The Light-Front Schr\"odinger Equation and Determination of the Perturbative QCD Scale from Color Confinement

The valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a relativistic equation of motion 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 light front 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 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 with the same slope in the radial quantum number $n$ and orbital angular momentum $L$. Only one mass parameter $\kappa$ appears. 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 show how the mass scale $\kappa$ underlying confinement and hadron masses determines the scale $\Lambda_{\overline{MS}} $ 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 light-front and its embedding in AdS space, to the perturbative QCD regime computed to four-loop order. The result is an effective coupling defined at all momenta. The predicted value $\Lambda_{\overline{MS}} = 0.328 \pm 0.034$ GeV is in agreement with the world average $0.339 \pm 0.010$ GeV. The analysis applies to any renormalization scheme.