Computing Four-Point Functions with Integrability, Bootstrap and Parity Symmetry

TL;DR

This paper develops a numerical bootstrap approach combining integrability, crossing symmetry, and parity to accurately bound and compute four-point functions in $ ext{N}=4$ SYM, especially at higher spectrum levels and weak coupling.

Contribution

It introduces an adapted semidefinite programming bootstrap method incorporating parity symmetry to improve bounds on four-point functions in $ ext{N}=4$ SYM.

Findings

Achieves high-precision bounds on four-point functions at weak coupling.

Demonstrates the effectiveness of parity symmetry in reducing parameter space.

Provides numerical results with up to 11-digit accuracy at strong coupling.

Abstract

The combination of integrability and crossing symmetry has proven to give tight non-perturbative bounds on some planar structure constants in $\mathcal{N}$=4 SYM, particularly in the setup of defect observables built on a Wilson-Maldacena line. Whereas the precision is good for the low lying states, higher in the spectrum it drops due to the degeneracies at weak coupling when considering a single correlator. As this could be a clear obstacle in restoring higher point functions, we studied the problem of bounding directly a 4-point function at generic cross ratio, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. Another tool we are using to further narrow the bounds is a parity symmetry descending from the $\mathcal{N}$=4 SYM theory, which allowed us to reduce the number of parameters. We also give an interpretation for the parity in terms of the Quantum Spectral Curve at weak coupling. Our numerical bounds give an accurate determination of the 4-point function for physical values of the cross ratio, with at worst 5-6 digits precision at weak coupling and reaching more than 11 digits for 't Hooft coupling $\frac{\sqrt{\lambda}}{4 \pi} \sim 4$.