The Three-Point Form Factor of $\textrm{Tr}\,\phi^3$ to Six Loops

TL;DR

This paper computes the three-point form factor of a specific operator in planar $ extrm{N}=4$ SYM theory up to six loops, revealing its mathematical structure and behavior in various kinematic limits.

Contribution

It demonstrates that the form factor functions reside in the same polylogarithmic space as the stress-tensor form factor and shows that leading collinear data suffices to determine the form factor to six loops.

Findings

Form factor functions are in the same polylogarithmic space as the stress-tensor form factor.

Leading collinear limit data uniquely determines the form factor up to six loops.

Derived a compact all-orders representation in the Regge limit.

Abstract

We study the three-point form factor of the length-three half-BPS operator ($\textrm{Tr}\,\phi^3$) in planar $\mathcal{N}=4$ Super-Yang-Mills theory, using analyticity and integrability methods. We find that the functions describing the form factor in perturbation theory live in the same restrictive space of multiple polylogarithms as the one describing the form factor of the stress-tensor operator ($\textrm{Tr}\,\phi^2$). Furthermore, we find that the leading-order data in the collinear limit provided by the form factor operator product expansion (FFOPE) is enough to fix the form factor uniquely, at least through six loops. We perform various tests of our results using the subleading FFOPE corrections. We also analyze the form factor in the Regge limit where two Mandelstam invariants are large; we obtain a compact representation for the form factor in this limit which is valid to all orders in the coupling.