Non linear gluon evolution in path-integral form

TL;DR

The paper clarifies how nonlinear small-x gluon evolution can be described by two equivalent renormalization-group formalisms: a U-representation based on Wilson-line correlators and a Langevin/path-integral framework, and an alpha-representation based on a stochastic color source with its own FP/Langevin structure. By formulating both approaches as a random walk in the space of SU(N) Wilson lines and establishing explicit connections via path integrals, the authors provide a unified statistical picture and practical Langevin algorithms for numerical simulation. The work yields FP equations in both representations, derives their path-integral solutions, and discusses discretization choices, setting the stage for systematic analytical and numerical study of nonlinear small-x QCD evolution and saturation phenomena.

Abstract

We explore and clarify the connections between two different forms of the renormalisation group equations describing the quantum evolution of hadronic structure functions at small $x$. This connection is established via a Langevin formulation and associated path integral solutions that highlight the statistical nature of the quantum evolution, pictured here as a random walk in the space of Wilson lines. The results confirm known approximations, form the basis for numerical simulations and widen the scope for further analytical studies.