Three loop anomalous dimensions of twist-3 gauge operators in N=4 SYM

TL;DR

This paper determines the three-loop anomalous dimension for finite-spin twist-3 gauge operators in $\mathcal{N}=4$ SYM by solving the long-range Bethe Ansatz equations perturbatively, starting from the one-loop reduction to the $XXX_{-3/2}$ spin chain. It introduces a weak KLOV-like Ansatz with subleading transcendentality, yielding closed expressions for $\gamma_1$, $\gamma_2$, and $\gamma_3$ in terms of harmonic sums with $n=N/2+1$, and confirms that the large-$N$ limit reproduces the universal cusp scaling function $f(g)$; additional subleading terms satisfy generalized Moch-Vermaseren-Vogt reciprocity relations. The analysis relies on the tripleton decomposition of the ${\cal N}=4$ singleton tensor product and duality transformations that map twist-3 gauge-sector Bethe equations to the $\mathrm{XXX}_{-3/2}$ chain, enabling a controlled perturbative expansion. The results provide a concrete, testable structure for twist-3 gauge dynamics in a maximally supersymmetric gauge theory and offer insights into reciprocity properties and potential wrapping effects for higher-loop orders.

Abstract

We propose a closed expression for the three loop anomalous dimension of a class of twist-3 operators built with gauge fields and covariant derivatives. To this aim, we solve the long-range Bethe Ansatz equations at finite spin and provide a consistent analytical formula obtained assuming maximal transcendentality violation as suggested by the known one-loop anomalous dimension. The final result reproduces the universal cusp anomalous dimension and obeys recursion relations inspired by the principle of reciprocity invariance.