The Knizhnik--Zamolodchikov structure of lattice BFKL evolution and the twist-two anomalous dimension

TL;DR

This paper explores a lattice regularization of BFKL evolution, revealing its connection to Knizhnik--Zamolodchikov equations and providing insights into the twist-two anomalous dimension through harmonic polylogarithms and transcendentality structures.

Contribution

It introduces a lattice framework for BFKL evolution governed by an abelian KZ equation, linking it to harmonic polylogarithms and the algebraic structure of twist-two anomalous dimensions.

Findings

Bulk dynamics governed by abelian KZ equation

Exact walk expansion shows Reggeisation at finite size

Small-$oldsymbol{ extomega}$ expansion matches known transcendentality structures

Abstract

We study a lattice regularization of the BFKL evolution, showing its bulk dynamics is governed by an abelian Knizhnik--Zamolodchikov equation. The Hamiltonian combines long-range hopping with virtual corrections encoded by harmonic numbers. An exact walk expansion renders Reggeisation manifest at finite system size. In the bulk continuum limit, evolution reduces to a connection on $\mathbb{P}^1\setminus\{0,1,\infty\}$: $\Omega(x) = -2\,dx/x - 4\,dx/(1-x)$, with solutions in $\{0,1\}$-alphabet harmonic polylogarithms. Projecting to the collinear sector via Brown's single-valued map organizes the twist-two anomalous dimension's small-$\omega$ expansion, generating polynomials in odd zeta values, matching the transcendentality structure of planar $\mathcal{N}=4$ SYM and multi-Regge kinematics. The lattice thus isolates the algebraic core of BFKL evolution.