Permutations and the Loop

TL;DR

This paper derives a group-theoretic expression for the one-loop two-point function of multi-trace operators in the U(2) sector of =4 SYM, revealing specific mixing conditions at finite N.

Contribution

It provides a novel formula linking the one-loop correlator to U(N) and symmetric group data, clarifying operator mixing in the multi-trace sector.

Findings

Operators only mix at one loop if their Young diagrams differ by adding a single box.

The derived expression involves group theory data from U(N) and symmetric groups.

Results suggest similar mixing patterns may hold at higher loops and other sectors.

Abstract

We consider the one-loop two-point function for multi-trace operators in the U(2) sector of \cN=4 supersymmetric Yang-Mills at finite N. We derive an expression for it in terms of U(N) and S_{n+1} group theory data, where n is the length of the operators. The Clebsch-Gordan operators constructed in 0711.0176, which are diagonal at tree level, only mix at one loop if you can reach the same (n+1)-box Young diagram by adding a single box to each of the n-box Young diagrams of their U(N) representations (which organise their multi-trace structure). Similar results are expected for higher loops and for other sectors of the global symmetry group.