Analytic continuation of harmonic sums with purely imaginary indices near the integer values

TL;DR

This paper introduces a straightforward algebraic method for the analytic continuation of harmonic sums with purely imaginary indices near integers, providing a Mathematica implementation and applying it to the ABJM model's anomalous dimensions.

Contribution

It presents a new algebraic approach for analytic continuation of harmonic sums with imaginary indices and demonstrates its application to the ABJM model, including explicit code and predictions.

Findings

The method reproduces known results for the slope function.

In the BFKL-like limit, the behavior matches that of N=4 SYM and QCD.

No general BFKL Pomeron eigenvalue expression was found in the ABJM model.

Abstract

We present a simple algebraic method for the analytic continuation of harmonic sums with integer real or purely imaginary indices near negative and positive integers. We provide a MATHEMATICA code for exact expansion of harmonic sums in a small parameter near these integers. As an application, we consider the analytic continuation of the anomalous dimension of twist-1 operators in ABJM model, which contains the nested harmonic sums with purely imaginary indices. We found that in the BFKL-like limit the result has the same single-logarithmic behavior as in N=4 SYM and QCD, however, we did not find a general expression for the ``BFKL Pomeron'' eigenvalue in this model. For the slope function, we found full agreement with the expansion of the known general result and give predictions for the first three perturbative terms in the expansion of the next-to-slope function. The proposed method of analytic continuation can also be used for other generalization of the nested harmonic sums.