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Loop Tree Duality for multi-loop numerical integration

Zeno Capatti, Valentin Hirschi, Dario Kermanschah, Ben Ruijl

TL;DR

The paper derives a new multi-loop Loop Tree Duality (LTD) representation by iteratively applying the residue theorem while preserving Feynman causal prescription. It analyzes the singular surfaces governing dual propagators, establishing dual cancellations for H-surfaces and identifying E-surfaces that may require contour deformation. Numerical validation across eight finite scalar topologies up to four loops demonstrates agreement with known results at the sub-percent level and showcases the practicality of LTD for momentum-space integration. The work also lays foundational steps toward a general contour deformation scheme and discusses future directions for integrating LTD with real-emission contributions to improve virtual corrections computations. Overall, this study advances a scalable, deformation-aware LTD framework for higher-loop numerical computations in quantum field theory.

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.

Loop Tree Duality for multi-loop numerical integration

TL;DR

The paper derives a new multi-loop Loop Tree Duality (LTD) representation by iteratively applying the residue theorem while preserving Feynman causal prescription. It analyzes the singular surfaces governing dual propagators, establishing dual cancellations for H-surfaces and identifying E-surfaces that may require contour deformation. Numerical validation across eight finite scalar topologies up to four loops demonstrates agreement with known results at the sub-percent level and showcases the practicality of LTD for momentum-space integration. The work also lays foundational steps toward a general contour deformation scheme and discusses future directions for integrating LTD with real-emission contributions to improve virtual corrections computations. Overall, this study advances a scalable, deformation-aware LTD framework for higher-loop numerical computations in quantum field theory.

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.

Paper Structure

This paper contains 17 sections, 30 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Loop Tree Duality formalism
  3. Singular surfaces
  4. Discussion of previous work
  5. Numerical application
  6. General kinematics
  7. Conclusion
  8. Explicit example of a three-loop cut structure
  9. External kinematics considered for our numerical results
  10. Topology a)* and b): 1-loop deca- and triacontagon
  11. Topology c) and c)*: two-loop six-point diagram
  12. Topology d)*: two-loop eight-point diagram
  13. Topology e): four-loop four-point ladder diagram
  14. Topology f): three-loop two-point Mercedes diagram
  15. Topology g)*: three-loop six-point Mercedes diagram
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Numerical LTD results obtained for the scalar massless one-, two- and three-loop ladder box diagram with external kinematics satisfying (in GeV$^2$) $p_1^2=-5$, $p_{i=2,3,4}^2=s=-1$ and values of the Mandelstam invariant $t$ ranging from $t=-7$ (loop threshold) to $t=100$. The analytic results are taken from ref. Usyukina:1992jd.
  • Figure 2: A 2-point 3-loop ladder diagram and its reduced diagram, obtained by merging propagator lines that share identical signatures and removing external legs.