Real Valued Functions for the BFKL Eigenvalue

We consider known expressions for the eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in N=4 super Yang-Mills theory as a real valued function of two variables. We define new real valued functions of two complex conjugate variables that have a definite complexity analogous to the weight of the nested harmonic sums. We argue that those functions span a general space of functions for the BFKL eigenvalue at any order of the perturbation theory.

Regge Cuts and NNLLA BFKL

In the leading and next-to-leading logarithmic approximations, QCD amplitudes with gluon quantum numbers in cross-channels and negative signature have the pole form corresponding to a reggeized gluon. The famous BFKL equation was derived using this form. In the next-to-next-to-leading approximation (NNLLA), the pole form is violated by contributions of Regge cuts. We discuss these contributions and their impact on the derivation of the BFKL equation in the NNLLA.

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

We analyze the structure of the eigenvalue of the color-singlet Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation in N[Formula: see text]=[Formula: see text]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[Formula: see text]=4[Formula: see text]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 nonlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.