Reflection Identities of Harmonic Sums and pole decomposition of BFKL eigenvalue

TL;DR

This paper investigates the pole structure of NNLO BFKL eigenvalues in N=4 SYM using harmonic sums and reflection identities, providing a method to analyze their complexity and compatibility with Bethe-Salpeter approaches.

Contribution

It introduces a systematic approach to decompose harmonic sums' poles and identify the most complex terms of NNLO BFKL eigenvalues for any conformal spin.

Findings

Harmonic sums' pole decomposition reveals the structure of NNLO BFKL eigenvalues.

The method identifies the most complex terms for arbitrary conformal spin.

Results align with the Bethe-Salpeter approach to BFKL evolution.

Abstract

We analyze known results of next-to-next-to-leading(NNLO) singlet BFKL eigenvalue in $N=4$ SYM written in terms of harmonic sums. The nested harmonic sums building known NNLO BFKL eigenvalue for specific values of the conformal spin have poles at negative integers. We sort the harmonic sums according to the complexity with respect to their weight and depth and use their pole decomposition in terms of the reflection identities to find the most complicated terms of NNLO BFKL eigenvalue for an arbitrary value of the conformal spin. The obtained result is compatible with the Bethe-Salpeter approach to the BFKL evolution.