Analytic three-loop Solutions for N=4 SYM Twist Operators

TL;DR

This work presents an analytic, perturbative solution to the higher-order Baxter equation for twist-two and twist-three operators in planar $N=4$ SYM. By deforming the one-loop Baxter solution and employing Mellin-space and hypergeometric representations, the authors derive closed-form expressions for the Baxter functions up to three loops and extract the corresponding anomalous dimensions. They reproduce the three-loop twist-two anomalous dimension in terms of nested harmonic sums, confirming the Kotikov conjecture, and obtain the twist-three results that agree with Dressing–Beccaria predictions. The approach demonstrates how integrability can generate the maximally transcendental parts of multi-loop QCD results and points toward extending these methods to finite-volume, higher-loop regimes.

Abstract

We introduce a method to obtain the analytic solution of the higher-order Baxter equation for twist-two and twist-three operators of planar N=4 SYM. Our result proofs the conjectured formula for the three-loop anomalous dimension of twist-two operators. As such we derive the maximally transcendental part of the corresponding three-loop QCD result from the maximal supersymmetric gauge theory in four dimension purely by methods of integrability.