Six-Loop Anomalous Dimension of Twist-Three Operators in N=4 SYM

TL;DR

The paper computes the six-loop anomalous dimension for twist-three operators in planar $\mathcal{N}=4$ SYM by combining the asymptotic Bethe Ansatz (ABA) with wrapping corrections via weak-coupling Lüscher methods. It expresses the ABA contribution in a reciprocity-respecting basis of binomial harmonic sums, including explicit rational and multiple zeta-value terms, and then adds the finite-size corrections using a modified $Y_Q$ integrand for twist-three. The full $M$-dependence is reconstructed under reciprocity constraints using high-precision numerics and lattice-reduction techniques, with cross-checks from large-$M$ asymptotics and analytic continuation to $M=-2+\omega$ that agree with resummation predictions. The work tests proposed spectral equations (Y-system/TBA) at six loops and provides a detailed framework for twist-three wrapping corrections, including explicit structures and bases for the contributing sums. This advances perturbative tests of AdS/CFT integrability and refines the understanding of finite-size effects in the planar theory.

Abstract

The result for the six-loop anomalous dimension of twist-three operators in the planar N=4 SYM theory is presented. The calculations were performed along the paper arXiv:0912.1624. This result provides a new data for testing the proposed spectral equations for planar AdS/CFT correspondence.