Parton distributions in the shockwave formalism

TL;DR

This work develops a unified framework to compute multi‑dimensional parton distributions directly from operator definitions in the shockwave CGC limit, enabling PDFs, TMDs, GPDs, GTMDs, and diffractive distributions to be evaluated at leading order via Wilson‑line correlators. A complete set of Feynman rules is provided to connect operator definitions to their CGC realization, with LO results for quark PDFs and GPDs including finite terms, and comprehensive GTMDs for gluons and quarks across general gauge links. The study also derives both gluon and quark diffractive distributions, clarifying the role of color neutrality and gauge‑link structures, and demonstrates explicit renormalization for quark GPDs and PDFs within the shockwave framework. By linking small‑x CGC dynamics with the partonic imaging of hadrons, the paper lays groundwork for NLO matching, sub‑eikonal corrections, and phenomenology relevant to the EIC and high‑energy colliders. Overall, it provides a systematic, operator‑level route to bridge high‑energy saturation physics with the multi‑dimensional nucleon structure probed in experiments.

Abstract

In this work, we calculate a broad class of parton distributions - including parton distribution functions (PDFs), transverse-momentum-dependent distributions (TMDs), generalized parton distributions (GPDs), generalized transverse-momentum-dependent distributions (GTMDs), and diffractive parton distributions - directly from their operator-level definition in the shockwave approximation for the target nucleon. This approximation is valid in the high-energy limit of scattering, corresponding to the small-$x$ regime. The shockwave framework allows us to employ the eikonal approximation and express the parton distributions in terms of Wilson-line correlators, naturally formulated within the color-glass condensate effective field theory. We present a comprehensive set of Feynman rules for evaluating parton distributions in this limit, and demonstrate how they can be systematically applied to calculate all phenomenologically relevant leading-twist parton distributions at leading order. This work establishes a unified starting point for future studies that aim to bridge the color-glass condensate approach with the partonic description of the nucleon.

Figures (11)

• Figure 1: Feynman rules for the shockwave with the target at the coordinate $b$. In practice, we can shift the coordinates such that in the actual computations we take $b =0$.
• Figure 2: Feynman rule for an eikonal quark line crossing through the shockwave. For antiquarks we get $-V^\dag(\mathbf{x})_{ij}$ and for gluons $U(\mathbf{x})_{ji}$ instead.
• Figure 3: Feynman rule for a gluon interacting with an eikonal line in the fundamental representation. Here $x_i$ and $x_f$ denote the light-cone times for the start and end points of the eikonal line.
• Figure 4: Feynman diagrams for computing the quark GTMD.
• Figure 5: An example contribution to the quark GTMD that evaluates to zero.