Bootstrapping form factor squared in ${\cal N}=4$ super-Yang-Mills

Abstract

We propose a bootstrap program for the {\it form factor squared} with operator ${\rm tr}(\phi^2)$ in maximally supersymmetric Yang-Mills theory in the planar limit, which plays a central role for perturbative calculations of important physical observables such as energy correlators. The tree-level $N$-point form factor (FF) squared can be obtained by cutting $N$ propagators of a collection of two-point ``master diagrams" at $(N{-}1)$ loops: for $N=3,4,5,6$ there are merely $1, 2, 4, 13$ topologies of such diagrams respectively, and their numerators are strongly constrained by power-counting (including ``no triangle" property) and other constraints such as the ``rung rule". Moreover, these two-point diagrams provide a ``unification" of FF squared at different numbers of loops and legs, which is similar to extracting (planar) amplitude squared from vacuum master diagrams (dual to $f$-graphs): by cutting $2\leq n<N$ propagators, one can also extract the planar integrand of $n$-point FF squared at $(N-n)$ loops, thus our results automatically include integrands of 2-point (Sudakov) FF up to four loops (where the squaring is trivial), 3-point FF squared up to three loops, and so on. Our ansatz is completely fixed using soft limits of (tree and loop) FF squared and the multi-collinear limit which reduces it to the splitting function, without any other inputs such as unitarity cuts. This method opens up the exciting possibility of a {\it graphical bootstrap} for FF squared for higher $N$ (which contains {\it e.g.} planar Sudakov FF to $N{-}2$ loops) similar to that for the amplitude squared via $f$-graphs. We also comment on applications to the computation of leading order energy correlators where new structures are expected after performing phase-space integrations.