One-Loop Corrections from Higher Dimensional Tree Amplitudes

Abstract

We show how one-loop corrections to scattering amplitudes of scalars and gauge bosons can be obtained from tree amplitudes in one higher dimension. Starting with a complete tree-level scattering amplitude of n+2 particles in five dimensions, one assumes that two of them cannot be "detected" and therefore an integration over their LIPS is carried out. The resulting object, function of the remaining n particles, is taken to be four-dimensional by restricting the corresponding momenta. We perform this procedure in the context of the tree-level CHY formulation of amplitudes. The scattering equations obtained in the procedure coincide with those derived by Geyer et al from ambitwistor constructions and recently studied by two of the authors for bi-adjoint scalars. They have two sectors of solutions: regular and singular. We prove that the contribution from regular solutions generically gives rise to unphysical poles. However, using a BCFW argument we prove that the unphysical contributions are always homogeneous functions of the loop momentum and can be discarded. We also show that the contribution from singular solutions turns out to be homogeneous as well.

21 pages v.2: ref added, typos fixed

References in corpus (8)

• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• One-loop amplitudes on the Riemann sphere
• From loops to trees by-passing Feynman's theorem
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• Loops and trees
• On BCFW shifts of integrands and integrals
• The Q-cut Representation of One-loop Integrands and Unitarity Cut Method

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