A semi-infinite matrix analysis of the BFKL equation

TL;DR

This paper introduces a semi-infinite matrix approach to analyze the BFKL equation, revealing insights into infrared and ultraviolet diffusion, and proposes a modification to ensure unitarity in high-energy evolution.

Contribution

It presents a novel matrix discretization of the BFKL equation and links it to spin chain models, offering new analytical tools for high-energy QCD analysis.

Findings

Matrix truncation reproduces BFKL eigenstates

Connection to SL(2) spin chain models

Modified matrix suppresses infrared modes for unitarity

Abstract

The forward BFKL equation is discretised in virtuality space and it is shown that the diffusion into infrared and ultraviolet momenta can be understood in terms of a semi-infinite matrix. The square truncation of this matrix can be exponentiated leading to asymptotic eigenstates sharing many features with the BFKL gluon Green's function in the limit of large matrix size. This truncation is closely related to a representation of the XXX Heisenberg spin $= - \frac{1}{2}$ chain with SL(2) invariance where the Hamiltonian acts on a symmetric double copy of the harmonic oscillator. A simple modification of the BFKL matrix suppressing the infrared modes generates evolution with energy compatible with unitarity.