Abstract
We extend the four-dimensional unsubtraction method, which is based on the loop-tree duality (LTD), to deal with processes involving heavy particles. The method allows to perform the summation over degenerate IR configurations directly at integrand level in such a way that NLO corrections can be implemented directly in four space-time dimensions. We define a general momentum mapping between the real and virtual kinematics that accounts properly for the quasi-collinear configurations, and leads to an smooth massless limit. We illustrate the method first with an scalar toy example, and then analyse the case of the decay of a scalar or vector boson into a pair of massive quarks. The results presented in this paper are suitable for the application of the method to any multipartonic process.
Highlights
In order to make these divergences manifest explicitly, the standard approach relies in the introduction of a convenient regularisation method
We extend the four-dimensional unsubtraction method, which is based on the loop-tree duality (LTD), to deal with processes involving heavy particles
We present the first analytical application of LTD with massive particles
Summary
We summarise the key concepts of the LTD theorem at one-loop. So, let’s consider a generic one-loop scalar integral for an N -particle process, as depicted in figure 1. Is the scalar Feynman propagator associated to a virtual particle with mass mi and fourmomentum qi,μ = (qi,0, qi) (qi,0 is the energy and qi are the spatial components). According to the LTD theorem, any loop contribution to scattering amplitudes in any relativistic, local and unitary quantum field theory can be computed through dual integrals, which are built from single cuts of the virtual diagrams at one-loop [28]. Considering the one-loop scalar integral, the LTD theorem establishes that its dual representation is given by. Assuming that there are only single powers of the Feynman propagators, the dual representation in eq (2.6) is straightforwardly valid for loop scattering amplitudes. The single-cuts do not affect numerators, the dual representation of scattering amplitudes is obtained by adding all possible dual single-cuts of the original loop diagram, and replacing the uncut Feynman propagators by dual propagators. The explicit form of the scattering amplitude is relevant because the numerator is affected by the derivative
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