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

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.

arXiv comment24 pages, 8 figures. v2: references added; matches published version

References in corpus (35)

• Asymptotic expansion of Feynman integrals near threshold
• An automatized algorithm to compute infrared divergent multi-loop integrals
• A novel approach to integration by parts reduction
• FIESTA 4: optimized Feynman integral calculations with GPU support
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop
• Sector Decomposition
• To $d$, or not to $d$: Recent developments and comparisons of regularization schemes
• High-Energy Approximation of One-Loop Feynman Integrals
• From loops to trees by-passing Feynman's theorem
• Loop Tree Duality for multi-loop numerical integration
• Four-dimensional unsubtraction with massive particles
• Chiral Anomaly and CPT invariance in an implicit momentum space regularization framework
• Towards gauge theories in four dimensions
• A Tree-Loop Duality Relation at Two Loops and Beyond
• Four-dimensional unsubtraction from the loop-tree duality
• Open loop amplitudes and causality to all orders and powers from the loop-tree duality
• A four-dimensional approach to quantum field theories
• On the singular behaviour of scattering amplitudes in quantum field theory
• Consistency Relations for an Implicit n-dimensional Regularization Scheme
• Universal four-dimensional representation of $H\to \gamma \gamma$ at two loops through the Loop-Tree Duality
• Numerical Loop-Tree Duality: contour deformation and subtraction
• The Compact Linear e${}^{+}$e${}^{-}$ Collider (CLIC): Physics Potential
• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
• Causality and Unitarity via the Tree-Loop Duality Relation
• Integrands of loop amplitudes within loop-tree duality
• ${\gamma }_5$ in FDH
• Universal dual amplitudes and asymptotic expansions for $gg\to H$ and $H\to \gamma \gamma$ in four dimensions
• Theoretical foundations and applications of the Loop-Tree Duality in Quantum Field Theories
• Rational Terms of UV Origin at Two Loops
• NNLO final-state quark-pair corrections in four dimensions
• Four-dimensional representation of scattering amplitudes and physical observables through the application of the Loop-Tree Duality theorem
• Generalised unitarity for dimensionally regulated amplitudes within FDF
• LHC at 10: the physics legacy

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• Collider Physics at the Precision Frontier
• May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations
• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
• Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order
• Geometrical approach to causality in multi-loop amplitudes
• Loop-tree duality from vertices and edges
• Quantum algorithm for Feynman loop integrals
• Numerical integration of loop integrals through local cancellation of threshold singularities
• Universal opening of four-loop scattering amplitudes to trees
• A scattering amplitudes approach to hard thermal loops
• Rewording Theoretical Predictions at Colliders with Vacuum Amplitudes
• Loop Feynman integration on a quantum computer
• Quantum integration of decay rates at second order in perturbation theory
• A stroll through the loop-tree duality