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 paper, 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 N4MLT 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 N4MLT universal topology is manifestly free of noncausal thresholds, thus pointing towards a remarkably more stable numerical implementation of multiloop scattering amplitudes.
Highlights
JHEP04(2021)129 features that convert it into a promising technique for tackling higher-order computations
This pattern was explicitly proven for a series of multiloop topologies, the maximal loop topology (MLT), next-to-maximal (NMLT) and next-to-next-to-maximal (N2MLT) that are characterized by L + 1, L + 2 and L + 3 sets of propagators, respectively, with each set categorized by the dependence on a specific loop momentum or a linear combination of the L independent loop momenta
We have analized the multiloop topologies that appear for the first time at four loops and have found a general representation, the N4MLT universal topology, which describes their opening to nondisjoint trees through the loop-tree duality
Summary
The LTD representation is obtained by integrating out one degree of freedom per loop through the Cauchy residue theorem. This results in a modification of the infinitesimal complex prescription of the Feynman propagators [18], that needs to be considered carefully to preserve the causal structure of the amplitude. The Cauchy countours are always closed on the lower half plane such that the poles with negative imaginary components are selected This is implemented through the future-like vector η that selects which components of the loop momenta are integrated. More details can be found in refs. [24, 50, 53]
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