. It is important to analyze whether the momentum and other sum rules derived from the OPE expansion in terms of local operators remain valid when these dynamical rescattering corrections to the nuclear PDF are included. The OPE is derived assuming that the LF time separation between the virtual photons in the forward virtual Compton amplitude

The light-front Hamiltonian equation

The full QCD LF equation can be reduced for massless quarks to an effective LF Shrödinger radial equation for the valence

The color-confining potential

One obtains new insights into the hadronic spectrum, light-front wavefunctions, and the

Other LF Holographic predictions include the following:

Universal Regge slopes in

The pion eigenstate is a massless

Empirically viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions [

Superconformal extensions to heavy-light quark mesons and baryons.

In addition, superconformal algebra leads to remarkable supersymmetric relations between mesons and baryons of the same parity. The mass scale

Future work will include the extension of superconformal representations to pentaquark and other exotic hadrons, comparisons with lattice gauge theory predictions, the construction of an AdS/QCD orthonormal basis to diagonalize the QCD light-front Hamiltonian, hadronization at the amplitude level, and the computation of intrinsic heavy quark higher Fock states.

The author declares that they have no conflicts of interest.

The author thanks Ralf Hofmann for organizing an outstanding 5th Winter Workshop on Nonperturbative Quantum Field Theory at the Université de Nice. The results presented here are based on collaborations and discussions with Kelly Chiu, Alexandre Deur, Guy de Téramond, Guenter Dosch, Susan Gardner, Fred Goldhaber, Paul Hoyer, Dae Sung Hwang, Rich Lebed, Simonetta Liuti, Cedric Lorcé, Valery Lyubovitskij, Matin Mojaza, Michael Peskin, Craig Roberts, Robert Shrock, Ivan Schmidt, Peter Tandy, and Xing-Gang Wu. This research was supported by the Department of Energy, Contract DE–AC02–76SF00515. SLAC-PUB-17012.

^{+}

^{−})

^{+}e

^{−}annihilation by perturbation theory in quantum chromodynamics