Higher-rank sectors in the hexagon formalism and marginal deformations

The hexagon approach provides an integrability framework for the computation of structure constants in $\mathcal{N}=4$ super Yang--Mills theory in four dimensions. Three-point functions are cut into two hexagonal patches, on which the excitations of the long-range Bethe ansatz of the spectrum problem scatter. To this end, the Bethe states representing the operators also need to be cut into two parts. In rank-one sectors such entangled states are fairly straightforward to construct so that most applications of the method have so far been restricted to this simplest case. In this article we construct entangled states for operators in $psu(1,1|2)$ sectors, importing a minimum of information from the nested Bethe ansatz. The idea is successfully tested against free field theory for a sample set of correlators with up to three higher-rank operators. Further, we take a look at the same correlators in the presence of marginal deformations of the theory. While a systematic modification of the hexagon procedure remains out of reach for now, in practical applications the undeformed amplitudes are surprisingly efficient, especially for a certain one-parameter deformation.