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

TL;DR

This work presents a lattice regularization of the forward BFKL kernel, revealing that bulk evolution is governed by an abelian Knizhnik–Zamolodchikov equation. A walk expansion with vertex-dressed propagators connects Reggeisation to the spectrum of a finite-dimensional Hamiltonian, while a continuum limit produces a KZ operator built from Euler kernels, whose solutions are harmonic polylogarithms. Projecting to the single-logarithmic collinear sector via Brown’s single-valued map yields a generating function for twist-two anomalous-dimension coefficients expressed as polynomials in odd zeta values, matching transcendentality patterns in planar N=4 SYM and multi-Regge kinematics. The lattice model thus isolates the algebraic core of BFKL evolution and unifies Regge poles, non-compact spin chains, KZ monodromy, and single-valued polylogarithms in a transparent, finite-dimensional setting.

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 $\mathbb{P}^1\setminus\{0,1,\infty\}$: $Ω(x) = -2\,dx/x - 4\,dx/(1-x)$, 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-$ω$ 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.

Figures (3)

• Figure 1: Continuum approximation of the lattice potential $V_i = H_{N-i} - H_{i-1}$ by $V(x) \simeq \log\frac{1-x}{x}$ with $x=i/N$. The shaded region indicates a bulk window $\varepsilon < x < 1-\varepsilon$ where $V_i$ stays bounded, while the divergences at $x\to 0$ and $x\to 1$ are associated with the infrared and ultraviolet ends of the lattice.
• Figure 2: Logarithmic lattice coordinate $t=\log i$. The bulk window $[T_-,T_+]$ acts as a one-dimensional Fabry--Perot cavity in log-space. Mellin waves $i^{-1/2+\mathrm{i}\nu}$ become standing waves with nodes near the effective walls, leading to the quantization condition \ref{['eq:nu-quant-fp']}. This is an illustrative sketch, in an ideal optical cavity the mode spacing is uniform and the levels are strictly parallel, since the length $L$ and the reflection phase $\delta$ are constant. In the lattice case, however, the effective cavity is only approximate and the drift potential remains slowly varying in the bulk and the reflection phase $\delta(\nu)$ acquires a mild $\nu$--dependence determined by the infrared and ultraviolet cutoffs. As a result, the discrete levels need not be perfectly parallel and small deviations from parallelism simply encode these boundary effects and do not affect the leading Fabry--Perot interpretation.
• Figure 3: A path $(m_0,m_1,m_2,m_3)$ contributing to the walk expansion \ref{['eq:walk-expansion']}. Each hop carries an edge factor $1/|m_i-m_{i+1}|$, while all virtual insertions at a given site $m_i$ are resummed into the vertex factor $(1+2zH_{m_i-1})^{-1}$ in the resolvent \ref{['eq:resolvent-walk-main']}.