Solution of the NLO BFKL Equation and a Strategy for Solving the All-Order BFKL Equation

TL;DR

This work derives the NLO BFKL solution by perturbatively correcting the LO conformal eigenfunctions, yielding a complete LO+NLO eigenbasis and an explicit Green function that is μ-independent to the stated order.The method generalizes to all-order BFKL by proposing a structured eigenvalue expansion with higher-order kernels, and clarifies the roles of running coupling and conformal symmetry breaking in shaping the high-energy evolution.A detailed treatment of completeness, orthogonality, and phase/nu-reparametrization freedoms ensures a consistent eigenfunction basis, while connections to DIS and DGLAP anomalous dimensions validate the approach against established QCD scales.The paper also analyzes NNLO prospects, showing that simple ansätze fail and that a full NNLO construction requires perturbative NNLO eigenfunctions, paving the way for future refinements and applications to γ*γ* scattering.

Abstract

We derive the solution of the NLO BFKL equation by constructing its eigenfunctions perturbatively, using an expansion around the LO BFKL (conformal) eigenfunctions. This method can be used to construct a solution of the BFKL equation with the kernel calculated to an arbitrary order in the coupling constant.