Higher loops, integrability and the near BMN limit

TL;DR

This paper explores higher-loop corrections to the dilatation operator in N=4 SYM, demonstrating how integrability and string theory constraints shape its form and lead to conjectures about all-loop energies in the near BMN limit.

Contribution

It provides a detailed analysis of higher-loop contributions in the su(2) sector, proposing an all-loop expression for energies based on integrability and string theory constraints.

Findings

Higher-loop integrability constrains the dilatation operator up to four loops.

An all-loop conjecture for the energy in the near BMN limit is proposed.

Differences between integrability-based and string theory-based constraints are discussed.

Abstract

In this note we consider higher-loop contributions to the planar dilatation operator of N=4 SYM in the su(2) subsector of two complex scalar fields. We investigate the constraints on the form of this object due to interactions of two excitations in the BMN limit. We then consider two scenarios to uniquely fix some higher-loop contributions: (i) Higher-loop integrability fixes the dilatation generator up to at least four-loops. Among other results, this allows to conjecture an all-loop expression for the energy in the near BMN limit. (ii) The near plane-wave limit of string theory and the BMN correspondence fix the dilatation generator up to three-loops. We comment on the difference between both scenarios.