The analytic structure of the BFKL equation and reflection identities of harmonic sums at weight five

TL;DR

This paper investigates the structure of the BFKL eigenvalue in N=4 SYM using harmonic sums, deriving reflection identities at weight five, and demonstrating how these can reconstruct the eigenvalue's functional form across the complex plane.

Contribution

It introduces minimal irreducible reflection identities at weight five for harmonic sums and shows their application in restoring the BFKL eigenvalue's functional form.

Findings

Derived reflection identities for harmonic sums at weight five.

Demonstrated the reconstruction of the eigenvalue's full functional form.

Connected reflection identities with quasi-shuffle relations.

Abstract

We analyze the structure of the eigenvalue of the color-singlet Balitsky-Fadin-Kuraev-Lipatov~(BFKL) equation in N=4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N=4 SYM , i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how non-linear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.