Frame-independent spatial coordinate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mi>z</mml:mi><mml:mo></mml:mo></mml:mover></mml:math>: Implications for light-front wave functions, deep inelastic scattering, light-front holography, and lattice QCD calculations
Gerald A. Miller, Stanley J. Brodsky
Abstract
A general procedure for obtaining frame-independent three-dimensional light-front coordinate-space wave functions is introduced. The third spatial coordinate $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}$ is the boost and Lorentz 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 a very large extent in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}$.