Abstract
In attempt to find a proper space of function expressing the eigenvalue of the color-singlet BFKL equation in N = 4 SYM, we consider an analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The resulting meromorphic functions have pole singularities at negative integers. We derive the reflection identities for harmonic sums at weight four 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 four, which represents the main result of the paper. We also discuss how other trilinear and quadlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.
Highlights
The Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation was formulated about four decades ago [1,2,3,4,5] in attempt to describe the leading Regge trajectory (Pomeron) in the framework of the perturbative gauge theory, in particular the Quantum Chromodynamics (QCD)
Our goal is to find a closed expression of the BFKL eigenvalue for all possible values of anomalous dimension and conformal spin, so that we follow the notation of Gromov, Levkovich-Maslyuk and Sizov [9], and use S+ (z) throughout the text
We consider meromorphic functions obtained by analytic continuation of the harmonic sums from positive integers to the complex plane except for isolated pole singularities at negative integer values of the argument
Summary
The Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation was formulated about four decades ago [1,2,3,4,5] in attempt to describe the leading Regge trajectory (Pomeron) in the framework of the perturbative gauge theory, in particular the Quantum Chromodynamics (QCD). The BFKL approach is based on identifying the leading contributions in the perturbative expansion, namely the terms that are accompanied by the large logarithm of center-of-mass energy. This way one separates the dynamics of the longitudinal and transverse degrees of freedom, where the longitudinal momentum contributes to the large parameter (logarithm of center-of-mass energy) and plays a role of the “time”. QCD selecting those that give the leading order (LO) in power of logarithm of the center-of-mass energy. The BFKL equation initiated a lot of activity in the field of analytic perturbative calculations as well phenomenological studies and comparison to the experimental data
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