The Light-Front Schrodinger Equation and the Determination of the Perturbative QCD Scale from Color Confinement: A First Approximation to QCD

The valence Fock-state wavefunctions of the light-front (LF) QCD Hamiltonian satisfy a relativistic equation of motion, analogous to the nonrelativistic radial Schrodinger equation, with an effective confining potential U which systematically incorporates the effects of higher quark and gluon Fock states. If one requires that the effective action which underlies the QCD Lagrangian remains conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to LF Hamiltonian theory, the potential U has a unique form of a harmonic oscillator potential, and a mass gap arises. The result is a nonperturbative relativistic LF 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 with the same slope in the radial quantum number n and orbital angular momentum L. Only one mass parameter kappa appears. The corresponding LF Dirac equation provides a dynamical and spectroscopic model of nucleons. The same LF equations arise from the holographic mapping of the soft-wall model modification of AdS(5) space with a unique dilaton profile to QCD (3+1) at fixed LF time. LF holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of Anti-de Sitter (AdS) space and the boost-invariant LFWFs describing the internal structure of hadrons in physical space-time. We also show how the mass scale underlying confinement and the masses of light-quark hadrons determines the scale controlling the evolution of the perturbative QCD coupling. The relation between scales is obtained by matching the nonperturbative dynamics, as described by an effective conformal theory mapped to the LF and its embedding in AdS space, to the perturbative QCD regime computed to four-loop order. The data for the effective coupling defined from the Bjorken sum rule are remarkably consistent with the Gaussian form predicted by LF holographic QCD. The result is an effective coupling defined at all momenta. The predicted value GeV is in agreement with the world average GeV. We thus can connect to hadron masses. The analysis applies to any renormalization scheme.