Asymptotic Four Point Functions

TL;DR

This work tackles four-point functions of large BPS operators in planar $\ ext{N}=4$ SYM by combining an OPE decomposition with integrability, enabling a nonperturbative handle on these correlators. The authors cast the problem into a sum over superconformal primaries and compute the coupling-dependent structure constants via an integrable framework that merges Beisert-Staudacher Bethe roots with the hexagon formalism, extended to nested (higher-rank) sectors. A central result is a compact all-loop formula for the asymptotic structure constants in higher rank sectors, expressed in terms of wing Gaudin norms, a partition sum $\mathcal{A}$, and a Pfaffian/determinant reformulation, with a Yangian symmetry enforcing a root-by-root equality between left and right wings. They validate the construction by matching tree-level and one-loop OPE data and discuss implications for strong coupling, potential AdS bulk locality, and the relationship between this approach and hexagonalization, highlighting a path toward a more complete, nonperturbative understanding of four-point functions in AdS/CFT.

Abstract

We initiate the study of four-point functions of large BPS operators at any value of the coupling. We do it by casting it as a sum over exchange of superconformal primaries and computing the structure constants using integrability. Along the way, we incorporate the nested Bethe ansatz structure to the hexagon formalism for the three-point functions and obtain a compact formula for the asymptotic structure constant of a non-BPS operator in a higher rank sector.

Figures (13)

• Figure 1: Various contributions to the 4pt function.
• Figure 2: $PSU(2,2|4)$ Dynkin diagram and Bethe roots.
• Figure 3: Splitting a two-particle state with indices. When the first particle passes through the second particle, it gets multiplied by a S-matrix $S(u_1,u_2)$. The resulting state is a complicated object which includes a summation over indices ($C$ and $D$ in the figure).
• Figure 4: Matrix part for $\bar{\alpha}=\varnothing$: One can simply act the S-matrix to the right wave function $\psi_{R}$ and simplify the structure. Note we are using the normalization of the matrix part, in which the abelian part ($\mathfrak{sl}(2)$$S$-matrix) is unity.
• Figure 5: Matrix part for $\bar{\alpha}\neq \varnothing$: The magnons for the right wing are in a different order from those in the wave function. To contract the wave function with the hexagon, we have to rewrite it using the "nested periodicity" \ref{['nestedperiodicity']}.