Causal representation of multi-loop Feynman integrands within the loop-tree duality

TL;DR

This paper develops a method to express multi-loop Feynman integrals using only causal propagators within the loop-tree duality framework, simplifying their structure and enabling efficient numerical evaluation up to four loops.

Contribution

It introduces a way to represent multi-loop Feynman integrals solely with causal propagators, removing unphysical singularities and facilitating numerical computations.

Findings

Explicit dual representations for three topologies

Implicit cancellation of unphysical singularities

Numerical integration up to four loops

Abstract

The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.