The Frame-Independent Spatial Coordinate $\tilde{z}$: Implications for Light-Front Wave Functions, Deep Inelastic Scattering, Light-Front Holography, and Lattice QCD Calculations

Abstract

A general procedure for obtaining frame-independent, three-dimensional light-front coordinate-space wave functions is introduced. The third spatial coordinate, $\tilde{z}$ , is the frame independent coordinate conjugate to the light-front momentum coordinate $x=\frac{k^{+}}{P^{+}}$ which appears in the momentum-space light-front wave functions underlying generalized parton distributions, structure functions, distribution amplitudes, form factors, and other hadronic observables. These causal light-front coordinate-space wave functions are used to derive a general expression for the quark distribution function of hadrons as an integral over the frame-independent longitudinal distance (the Ioffe time) between virtual-photon absorption and emission appearing in the forward virtual photon-hadron Compton scattering amplitude. Specific examples using models derived from light-front holographic QCD show that the spatial extent of the proton eigenfunction in the longitudinal direction can have very large extent in $\tilde{z}$.

Comments: 6 pages, 6 figures, this version adds references, corrects a typographical error and contains more explanations regarding the use of the variable $\tilde{z}$