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Causality and loop-tree duality at higher loops

Robert Runkel, Zoltán Szőr, Juan Pablo Vesga, Stefan Weinzierl

TL;DR

This paper develops a general loop-tree duality that relates an $l$-loop Feynman integral to a sum of $l$-fold phase-space integrals labeled by spanning trees of the original graph. Propagators that remain uncut are endowed with a causality-based dual $i\delta$-prescription, with the sign function $s_j(\sigma)$ determined from the on-shell conditions of the cut edges; the main result reduces to a transparent residue-based formula, and a covariant, frame-independent expression for the dual prescription is provided. Specializing to unit propagator powers yields an explicit representation that facilitates numerical higher-loop calculations, clarifies infrared structure via on-shell tree diagrams, and offers a path toward extending scattering-form ideas from trees to loops. The approach is mass- and spin-agnostic, consistent with causality, and contrasts with other methods (e.g., Q-cut) by using only quadratic propagators and a direct dual prescription. Collectively, these results advance higher-loop computations, improve understanding of singularities, and support potential extensions to loop-level scattering form frameworks.

Abstract

We relate a $l$-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original $l$-loop graph. Causality requires that the propagators of the trees have a modified $iδ$-prescription and we present a simple formula for the correct $iδ$-prescription.

Causality and loop-tree duality at higher loops

TL;DR

This paper develops a general loop-tree duality that relates an -loop Feynman integral to a sum of -fold phase-space integrals labeled by spanning trees of the original graph. Propagators that remain uncut are endowed with a causality-based dual -prescription, with the sign function determined from the on-shell conditions of the cut edges; the main result reduces to a transparent residue-based formula, and a covariant, frame-independent expression for the dual prescription is provided. Specializing to unit propagator powers yields an explicit representation that facilitates numerical higher-loop calculations, clarifies infrared structure via on-shell tree diagrams, and offers a path toward extending scattering-form ideas from trees to loops. The approach is mass- and spin-agnostic, consistent with causality, and contrasts with other methods (e.g., Q-cut) by using only quadratic propagators and a direct dual prescription. Collectively, these results advance higher-loop computations, improve understanding of singularities, and support potential extensions to loop-level scattering form frameworks.

Abstract

We relate a -loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original -loop graph. Causality requires that the propagators of the trees have a modified -prescription and we present a simple formula for the correct -prescription.

Paper Structure

This paper contains 3 sections, 2 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Loop-tree duality

Key Result

Theorem 1

With $f$ as in eq. (def_f) we have where the contour of integration on the left-hand side is along the real axes separating the poles at $+\sqrt{...}$ from the poles at $-\sqrt{...}$.

Figures (2)

  • Figure 1: A two-loop eight-point function with $11$ propagators.
  • Figure 2: A cut diagram corresponding to $\sigma=\{3,9\}$. We also indicate an orientation for the edges $e_3$, $e_5$, $e_6$ and $e_9$.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof