The Dilatation Operator of Conformal N=4 Super Yang-Mills Theory

TL;DR

This work reframes perturbative anomalous dimensions in N=4 SYM as eigenvalues of the dilatation operator, enabling algebraic diagonalization instead of laborious two-point-function renormalization. It derives the one-loop ($D_2$) and two-loop ($D_4$) dilatation generators for pure scalar operators, yielding numerous new anomalous dimensions and clarifying large-N degeneracies. The authors reveal planar integrability, uncover a planar axial symmetry connecting parity pairs, and construct higher spin charges, arguing for all-loop integrability in the planar limit and proposing BMN-consistent all-loop relations. They also explore BMN limits, non-planar effects, and the implications of integrability for the gauge/string duality, suggesting that the dilatation operator approach may unlock exact planar spectra and new spin-chain deformations in N=4 SYM.

Abstract

We argue that existing methods for the perturbative computation of anomalous dimensions and the disentanglement of mixing in N = 4 gauge theory can be considerably simplified, systematized and extended by focusing on the theory's dilatation operator. The efficiency of the method is first illustrated at the one-loop level for general non-derivative scalar states. We then go on to derive, for pure scalar states, the two-loop structure of the dilatation operator. This allows us to obtain a host of new results. Among these are an infinite number of previously unknown two-loop anomalous dimensions, new subtleties concerning 't Hooft's large N expansion due to mixing effects of degenerate single and multiple trace states, two-loop tests of various protected operators, as well as two-loop non-planar results for two-impurity operators in BMN gauge theory. We also put to use the recently discovered integrable spin chain description of the planar one-loop dilatation operator and show that the associated Yang-Baxter equation explains the existence of a hitherto unknown planar ``axial'' symmetry between infinitely many gauge theory states. We present evidence that this integrability can be extended to all loops, with intriguing consequences for gauge theory, and that it leads to a novel integrable deformation of the XXX Heisenberg spin chain. Assuming that the integrability structure extends to more than two loops, we determine the planar three-loop contribution to the dilatation operator.

Figures (5)

• Figure 1: One-loop Green functions. The diagrams display the combinatorial structure with respect to the gauge group rather than their space-time configuration. The solid, wiggly, dashed lines represent scalars, gluons, fermions, respectively. The dotted lines correspond to a local interaction of four scalars/gluons.
• Figure 2: Sample two-loop Green functions.
• Figure 3: Two-loop Green functions with three pairs of legs.
• Figure 4: Graphical representation of the planar one-loop interaction $D_2$.
• Figure 5: Graphical representation of the generator $U_2$.