Anomalous dimensions at twist-3 in the sl(2) sector of N=4 SYM

TL;DR

This paper determines the lowest anomalous dimensions for twist-3 operators in the ${\frak{sl}}(2)$ sector of planar ${\cal N}=4$ SYM by solving asymptotic Bethe Ansatz equations and, where possible, Baxter equations. It provides closed expressions for the spin dependence of the anomalous dimension $\gamma_3(s)$ up to four loops, including the dressing phase, and introduces a twist-3 transcendentality principle that organizes these results in terms of nested harmonic sums evaluated at $s/2$. A key nontrivial check is the reproduction of the universal cusp anomalous dimension in the large-spin limit, confirming consistency with the Beisert–Eden–Staudacher framework. The findings strengthen evidence for integrability-based control in ${\cal N}=4$ SYM, offer precise finite-spin data that can test relatedQCD-like twist-3 sectors, and illuminate the role of the dressing phase at finite spin.

Abstract

We consider twist-3 operators in the sl(2) sector of N=4 SYM built out of three scalar fields with derivatives. We extract from the Bethe Ansatz equations of this sector the exact lowest anomalous dimension gamma(s) of scaling fields for several values of the operator spin s. We propose compact closed expressions for the spin dependence of gamma(s) up to the four loop level and show that they obey a simple new twist-3 transcendentality principle. As a check, we reproduce the four loop universal cusp anomalous dimension governing the logarithmic large spin limit of gamma(s).