Determinant Formula for the Octagon Form Factor in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory

We present a closed expression for the octagon form factor which appears as a building block in a class of four-point correlation functions in $\mathcal{N}=4$ supersymmetric Yang-Mills theory considered recently by Coronado. The octagon form factor is expressed, to all loop orders, as the determinant of a semi-infinite matrix. We find that perturbatively at weak coupling the entries of this matrix are linear combinations of ladder functions with simple rational coefficients.

Figure 1

A sketch of the octagon ${\mathbb{O}}_{\ell }$. The red lines symbolize the mirror particles propagating between the two hexagons, each one characterized by a rapidity $u$ and a bound state number $a$. The two hexagons are separated by a bridge of length $\ell$.