Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory

TL;DR

This work introduces a non-perturbative hexagon bootstrap for structure constants in planar $ ext{N}=4 ext{ SYM}$, recasting a three-point function as two hexagon patches glued along three seams. The central axiom is that hexagon form factors factorize into a dynamical part built from two-particle factors $h(u_i,u_j)$ and a matrix part given by Beisert’s $SU(2|2)^2$ S-matrix, with a finite-sum over bipartitions of Bethe roots and mirror-state corrections accounting for finite-size effects. The authors provide an all-loop asymptotic formula for rank-one sectors, derive leading finite-size corrections, and perform extensive weak- and strong-coupling tests against perturbative OPE data and classical string results, respectively. The framework reproduces known weak- and strong-coupling data, offering a scalable, integrable route to arbitrary three-point functions and hinting at extensions to higher-point correlators and connections to broader bootstrap programs.

Abstract

We introduce a non-perturbative framework for computing structure constants of single-trace operators in the N=4 SYM theory at large N. Our approach features new vertices, with hexagonal shape, that can be patched together into three- and possibly higher-point correlators. These newborn hexagons are more elementary and easier to deal with than the three-point functions. Moreover, they can be entirely constructed using integrability, by means of a suitable bootstrap program. In this letter, we present our main results and conjectures for these vertices, and match their predictions for the three-point functions with both weak and strong coupling data available in the literature.

Figures (24)

• Figure 1: A pair of pants cut into two hexagons. Each closed spin chain operator is split into two open chains. Its excitations can end up on either half. We should sum over those possibilities. Stitching the hexagons back into the pair of pants amounts to integrating over all possible states at the gluing segments. The whole construction is reminiscent of the standard folklore: $\text{closed string}=(\text{open string})^2$.
• Figure 2: When cutting the pair of pants into two hexagons, we should keep record of the structure of the Bethe wave function on each cuff. More technically, it means that, for each chain/string, we should sum over all possible bipartite partitions $\alpha, \bar{\alpha}$ of the set of magnons' rapidities $\{u\}=\alpha \cup \bar{\alpha}$. The weight $w(\alpha,\bar{\alpha})$ dressing each term in this sum ought to take into account that for an excitation to end on the second hexagon it needs to propagate through the partial length $\ell$ of the (corresponding edge of the) first hexagon and, potentially, scatter with other excitations along the way, that is $w(\alpha,\bar{\alpha}) = \prod_{ u_j \in \bar{\alpha}} (e^{i p(u_j) \ell} \prod_{u_i \in \alpha \text{ with } i > j} S(u_j,u_i) )$. At the level of the asymptotic wave function, this is the same cutting procedure as discussed in section 3.1 of tailoring1 at tree level.
• Figure 3: There are two kinds of finite size corrections in a three-point correlator. One is the standard wrapping corrections which encircle each of the three operators Ambjorn:2005wa. One such correction is indicated at the top on the left. They yield corrections of order $\mathcal{O}(g^{2L_i})$ which can be dropped for long operators. Then we have a new kind of wrapping effect corresponding to virtual excitations propagating from an hexagon to another or, equivalently, from the inside to the outside of the right diagram. These effects are of order $\mathcal{O}(g^{2l_{ij}})$ and can be dropped provided the bridge length $l_{ij}$ between operators $\mathcal{O}_i$ and $\mathcal{O}_j$ is large. In particular, it is important to note that these new wrapping corrections typically show up earlier than the conventional ones. A more detailed analysis will show that they can appear as early as 2 loops, compared to 4 loops for usual wrapping effects.
• Figure 4: A mirror transformation $\gamma: u\to u^\gamma$ moves an excitation to a neighbouring edge. As illustrated here on a simple example, we can iterate it to relate a creation amplitude $h$ with all particles at the top to the most general hexagon process $H$ where excitations can inhabit any of the six edges.
• Figure 5: The multi-particle conjecture relates the hexagon creation amplitude to a multi-particle scattering process as depicted here.