Anomalous dimensions of twist 2 operators and $\mathcal{N}=4$ SYM quantum spectral curve

TL;DR

This paper introduces an algorithmic method to compute anomalous dimensions of twist 2 operators in $ ext{N}=4$ SYM using the quantum spectral curve, valid to high loop orders and arbitrary operator spins.

Contribution

It develops a novel, closed-form function class and an algorithmic approach for solving the spectral curve equations at arbitrary coupling and spin, enabling high-order perturbative calculations.

Findings

Computed anomalous dimensions up to four loops.

Established a closed function class under key operations.

Validated the method for arbitrary operator spins.

Abstract

We present algorithmic perturbative solution of $\mathcal{N}=4$ SYM quantum spectral curve in the case of twist 2 operators, valid to in principle arbitrary order in coupling constant. The latter treats operator spins as arbitrary integer values and is written in terms of special class of functions -- products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions. It is shown that this class of functions is closed under elementary operations, such as shifts, partial fractions, multiplication by spectral parameter and differentiation. Also, it is fully sufficient to solve arising non-homogeneous multiloop Baxter and first order difference equations. As an application of the proposed method we present the computation of anomalous dimensions of twist 2 operators up to four loop order.