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Self-duality of massless scalar three-point amplitudes

Oliver Schnetz

Abstract

We prove that massless scalar three-point amplitudes are self-dual under Fourier transformation. This implies that the momentum space amplitude can be expressed as the position space amplitude of the same graph with transformed edge-weights (not the dual graph) if external vertices are labeled accordingly. In particular, a massless scalar three-point 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 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 $φ^4$ theory.

Self-duality of massless scalar three-point amplitudes

Abstract

We prove that massless scalar three-point amplitudes are self-dual under Fourier transformation. This implies that the momentum space amplitude can be expressed as the position space amplitude of the same graph with transformed edge-weights (not the dual graph) if external vertices are labeled accordingly. In particular, a massless scalar three-point 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 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 theory.
Paper Structure (5 sections, 1 theorem, 55 equations, 3 figures)

This paper contains 5 sections, 1 theorem, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Integer dimensions
  3. Parametric representation
  4. A new identity for graphical functions
  5. A new identity for scalar integrals in $\phi^4$ theory

Key Result

Lemma 1

Let $1\leq D\in{\mathbb Z}$ and let $G$ be a connected weighted graph with three external vertices $0,z_1,z_2$ and $|\mathcal{V}_G^{\mathrm{int}}|$ internal vertices such that the position space Feynman integral $A_G(0,z_1,z_2)$ in (AG) exists. If $N_G=D/2$ (see (eqNG)), then $A_G(0,z_1,z_2)$ is inv

Figures (3)

  • Figure 1: The three external vertices $z_0,z_1,z_2$ span the complex plane (picture by M. Borinsky).
  • Figure 2: The momentum routing of a three-point integral $A_G^p(p_1,p_2)$ and the corresponding position space integral $A_G(0,p_1,p_2)$. Note that the weights are connected via $\lambda\nu_e=\lambda+1-\nu_e^p$; see (\ref{['nunu']}). Except for the weights, the graphs $G^p$ and $G$ are identical.
  • Figure 3: The bubble and the tetrahedron are the smallest primitive graphs in $\phi^4$ theory.

Theorems & Definitions (2)

  • Lemma 1
  • proof