BFKL eigenvalue and maximal alternation of harmonic sums

Alex Prygarin

doi:10.1016/j.nuclphysb.2019.114734

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 weight - a sum of inverse powers of the summation indices; b) the depth - a number of nested summations. We introduce the third parameter: the 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 can be 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.

Highlights

The Balitsky-Fadin-Kuraev-Lipatov (BFKL) [1] approach is used to resum leading logarithms of energy in the framework of the perturbative approach to the gauge field theories
The QCD version of the BFKL equation was shown to comply with experimental data, while its supersymmetric extension was found to be very useful in predicting the high energy behavior of helicity amplitudes
In this paper we consider the analytic continuation of the harmonic sums from positive integer values of the argument to the complex plane denoted by Sa+1,a2,...(z)

Summary

Introduction

The Balitsky-Fadin-Kuraev-Lipatov (BFKL) [1] approach is used to resum leading logarithms of energy in the framework of the perturbative approach to the gauge field theories. The main motivation for the BFKL equation was to derive a perturbative representation of the leading Regge trajectory (Pomeron) in the framework of the gauge theory, mainly in QCD As such most of attention was attracted to the color singlet representation, while the color adjoint state was used only to show the self consistency of the approach and demonstrate the gluon reggeization. In the present discussion we are focused only on the commonly used BDS-like prescription for removing infrared divergent parts of the BFKL equation projected on the color adjoint state of the reggeized gluons in the t-channel In this regularization prescription the resulting final part builds Hamiltonian that has conformal symmetry in the dual momentum space. The maximal sign alternation being equal one is the artifact of the integrability of the closed spin chain

Harmonic Sums

LO and NLO BFKL Eigenvalue through Harmonic Sums

Conclusion and Discussions