The Bethe-Ansatz for N=4 Super Yang-Mills

TL;DR

This paper derives the one-loop anomalous dimension matrix in N=4 Super Yang-Mills, showing its integrability via the Bethe ansatz and providing exact and approximate results for various operators.

Contribution

It introduces the integrable SO(6) spin chain model for computing anomalous dimensions and applies the Bethe ansatz to obtain explicit results for BMN operators.

Findings

Exact anomalous dimensions for BMN operators with two impurities

First-order 1/J corrections for operators with many impurities

Anomalous dimension proportional to the square root of string level in the weak coupling limit

Abstract

We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.