Four loop reciprocity of twist two operators in N=4 SYM

TL;DR

This paper proves that the four-loop twist-2 anomalous dimension in $\mathcal{N}=4$ SYM respects generalized Gribov-Lipatov reciprocity, including wrapping corrections. By inverting the relation between the anomalous dimension $\gamma(N)$ and the $P$-kernel and recasting the result in a basis of definite-parity harmonic sums (Omega combinations), the authors show that the $P$-kernel at four loops is reciprocity respecting, with separate confirmations for the asymptotic and wrapping parts. However, in the large-$N$ regime, the expected simple inheritance from the cusp anomaly breaks down at four loops: $P$ acquires nontrivial subleading terms like $\log^2 N / N^2$, indicating a loss of simplicity in the $P$-kernel despite reciprocity holding. The results support reciprocity as a robust, though not universally simplifying, organizing principle, consistent with strong-coupling AdS/CFT insights and analogous higher-loop twist results.

Abstract

The four loop universal anomalous dimension of twist-2 operators in N=4 SYM has been recently conjectured. In this paper, we prove that it obeys a generalized Gribov-Lipatov reciprocity, previously known to hold at the three loop level.