A three-loop test of the dilatation operator in N=4 SYM

TL;DR

This work develops and applies a superspace-based method to test the three-loop dilatation operator in N=4 SYM by focusing on the BMN family, specifically the Konishi (J=0) and J=1 multiplets. By using superspace differentiation and ratios of descendants’ two-point functions, the authors effectively reduce the loop order to two for the extraction of higher-loop anomalous dimensions, carefully addressing quantum anomalies and operator mixing. They perform explicit perturbative calculations up to three loops, obtaining gamma values that precisely match the BMN-scaling-compatible dilatation-operator predictions: for the Konishi multiplet, $\gamma_1 = 3N/(4\pi^2)$, $\gamma_2 = -3N^2/(4\pi^2)^2$, $\gamma_3 = 21N^3/(4\pi^2)^3$; for the BMN $J=1$ multiplet, $\gamma_1 = 2N/(4\pi^2)$, $\gamma_2 = -3N^2/(2(4\pi^2)^2)$, $\gamma_3 = 17N^3/(8(4\pi^2)^3)$. These results provide nontrivial, nonplanar-consistent evidence supporting BMN scaling at three loops and help fix the form of the three-loop dilatation operator in Beisert’s framework. The methods and conclusions reinforce the integrability-based dilatation operator program and point toward broader checks at higher twists or spins.

Abstract

We compute the three-loop anomalous dimension of the BMN operators with charges J=0 (the Konishi multiplet) and J=1 in N=4 super-Yang-Mills theory. We employ a method which effectively reduces the calculation to two loops. Instead of using the superconformal primary states, we consider the ratio of the two-point functions of suitable descendants of the corresponding multiplets. Our results unambiguously select the form of the N=4 SYM dilatation operator which is compatible with BMN scaling. Thus, we provide evidence for BMN scaling at three loops.