On the singular behaviour of scattering amplitudes in quantum field theory

Sebastian Buchta,Petros Draggiotis,Germán Rodrigo,Ioannis Malamos,Grigorios Chachamis

doi:10.1007/jhep11(2014)014

Abstract

We analyse the singular behaviour of one-loop integrals and scattering amplitudes in the framework of the loop--tree duality approach. We show that there is a partial cancellation of singularities at the loop integrand level among the different components of the corresponding dual representation that can be interpreted in terms of causality. The remaining threshold and infrared singularities are restricted to a finite region of the loop momentum space, which is of the size of the external momenta and can be mapped to the phase-space of real corrections to cancel the soft and collinear divergences.

Highlights

JHEP11(2014)[014] at the same time
We show that there is a partial cancellation of singularities at the loop integrand level among the different components of the corresponding dual representation that can be interpreted in terms of causality
The remaining threshold and infrared singularities are restricted to a finite region of the loop momentum space, which is of the size of the external momenta and can be mapped to the phase-space of real corrections to cancel the soft and collinear divergences

Summary

The singular behaviour of the loop integrand

In Cartesian coordinates, the Feynman propagator in (2.2) becomes singular at hyperboloids with origin in −ki, where the minimal distance between each hyperboloid and its origin is determined by the internal mass mi This is illustrated, where for simplicity we work in d = 2 space-time dimensions. In the massless case, (2.7) and (2.8) are the equations of conic sections in the loop three-momentum space; qi(,+0) and qj(,+0) are the distance to the foci located at −ki and −kj, respectively, and the distance between the foci is k2ji. The solution to (2.7) is an ellipsoid and clearly requires kji,0 < 0 Since it is the result of the intersection of a forward with a backward hyperboloid the distance between the two propagators has to be future-like, kj2i ≥ 0. The singularity appears for loop three-momenta with imaginary components

Cancellation of singularities among dual integrands

Cancellation of infrared singularities with real corrections

Conclusions and outlook