# BFKL Eigenvalue and Maximal Alternation of Harmonic Sums

## TL;DR

This paper investigates the eigenvalues of the BFKL equation using analytic continuation of harmonic sums, introducing a new parameter called 'alternation' and analyzing its preservation across perturbative loops, with implications for color configurations.

## Contribution

It introduces the concept of 'alternation' in harmonic sums and demonstrates its invariance in the perturbative expansion of the BFKL eigenvalue, providing a new classification scheme.

## Key findings

- Maximal alternation is preserved across loops in perturbative expansion.
- Color adjoint BFKL eigenvalues use harmonic sums with zero alternation and depth one.
- Color singlet BFKL eigenvalues involve harmonic sums with maximal alternation one.

## Abstract

We analyze the known results for the eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in the perturbative regime using the analytic continuation of harmonic sums from even positive arguments to the complex plane. The resulting meromorphic functions have poles at negative integer values of the argument. The typical classification of harmonic sums is determined by two major parameters: $a)$ the \textit{weight} - a sum of inverse powers of the summation indices; $b)$ the \textit{depth} - a number of nested summations. We introduce the third parameter: the \textit{alternation} - a number of nested sign-alternating summations in a given harmonic sum. We claim that the maximal alternation of the nested summation in the functions building the BFKL eigenvalue is preserved from loop to loop in the perturbative expansion. The BFKL equation is formulated for arbitrary color configuration of the propagating states in the $t$-channel. Based on known results one can state that color adjoint BFKL eigenvalue be can written using only harmonic sums with positive indices, maximal alternation zero, and at most depth one, whereas the singlet BFKL eigenvalue is constructed of harmonic sums with maximal sign alternation being equal one. We also note that for maximal alternation being equal unity the harmonic sums can be expressed through alternation zero harmonic sums with half-shifted arguments.