First Saturation Correction in High Energy Proton-Nucleus Collisions: I. Time evolution of classical Yang-Mills fields beyond leading order

TL;DR

This paper develops a perturbative analytical method to solve classical Yang-Mills equations beyond leading order, enabling the calculation of saturation corrections in high energy proton-nucleus collisions.

Contribution

It provides the first systematic derivation of higher order solutions to Yang-Mills equations in this context, advancing the understanding of saturation effects.

Findings

Explicit next-to-leading order solutions derived

Next-to-next-to-leading order solutions constructed

A framework for all higher order solutions outlined

Abstract

In high energy proton-nucleus collisions, the single- and double-inclusive soft gluon productions at the leading order have been calculated and phenomenologically studied in various approaches for many years. These studies do not take into account the saturation and multiple rescatterings in the field of the proton. The first saturation correction to these leading order results (the terms that are enhanced by the combination $\alpha_s^2 \mu^2$, where $\mu^2$ is the proton's color charge squared per unit transverse area) has not been completely derived despite recent attempts using a diagrammatic approach. This paper is the first in a series of papers towards analytically completing the first saturation correction to physical observables in high energy proton-nucleus collisions. Our approach is to analytically solve the classical Yang-Mills equations in the dilute-dense regime using the Color Glass Condensate effective theory and compute physical observables constructed from classical gluon fields. In the current paper, the Yang-Mills equations are solved perturbatively in the field of the dilute object (the proton). Next-to-leading order and next-to-next-to-leading order analytic solutions are explicitly constructed. A systematic way to obtain all higher order analytic solutions is outlined