ABJM quantum spectral curve at twist 1: algorithmic perturbative solution

TL;DR

This paper introduces an algorithmic perturbative method for solving the ABJM quantum spectral curve at twist 1, enabling calculation of anomalous dimensions at high loop orders using a novel class of functions.

Contribution

The paper develops a new recursive algorithm for solving Baxter equations within the ABJM spectral problem, applicable to arbitrary spin and perturbation order, and introduces a closed class of functions for these solutions.

Findings

Computed six-loop anomalous dimensions for twist 1 operators.

Identified a function class closed under elementary operations for spectral problems.

Proposed method is potentially extendable to higher twists and other theories.

Abstract

We present an algorithmic perturbative solution of ABJM quantum spectral curve at twist 1 in sl(2) sector for arbitrary spin values, which can be applied to, in principle, arbitrary order of perturbation theory. We determined the class of functions -- products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions -- closed under elementary operations, such as shifts and partial fractions, as well as differentiation. It turns out, that this class of functions is also sufficient for finding solutions of inhomogeneous Baxter equations involved. For the latter purpose we present recursive construction of the dictionary for the solutions of Baxter equations for given inhomogeneous parts. As an application of the proposed method we present the computation of anomalous dimensions of twist 1 operators at six loop order. There is still a room for improvements of the proposed algorithm related to the simplifications of arising sums. The advanced techniques for their reduction to the basis of generalized harmonic sums will be the subject of subsequent paper. We expect this method to be generalizable to higher twists as well as to other theories, such as N=4 SYM.