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Abstract
We propose a novel representation of differential scattering cross-sections that locally realises the direct cancellation of infrared singularities exhibited by its so-called real-emission and virtual degrees of freedom. We take advantage of the Loop-Tree Duality representation of each individual forward-scattering diagram and we prove that the ensuing expression is locally free of infrared divergences, applies at any perturbative order and for any process without initial-state collinear singularities. Divergences for loop momenta with large magnitudes are regulated using local ultraviolet counterterms that reproduce the usual Lagrangian renormalisation procedure of quantum field theories. Our representation is especially suited for a numerical implementation and we demonstrate its practical potential by computing fully numerically and without any IR counterterm the next-to-leading order accurate differential cross-section for the process e+e− → doverline{d} . We also show first results beyond next-to-leading order by computing interference terms part of the N4LO-accurate inclusive cross-section of a 1 → 2 + X scalar scattering process.
Highlights
The ever-increasing need for generic and accurate Monte-Carlo simulations for collider experiments spurred the emergence of an entire subfield of the high energy physics community whose research activities are to a large extent motivated by fulfilling this demand
When a supergraph features raised propagators, which in physical theories appear as a result of a self-energy insertion, the naive substitution of propagators with Dirac deltas leads to manifestly ill-defined interference diagrams, as performing the substitution for one of the raised propagators exactly evaluates the remaining repeated propagators on their mass shell
We remind the reader that the contour deformation is not the one constructed for the entire double-triangle supergraph, but it is constructed independently for each loop integral remaining in each Cutkosky cut contribution, so as to have real-valued momenta in the observable function and allow to separately accommodate the complex-conjugated causal prescription applying to loops appearing on the right-hand side of the Cutkosky cut
Summary
The ever-increasing need for generic and accurate Monte-Carlo simulations for collider experiments spurred the emergence of an entire subfield of the high energy physics community whose research activities are to a large extent motivated by fulfilling this demand. Another possible direction is to turn all loop integrals into phase-space integrals that can be performed numerically if the phase-space measure of resolved and unresolved degrees of freedom can be aligned so as to guarantee a local cancellation of all infrared divergences This is the main goal of our paper and, perhaps contrary to the expectations of many, we show that it is possible to write the differential cross-section for an arbitrary process (without initial-state singularities) and at any perturbative order as an expression that is locally free of any IR singularities. The LU representation realises local IR cancellation by construction so that there is no need for explicitly listing and regularising all singular limits and their overlaps, thereby rendering it de facto valid for arbitrary perturbative orders, both conceptually and in the practical context of numerical computations We view this characteristic as unique to Local Unitarity and it is directly responsible for LU’s universal applicability as the formalism does not depend on the particular theory, scattering process or observables considered.
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