Aguilera-Verdugo, J. J., Hernandez-Pinto, R. J., Rodrigo, G., Sborlini, G. F. R., & Torres Bobadilla, W. J. (2021). Mathematical properties of nested residues and their application to multi-loop scattering amplitudes. J. High Energy Phys., 02(2), 112–42pp.
Abstract: The computation of multi-loop multi-leg scattering amplitudes plays a key role to improve the precision of theoretical predictions for particle physics at high-energy colliders. In this work, we focus on the mathematical properties of the novel integrand-level representation of Feynman integrals, which is based on the Loop-Tree Duality (LTD). We explore the behaviour of the multi-loop iterated residues and explicitly show, by developing a general compact and elegant proof, that contributions associated to displaced poles are cancelled out. The remaining residues, called nested residues as originally introduced in ref. [1], encode the relevant physical information and are naturally mapped onto physical configurations associated to nondisjoint on-shell states. By going further on the mathematical structure of the nested residues, we prove that unphysical singularities vanish, and show how the final expressions can be written by using only causal denominators. In this way, we provide a mathematical proof for the all-loop formulae presented in ref. [2].
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Aguilera-Verdugo, J. J., Driencourt-Mangin, F., Plenter, J., Ramirez-Uribe, S., Rodrigo, G., Sborlini, G. F. R., et al. (2019). Causality, unitarity thresholds, anomalous thresholds and infrared singularities from the loop-tree duality at higher orders. J. High Energy Phys., 12(12), 163–12pp.
Abstract: We present the first comprehensive analysis of the unitarity thresholds and anomalous thresholds of scattering amplitudes at two loops and beyond based on the loop- tree duality, and show how non-causal unphysical thresholds are locally cancelled in an efficient way when the forest of all the dual on-shell cuts is considered as one. We also prove that soft and collinear singularities at two loops and beyond are restricted to a compact region of the loop three-momenta, which is a necessary condition for implementing a local cancellation of loop infrared singularities with the ones appearing in real emission; without relying on a subtraction formalism.
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Aguilera-Verdugo, J. J., Driencourt-Mangin, F., Hernandez-Pinto, R. J., Plenter, J., Ramirez-Uribe, S., Renteria-Olivo, A. E., et al. (2020). Open Loop Amplitudes and Causality to All Orders and Powers from the Loop-Tree Duality. Phys. Rev. Lett., 124(21), 211602–6pp.
Abstract: Multiloop scattering amplitudes describing the quantum fluctuations at high-energy scattering processes are the main bottleneck in perturbative quantum field theory. The loop-tree duality is a novel method aimed at overcoming this bottleneck by opening the loop amplitudes into trees and combining them at integrand level with the real-emission matrix elements. In this Letter, we generalize the loop-tree duality to all orders in the perturbative expansion by using the complex Lorentz-covariant prescription of the original one-loop formulation. We introduce a series of mutiloop topologies with arbitrary internal configurations and derive very compact and factorizable expressions of their open-to-trees representation in the loop-tree duality formalism. Furthermore, these expressions are entirely independent at integrand level of the initial assignments of momentum flows in the Feynman representation and remarkably free of noncausal singularities. These properties, that we conjecture to hold to other topologies at all orders, provide integrand representations of scattering amplitudes that exhibit manifest causal singular structures and better numerical stability than in other representations.
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Aguilera-Verdugo, J. D., Driencourt-Mangin, F., Hernandez-Pinto, R. J., Plenter, J., Prisco, R. M., Ramirez-Uribe, N. S., et al. (2021). A Stroll through the Loop-Tree Duality. Symmetry-Basel, 13(6), 1029–37pp.
Abstract: The Loop-Tree Duality (LTD) theorem is an innovative technique to deal with multi-loop scattering amplitudes, leading to integrand-level representations over a Euclidean space. In this article, we review the last developments concerning this framework, focusing on the manifestly causal representation of multi-loop Feynman integrals and scattering amplitudes, and the definition of dual local counter-terms to cancel infrared singularities.
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