Abstract
We extend useful properties of the H → γγ unintegrated dual amplitudes from one- to two-loop level, using the Loop-Tree Duality formalism. In particular, we show that the universality of the functional form — regardless of the nature of the internal particle — still holds at this order. We also present an algorithmic way to renormalise two-loop amplitudes, by locally cancelling the ultraviolet singularities at integrand level, thus allowing a full four-dimensional numerical implementation of the method. Our results are compared with analytic expressions already available in the literature, finding a perfect numerical agreement. The success of this computation plays a crucial role for the development of a fully local four-dimensional framework to compute physical observables at Next-to-Next-to Leading order and beyond.
Highlights
The Loop-Tree Duality at two loopsThe Loop-Tree Duality (LTD) theorem [21,22,23] transforms any loop integral or loop scattering amplitude into a sum of tree-level like objects that are constructed by setting on shell a number of internal loop propagators equal to the number of loops
Basis of integrals is known and their evaluation has been implemented in several codes
We remark that the calculation of this amplitude is the first two-loop application to a physical process done through Loop-Tree Duality (LTD), and it is computed below the mass threshold limit in the MS renormalisation scheme
Summary
The Loop-Tree Duality (LTD) theorem [21,22,23] transforms any loop integral or loop scattering amplitude into a sum of tree-level like objects that are constructed by setting on shell a number of internal loop propagators equal to the number of loops. It is interesting to note that the integration over the loop three-momenta is unrestricted, after analysing the singular behaviour of the loop integrand one realises that thanks to a partial cancellation of singularities among different dual components, all the physical threshold and IR singularities remain confined to a compact region of the loop three-momentum [40, 41] This relevant fact allows to construct mappings between the virtual and real kinematics, which are based on the factorisation properties of QCD, to implement the summation over degenerate soft and collinear states for physical observables in the Four-Dimensional Unsubtraction (FDU) formalism [31, 32, 42].
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