# Correlation Functions on the Half-BPS Wilson Loop: Perturbation and Hexagonalization

## TL;DR

This paper computes correlation functions of protected operators on the 1/2-BPS Wilson loop in N=4 SYM at weak coupling, confirming results with localization and proposing an integrability-based hexagonalization approach involving boundary states.

## Contribution

It introduces a novel method to compute Wilson loop correlators using hexagonalization with boundary states, extending integrability techniques to Wilson loop contexts.

## Key findings

- Perturbative one-loop calculations match localization results.
- Conjecture that correlators can be computed via hexagon boundary state contractions.
- Predictions for large-charge asymptotics of structure constants.

## Abstract

We compute correlation functions of protected primaries on the $1/2$-BPS Wilson loop in ${\cal N}$ = 4 super Yang-Mills theory at weak coupling. We first perform direct perturbative computation at one loop in the planar limit and present explicit formulae for general two-, three- and four-point functions. The results for two- and three-point functions as well as four-point functions in special kinematics are in perfect agreement with the localization computation performed in arXiv:1802.05201. We then analyze the results in view of the integrability-based approach called "hexagonalization", which was introduced previously to study the correlation functions in the absence of the Wilson loop. In this approach, one decomposes the correlator into fundamental building blocks called "hexagons", and glues them back together by summing over the intermediate states. Through the comparison, we conjecture that the correlation functions on the Wilson loop can be computed by contracting hexagons with boundary states, where each boundary state represents a segment of the Wilson loop. As a byproduct, we make predictions for the large-charge asymptotics of the structure constants on the Wilson loop. Along the way, we refine the conjecture for the integrability-based approach to the general non-BPS structure constants on the Wilson loop, proposed originally in arXiv:1706.02989.