Self-duality of massless scalar three-point amplitudes

TL;DR

This paper proves that off-shell massless scalar three-point Feynman integrals are self-dual under Fourier transformation, enabling their expression as graphical functions and revealing new identities in quantum field theory.

Contribution

It generalizes Jiang's 2025 observation to all off-shell massless scalar three-point integrals, establishing their self-duality and deriving new identities for graphical functions.

Findings

Off-shell massless scalar three-point integrals are self-dual under Fourier transform.

Any such integral can be expressed as a graphical function.

New identities and twist relations for scalar integrals in $\

Abstract

We prove that off-shell massless scalar three-point Feynman integrals are self-dual under Fourier transformation. This implies that a momentum space integral can be expressed as the position space integral of the same Feynman graph with transformed edge-weights (not the dual graph) if external vertices are labeled accordingly. In particular, any off-shell massless scalar three-point Feynman integral can be expressed as a graphical function. The result follows immediately from a theorem by M. Golz, E. Panzer and the author on parametric representations of position space integrals (2015), but it was only observed by X. Jiang in 2025 in the context of four-dimensional $\mathcal{N}=4$ Super-Yang-Mills theory. We generalize Jiang's result and discuss the consequences of the self-duality in the context of graphical functions. In particular, we derive a new identity for graphical functions and a new twist relation for scalar integrals (Feynman periods) in $\phi^4$ theory.