Universal opening of four-loop scattering amplitudes to trees

Abstract

The perturbative approach to quantum field theories has made it possible to obtain incredibly accurate theoretical predictions in high-energy physics. Although various techniques have been developed to boost the efficiency of these calculations, some ingredients remain specially challenging. This is the case of multiloop scattering amplitudes that constitute a hard bottleneck to solve. In this Letter, we delve into the application of a disruptive technique based on the loop-tree duality theorem, which is aimed at an efficient computation of such objects by opening the loops to nondisjoint trees. We study the multiloop topologies that first appear at four loops and assemble them in a clever and general expression, the N 4 MLT universal topology. This general expression enables to open any scattering amplitude of up to four loops, and also describes a subset of higher order configurations to all orders. These results confirm the conjecture of a factorized opening in terms of simpler known subtopologies, which also determines how the causal structure of the entire loop amplitude is characterized by the causal structure of its subtopologies. In addition, we confirm that the loop-tree duality representation of the N 4 MLT universal topology is manifestly free of non-causal threshold singularities, thus pointing towards a remarkably more stable numerical implementation of multi-loop scattering amplitudes.

Geometrical approach to causality in multiloop amplitudes

“…a connected L-loop diagram with k + 1 cusps and L + k propagators as defined in Refs. [51][52][53]56]. We conjecture that its causal representation is given by…”

“…In this Section, we briefly explain the notation used to describe any generic L-loop scattering amplitude with N external legs following the ideas presented in Refs. [51][52][53]56]. First, we define L primitive loop momenta, { i } i=1,...,L , which exactly corresponds to the integration variables.…”

“…The application of the nested residues lead to a collection of diagrams with as many on-shell cuts as loops, in such a way that each loop diagram is open into a sum of non-disjoint trees. After adding together all the dual terms, the final result only involves same-sign combinations of on-shell energies in the denominator: these are the so-called causal propagators [51][52][53]56].…”

“…Using the notation of Refs. [51][52][53]56] a N k−1 MLT topological family has complexity k: MLT describes any one and two loop amplitude, N 2 MLT describes any possible three loop amplitude, and so on. Σ contains all the subsets of products of k causal propagators λ ± i which fulfill certain compatibility criteria; σ is obtained from σ by the replacement λ ± j ↔ λ ∓ j .…”

“…Regarding the casual structure of scattering amplitudes, there are several previous studies using different techniques [44][45][46][47]. Very recently, LTD was applied to remove unphysical threshold singularities and obtain a manifestly causal integrand-level definition of multi-loop scattering amplitudes [48][49][50][51][52][53][54]. All these LTD-based techniques were implemented in an automatized framework [55].…”