Sub-eikonal Structure of High-Energy Deep-Inelastic Scattering

TL;DR

This paper develops a mixed-space formalism for high-energy deep-inelastic scattering at sub-eikonal order, deriving new corrections to structure functions and analyzing divergence properties.

Contribution

It introduces a novel mixed-space formulation including sub-eikonal corrections and organizes the results in a gauge-invariant operator basis.

Findings

Rederived standard eikonal dipole cross sections for photon polarization.

Computed first sub-eikonal corrections to structure functions $F_L$, $F_T$, and $g_1$.

Found that the longitudinal structure function is finite, while transverse and helicity-dependent functions have logarithmic divergences.

Abstract

I develop a mixed-space formulation of high-energy deep-inelastic scattering in the shock-wave formalism at sub-eikonal order. Starting from the quark propagator in the background field, I derive the corresponding mixed-space Feynman rules from the LSZ reduction formula in the presence of a shock wave, including the instantaneous contributions generated by the presence of the shock-wave. As a first check of the formalism, I rederive the standard eikonal dipole cross sections for longitudinal and transverse photon polarization. I then use the same framework to compute the first sub-eikonal corrections to the dipole structure functions. In particular, I obtain the sub-eikonal contributions to the longitudinal and transverse structure functions $F_L$ and $F_T$, as well as to the helicity-sensitive asymmetry related to $g_1$, and organize the result in terms of a gauge-invariant operator basis. The resulting operator combinations are naturally written in dipole form and vanish in the zero-dipole-size limit, making the unitarity property and the small-dipole behavior manifest. Finally, I analyze the divergence structure of the sub-eikonal dipole corrections. I show that the longitudinal structure function is finite at this order, whereas the transverse and helicity-dependent structure functions contain only logarithmic divergences.