Twist-three at five loops, Bethe Ansatz and wrapping

TL;DR

This work computes the five-loop anomalous dimension of twist-three operators in the sl(2) sector of N=4 SYM by combining the asymptotic Bethe Ansatz with generalized Lüscher finite-size corrections. The authors constrain the ABA input using maximum transcendentality and reciprocity (parity invariance), and derive an explicit expression for the ABA part $\gamma^{\mathrm{ABA}}_{10}$, together with a leading wrapping correction $\gamma^{\mathrm wrapping}_{10}$, ensuring consistency with the universal five-loop scaling function and Y-system predictions. Wrapping effects enter at order $O(\log^2 M/M^2)$ and preserve reciprocity; their structure is organized in terms of harmonic sums with argument $M/2$ and zeta-values arising from the dressing phase. Analytic continuation near $M=-2$ supports the KLRSV resummation at NLO and suggests a controlled all-loop pattern for high-spin behavior, reinforcing the interplay between ABA, wrapping, and resummation constraints in planar $\mathcal{N}=4$ SYM.

Abstract

We present a formula for the five-loop anomalous dimension of N=4 SYM twist-three operators in the sl(2) sector. We obtain its asymptotic part from the Bethe Ansatz and finite volume corrections from the generalized Luescher formalism, considering scattering processes of spin chain magnons with virtual particles that travel along the cylinder. The complete result respects the expected large spin scaling properties and passes non-trivial tests including reciprocity constraints. We analyze the pole structure and find agreement with a conjectured resummation formula. In analogy with the twist-two anomalous dimension at four-loops, wrapping effects are of order log^2 M/M^2 for large values of the spin.