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

Giovanni Antonio Chirilli

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.

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

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 and , as well as to the helicity-sensitive asymmetry related to , 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.
Paper Structure (25 sections, 137 equations, 3 figures)

This paper contains 25 sections, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Quark propagator up to sub-eikonal corrections
  4. Eikonal contribution
  5. Sub-eikonal corrections
  6. Gluon field in the background
  7. Quark propagator with $\psi$ and ${\bar{\psi}}$ in the background field
  8. Summary of contributions to the quark propagator
  9. LSZ reduction formula in the shock-wave formalism
  10. Propagation outside the shock-wave
  11. Propagation crossing the shock-wave
  12. Dipole cross-section in the eikonal approximation
  13. Eikonal dipole cross-section with Longitudinal polarization
  14. Eikonal dipole cross-section with Transverse Polarization
  15. Dipole cross-section at sub-eikonal level
  16. ...and 10 more sections

Figures (3)

  • Figure 1: In the picture is shown a typical diagram contributing the quark propagator in the background of quark fields. As usual, we indicate in blue the quantum field while in red the background one.
  • Figure 2: Diagrams contributing to the transition amplitude $\gamma^*(q)\to q(k){\bar{q}}(p)$ in the eikonal approximation.
  • Figure 3: Diagrams for the dipole cross-section with sub-eikonal corrections. In the left panel we have the diagram with quark field in the background. In the right panel we have the diagram with sub-eikonal contribution due to the gluon field. In particular we will consider the ${\cal F}(z_\perp)$, ${\cal F}_2(z_\perp)$, ${\cal F}_{2'}(z_\perp)$, and ${\cal F}'(z_\perp)$ operators.