Tilings and Twist at 1/N^4

TL;DR

This paper investigates operator mixing in the SU(2) sector of N=4 super Yang-Mills theory, focusing on parity pairs with three excitations, and uses integrability methods to match large N eigenstates up to order 1/N^4, revealing insights into twist effects and admixtures.

Contribution

It introduces a detailed analysis of operator mixing and eigenstate construction in the SU(2) sector, incorporating twist effects and higher-trace admixtures using integrability techniques.

Findings

Reproduced tree-level norms of large N eigenstates up to 1/N^4.

Identified the impact of twist on double-trace admixtures and two-point functions.

Found success for transverse scalars but limitations for longitudinal magnons.

Abstract

We re-consider operator mixing in the so-called $SU(2)$ sector of ${\cal N} \, = \, 4$ super Yang-Mills theory with gauge group $SU(N)$. Where possible, single-trace operators of moderate length are completed by higher-trace admixtures so as to yield large $N$ tree level eigenstates. We are particularly interested in parity pairs with three excitations. Since parity is respected in the mixing, the odd single-trace operators at low length cannot receive too many admixtures. We reproduce the tree-level norms of a set of large $N$ eigenstates up to order $1/N^4$ by integrability methods. This involves evaluating two-point functions on the sphere, the torus, and the double-torus. A perfect match is found as long as descendents are absent from the mixing. Using twist to make the descendents appear in the integrability picture immediately leads to the question how to modify the entangled states occurring in the hexagon tessellations. We take a closer look at the double-trace admixtures to the parity even three-excitation operator at length seven, which are both products of a primary state and a descendent. Their two-point functions are sensitive to the twist introduced into the Bethe equations. For transverse scalar excitations we succeed in recovering the corresponding field theory results. For longitudinal magnons our methods fail, pointing at a potential weakness of the formalism.