One-Loop Corrections from Higher Dimensional Tree Amplitudes

TL;DR

The paper shows how to obtain four-dimensional one-loop amplitudes from forward limits of five-dimensional tree amplitudes by integrating out two undetected legs and reducing to four dimensions, all within the CHY framework. It identifies regular and singular sectors in the one-loop forward limit, proves that unphysical poles and ambiguities are confined to scaleless terms, and demonstrates that the resulting integrands are equivalent to ambitwistor-string formulations up to scaleless pieces. The analysis includes both scalar and gluon loops, clarifies ambiguities in the singular sector, and connects the CHY construction to Q-cut approaches. The results provide a regulator-friendly, dimensionally-augmented route to one-loop amplitudes and suggest extensions to gravity and broader theories.

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.