Wave Functions, Evolution Equations and Evolution Kernels from Light-Ray Operators of QCD

TL;DR

This work develops a gauge-invariant framework where nonperturbative QCD inputs are encoded as matrix elements of light-ray operators, connecting forward parton distributions and hadron wave functions through two-variable, nonforward distribution amplitudes. It derives a general evolution equation for these amplitudes from the renormalization group of light-ray operators, unifying the Altarelli–Parisi and Brodsky–Lepage evolutions as special limits of an extended BL-kernel. The authors construct the extended BL-kernel and demonstrate, through a detailed two-loop analysis and analytic continuation, that the Altarelli–Parisi kernel is recovered consistently from the extended framework. This approach provides a coherent, gauge-invariant method to treat both exclusive and inclusive QCD processes in a single evolution formalism, with practical cross-checks between forward and nonforward kernels. Mathematics is organized around the light-ray operator formalism, its anomalous dimensions, and the spectral representation linking two- and one-variable distributions.

Abstract

The widely used nonperturbative wave functions and distribution functions of QCD are determined as matrix elements of light-ray operators. These operators appear as large momentum limit of nonlocal hadron operators or as summed up local operators in light-cone expansions. Nonforward one-particle matrix elements of such operators lead to new distribution amplitudes describing both hadrons simultaneously. These distribution functions depend besides other variables on two scaling variables. They are applied for the description of exclusive virtual Compton scattering in the Bjorken region near forward direction and the two meson production process. The evolution equations for these distribution amplitudes are derived on the basis of the renormalization group equation of the considered operators. This includes that also the evolution kernels follow from the anomalous dimensions of these operators. Relations between different evolution kernels (especially the Altarelli-Parisi and the Brodsky-Lepage) kernels are derived and explicitly checked for the existing two-loop calculations of QCD. Technical basis of these results are support and analytically properties of the anomalous dimensions of light-ray operators obtained with the help of the $α$-representation of Green's functions.

Figures (4)

• Figure 1: Leading contribution for the virtual Compton scattering in the Bjorken region.
• Figure 2: Leading contribution for the virtual Compton amplitude at fixed angles.
• Figure 3: Support of $\gamma (t,t')$, where $f_{\pm\pm}=f(\pm t,\pm t')$, and $g_{\pm\mp}=g(\pm t,\mp t')$ are defined by Eqs. (\ref{['INTRE']}).
• Figure 4: Topology of the 1PI-Feynman graphs for the light-cone operators in light-cone gauge. The box symbolizes the connected $(N+2)$-point function. The lines $l_1$ and $l_2$ are contained in a loop $c$.