A Tree-Loop Duality Relation at Two Loops and Beyond

TL;DR

This work extends the loop–tree duality, originally linking one-loop integrals to phase-space integrals, to two and more loops by introducing a dual $i0$ prescription that suppresses multiple cuts. Through an iterative application of the duality, the authors derive explicit dual representations for two- and three-loop scalar master integrals, employing dual propagators $G_D$ and their set-based generalizations, and discuss how to manage the external-m momentum–dependent prescription to avoid momentum-flow ambiguities. They demonstrate the approach with concrete two-loop results (including the Sunrise example) and three-loop topologies (basket, zigzag, ladder, Mercedes star), showing that the loop diagram can be opened to a tree-level structure while accounting for the required cuts. The framework paves the way for recasting virtual corrections in a form closely aligned with real emissions, potentially aiding NNLO and higher-order cross-section calculations, while highlighting subtleties such as higher-order poles and gauge choices at higher loops.

Abstract

The duality relation between one-loop integrals and phase-space integrals, developed in a previous work, is extended to higher-order loops. The duality relation is realized by a modification of the customary +i0 prescription of the Feynman propagators, which compensates for the absence of the multiple-cut contributions that appear in the Feynman tree theorem. We rederive the duality theorem at one-loop order in a form that is more suitable for its iterative extension to higher-loop orders. We explicitly show its application to two- and three-loop scalar master integrals, and we discuss the structure of the occurring cuts and the ensuing results in detail.

Figures (5)

• Figure 1: Momentum configuration of the one--loop $N$--point scalar integral.
• Figure 2: Location of the particle poles of the Feynman (left) and advanced (right) propagators $G_F(q_i)$ and $G_A(q_i)$ in the complex plane of the variable $q_{i,0}$.
• Figure 3: Momentum configuration of the two--loop $N$--point scalar integral.
• Figure 4: Master topologies of three--loop scalar integrals. Each internal line $\alpha_k$ can be dressed with an arbitrary number of external lines, which are not shown here.
• Figure 5: Sunrise two--loop two--point function.