# Three-point functions at strong coupling in the BMN limit

## TL;DR

This paper calculates three-point functions involving BMN operators at strong coupling in planar N=4 SYM using the hexagon formalism, revealing explicit factorization and connections to string theory structures.

## Contribution

It explicitly evaluates sums over virtual magnons at strong coupling, demonstrating their factorization into Gamma functions and stringy prefactors, and explores the transition to the classical regime.

## Key findings

- Structure constants factorize into Gamma functions and stringy prefactors.
- Poles and zeros in structure constants relate to operator mixing and supersymmetry.
- Connections to Neumann coefficients in string field theory are observed.

## Abstract

We consider structure constants of single-trace operators at strong coupling in planar $\mathcal{N}=4$ SYM theory using the hexagon formalism. We concentrate on heavy-heavy-light correlators where the heavy operators are BMN operators, with large R-charges and finite anomalous dimensions, and the light one is a finite-charge chiral primary operator. They describe the couplings between two highly boosted strings and a supergravity mode in the bulk dual. In the hexagon framework, two sums over virtual magnons are needed to bind the hexagons together around the light operator. We evaluate these sums explicitly at strong coupling, for a certain choice of BMN operators, and show that they factorise into a ratio of Gamma functions and a simple stringy prefactor. The former originates from giant mirror magnons scanning the AdS geometry while the latter stems from small fluctuations around the BMN vacuum. The resulting structure constants have poles at positions where an enhanced mixing with double-trace operators is expected and zeros whenever the process is forbidden by supersymmetry. We also discuss the transition to the classical regime, when the length of the light operator scales like the string tension, where we observe similitudes with the Neumann coefficients of the pp-wave String Field Theory vertex.