On Hermitian separability of the next-to-leading order BFKL kernel for the adjoint representation of the gauge group in the planar N = 4 SYM

TL;DR

This paper investigates a proposed modification to the BFKL kernel in N=4 SYM theory aimed at achieving Hermitian separability, but finds that the modification conflicts with known kernel properties due to its complex nature.

Contribution

It critically analyzes a proposed modification to the BFKL kernel for the adjoint representation in N=4 SYM, revealing inconsistencies with established kernel properties.

Findings

Modification affects only the real part of the kernel.

The correction cannot be expressed as a single analytic function.

Contradicts known properties of the BFKL kernel.

Abstract

We analyze a modification of the BFKL kernel for the adjoint representation of the colour group in the maximally supersymmetric (N=4) Yang-Mills theory in the limit of a large number of colours, related to the modification of the eigenvalues of the kernel suggested by S. Bondarenko and A. Prygarin in order to reach the Hermitian separability of the eigenvalues. We restore the modified kernel in the momentum space. It turns out that the modification is related only to the real part of the kernel and that the correction to the kernel can not be presented by a single analytic function in the entire momentum region, which contradicts the known properties of the kernel.