Abstract
We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.
Highlights
The Large Hadron Collider (LHC) is entering its high luminosity data acquisition phase and is transitioning from being a discovery experiment to a precision measurement one
We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space
Contour deformations for numerical integration have been considered in the past [71, 75, 84], and we present a novel variant well-suited to our multiloop Loop-Tree Duality [76–78] (LTD) expression
Summary
The Large Hadron Collider (LHC) is entering its high luminosity data acquisition phase and is transitioning from being a discovery experiment to a precision measurement one. This energy integration yields the Loop-Tree Duality [76,77,78] (LTD) which provides an alternative representation for the loop integral containing terms with as many on-shell constraints as there are loops, making them effectively trees This aligns the measure of phase-space and LTD integrals, making LTD ideally suited to pursue the ambitious goal of directly combining real-emission and virtual contributions and compute them numerically at once by realising the local cancellation of their infrared singularities. We consider divergent scalar box and pentagon topologies and the one-loop correction to the ordered production of two and three photons from a quark line This amplitude involves soft and collinear singularities that correspond to pinched threshold singularities where no regulating contour deformation is allowed.
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