# Loop Tree Duality for multi-loop numerical integration

## TL;DR

This paper develops a formal multi-loop Loop Tree Duality (LTD) framework, enabling direct momentum space integration of complex multi-loop integrals, and demonstrates its potential for efficient numerical calculations of virtual corrections.

## Contribution

It provides a novel multi-loop LTD expression, analyzes its singularity structure, and applies it to up to four-loop topologies without contour deformation, advancing numerical methods in quantum field theory.

## Key findings

- Successfully applied LTD to four-loop topologies
- Identified threshold singularity structures in multi-loop LTD
- Laid groundwork for contour deformation in LTD

## Abstract

Loop Tree Duality (LTD) offers a promising avenue to numerically integrate multi-loop integrals directly in momentum space. It is well-established at one loop, but there have been only sparse numerical results at two loops. We provide a formal derivation for a novel multi-loop LTD expression and study its threshold singularity structure. We apply our findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed. We also lay down the ground work for constructing such a deformation. Our results serve as an important stepping stone towards a generalised and efficient numerical implementation of LTD, applicable to the computation of virtual corrections.