Light-cone limits of large rectangular fishnets

Basso-Dixon integrals evaluate rectangular fishnets – Feynman graphs with massless scalar propagators which form a $m\times n$ rectangular grid – which arise in certain one-trace four-point correlators in the `fishnet’ limit of $\mathcal{N}=4$ SYM. Recently, Basso {\it et al} explored the thermodynamical limit $m\to \mathrm{\infty }$ with fixed aspect ratio $n\mathrm{/}m$ of a rectangular fishnet and showed that in general the dependence on the coordinates of the four operators is erased, but it reappears in a scaling limit with two of the operators getting close in a controlled way. In this note I investigate the most general double scaling limit which describes the thermodynamics when one of two pairs of operators become nearly light-like. In this double scaling limit, the rectangular fishnet depends on both coordinate cross ratios. I show that all singular limits of the fishnet can be attained within the double scaling limit, including the null limit with the four points approaching the cusps of a null square. A direct evaluation of the fishnet in the null limit is presented any $m$ and $n$.