Advances in High Energy Physics

Volume 2018, Article ID 7236382, 16 pages

https://doi.org/10.1155/2018/7236382

## Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra

SLAC National Accelerator Laboratory, Stanford University, Stanford, CA, USA

Correspondence should be addressed to Stanley J. Brodsky; ude.drofnats.cals@htbjs

Received 12 June 2017; Accepted 31 January 2018; Published 16 April 2018

Academic Editor: Ralf Hofmann

Copyright © 2018 Stanley J. Brodsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The QCD light-front Hamiltonian equation derived from quantization at fixed LF time provides a causal, frame-independent method for computing hadron spectroscopy as well as dynamical observables such as structure functions, transverse momentum distributions, and distribution amplitudes. The QCD Lagrangian with zero quark mass has no explicit mass scale. de Alfaro, Fubini, and Furlan (dAFF) have made an important observation that a mass scale can appear in the equations of motion without affecting the conformal invariance of the action if one adds a term to the Hamiltonian proportional to the dilatation operator or the special conformal operator. If one applies the dAFF procedure to the QCD light-front Hamiltonian, it leads to a color-confining potential for mesons, where is the LF radial variable conjugate to the invariant mass squared. The same result, including spin terms, is obtained using light-front holography, the duality between light-front dynamics and , if one modifies the action by the dilaton in the fifth dimension . When one generalizes this procedure using superconformal algebra, the resulting light-front eigensolutions provide a unified Regge spectroscopy of meson, baryon, and tetraquarks, including remarkable supersymmetric relations between the masses of mesons and baryons and a universal Regge slope. The pion eigenstate has zero mass at The superconformal relations also can be extended to heavy-light quark mesons and baryons. This approach also leads to insights into the physics underlying hadronization at the amplitude level. I will also discuss the remarkable features of the Poincaré invariant, causal vacuum defined by light-front quantization and its impact on the interpretation of the cosmological constant. AdS/QCD also predicts the analytic form of the nonperturbative running coupling . The mass scale underlying hadron masses can be connected to the parameter in the QCD running coupling by matching the nonperturbative dynamics to the perturbative QCD regime. The result is an effective coupling defined at all momenta. One obtains empirically viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. Finally, I address the interesting question of whether the momentum sum rule is valid for nuclear structure functions.

#### 1. Introduction

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 . Thus QCD with can in principle only predict* ratios of masses* such as , not their absolute values. Similarly, given that color is confined, how does QCD set its range without a parameter with dimensions of length? It is hard to see how this mass scale problem could be solved by “dimensional transmutation,” since the mass scale determined by perturbative QCD such as is renormalization scheme dependent, whereas hadron masses are independent of the conventions chosen to regulate the UV divergences.

A remarkable principle, first demonstrated by de Alfaro, Fubini, and Furlan (dAFF) [8] for conformal theory in quantum mechanics, is that a mass scale can appear in a Hamiltonian and its equations of motion without affecting the conformal invariance of the action. The essential step introduced by dAFF is to add to the conformal Hamiltonian terms proportional to the dilation operator and the special conformal operator . The unique result is the addition of a harmonic oscillator potential to the Hamiltonian, The group algebra is maintained despite the fact that and have dimensions. In fact, the new Hamiltonian has “extended dilatation invariance” since the mass scale can be rescaled arbitrarily. This implies that only ratios of the mass eigenvalues can be determined, not their absolute values.

Brodsky et al. [9] have shown that a mass gap and color confinement appears when one extends the dAFF procedure to relativistic, causal, Poincaré invariant, light-front Hamiltonian theory for QCD. The resulting predictions for both hadronic spectroscopy and dynamics provide an elegant description of meson and baryon phenomenology, including Regge trajectories with universal slopes in the principal quantum number and the orbital angular momentum . In addition, the resulting quark-antiquark bound-state equation predicts a massless pion for zero quark mass.

In this contribution, I will review a number of recent advances in holographic QCD, extending earlier reviews given in [10–12] with a new emphasis on the impact of superconformal algebra and new applications. As I will discuss, the resulting light-front eigensolutions provide a unified Regge spectroscopy of meson, baryon, and tetraquarks, including remarkable supersymmetric relations between the masses of mesons and baryons and a universal Regge slope. The combination of light-front holography with superconformal algebra thus leads to the novel prediction that hadron physics has supersymmetric properties in both spectroscopy and dynamics. It also predicts the form of the QCD running coupling at all scales and provides new insights into the physics underlying hadronization at the amplitude level. I will also discuss the remarkable features of the Poincaré invariant, causal vacuum defined by light-front quantization and its impact on the interpretation of the cosmological constant. Finally, I address the interesting question of whether the momentum sum rule is valid for nuclear structure functions.

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 , analogous to a flash photograph, not at a single “instant time.” As shown by Dirac [13], boosts are kinematical in the “front form." Thus all formulae using the front form are independent of the observer’s motion [14]; that is, they are Poincaré invariant. The eigenstates of the light-front Hamiltonian derived from the QCD Lagrangian encode the entire the hadronic mass spectrum for both individual hadrons and the multihadron continuum. The eigenvalues of the LF Hamiltonian are the squares of the hadron masses : [14]. The evaluation of the Wilson line for gauge theories in the front form is discussed in [15]. In addition, I will discuss the advantages of perturbative QCD calculations using light-front-time-ordered perturbation theory, including the use of conservation.

The eigenfunctions of the light-front Hamiltonian derived from the QCD Lagrangian correspond to the single hadron and multihadronic continuum eigenstates. The eigenvalues of the LF Hamiltonian are the squares of the hadron masses : [14]. Here is the LF time evolution operator, and and are kinematical. The eigenfunctions of provide hadronic LF Fock state wavefunctions (LFWFs): , the projection of the hadronic eigenstate on the free Fock basis. The constituents’ physical momenta are and , and the label the spin projections . Remarkably one can reduce the LF Hamiltonian theory for mesons with to an effective LF Schrödinger equation in a single variable, the LF radial variable .

The LFWFs are Poincaré invariant: they are independent of and and are thus independent of the motion of the observer. Since the LFWFs are independent of the hadron’s momentum, there is no length contraction [16, 17]. Structure functions are essentially the absolute square of the LFWFs. One thus measures the same structure function in an electron-ion collider as in an electron-scattering experiment where the target hadron is at rest.

Light-front wavefunctions thus provide a direct link between the QCD Lagrangian and hadron structure. Since they are defined at a fixed , they connect the physical on-shell hadronic state to its quark and gluon parton constituents, not at off-shell energy, but off-shell in invariant mass squared They thus control the transformation of the quarks and gluons in an off-shell intermediate state into the observed final on-shell hadronic state. See Figure 1.