5 research outputs found
Numerical Loop-Tree Duality: contour deformation and subtraction
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.Comment: 87 page
Loop Tree Duality for multi-loop numerical integration
Loop Tree Duality (LTD) offers a promising avenue to numerically integrate
multi-loop integrals directly in momentum space. It is well-established at one
loop, but there have been only sparse numerical results at two loops. We
provide a formal derivation for a novel multi-loop LTD expression and study its
threshold singularity structure. We apply our findings numerically to a diverse
set of up to four-loop finite topologies with kinematics for which no contour
deformation is needed. We also lay down the ground work for constructing such a
deformation. Our results serve as an important stepping stone towards a
generalised and efficient numerical implementation of LTD, applicable to the
computation of virtual corrections.Comment: 13 page
Numerical integration of loop integrals through local cancellation of threshold singularities
We propose a new approach that allows for the separate numerical calculation of the real and imaginary parts of finite loop integrals. We find that at one-loop the real part is given by the Loop-Tree Duality integral supplemented with suitable counterterms and the imaginary part is a sum of two-body phase space integrals, constituting a locally finite representation of the generalised optical theorem. These expressions are integrals in momentum space, whose integrands were specially designed to feature local cancellations of threshold singularities. Such a representation is well suited for Monte Carlo integration and avoids the drawbacks of a numerical contour deformation around remaining singularities. Our method is directly applicable to a range integrals with certain geometric properties but not yet fully generalised for arbitrary one-loop integrals. We demonstrate the computational performance with examples of one-loop integrals with various kinematic configurations, which gives promising prospects for an extension to multi-loop integrals.ISSN:1126-6708ISSN:1029-847