# Integrands of loop amplitudes within loop-tree duality

## TL;DR

This paper explores the use of loop-tree duality to relate complex loop amplitudes in quantum field theory to more manageable phase-space integrals, enabling easier computation up to three loops.

## Contribution

It introduces a method to express loop amplitudes as phase-space integrals of UV-subtracted tree-like objects, with a recursive approach for calculations up to three loops.

## Key findings

- Loop amplitudes can be related to phase-space integrals via loop-tree duality.
- Up to three loops, the relevant objects are computable using recurrence relations.
- The method provides a global definition of loop momenta and incorporates on-shell renormalisation.

## Abstract

Using loop-tree duality, we relate a renormalised $n$-point $l$-loop amplitude in a quantum field theory to a phase-space integral of a regularised $l$-fold forward limit of a UV-subtracted $(n+2l)$-point tree-amplitude-like object. We show that up to three loops the latter object is easily computable from recurrence relations. This defines an integrand of the loop amplitude with a global definition of the loop momenta. Field and mass renormalisation are performed in the on-shell scheme.