Eur. Phys. J. C (2014) 74: 3197
https://doi.org/10.1140/epjc/s10052-014-3197-4

Regular Article - Theoretical Physics

On the four-dimensional formulation of dimensionally regulated amplitudes

¹ Departamento de Física, Universidad Nacional de Colombia, Ciudad Universitaria, Bogotá, D.C., Colombia
² Dipartimento di Fisica e Astronomia, Università di Padova, and INFN, Sezione di Padova, via Marzolo 8, 35131, Padua, Italy
³ Max-Planck-Institut für Physik, Föhringer Ring 6, 80805, Munich, Germany

^* e-mail: pierpaolo.mastrolia@cern.ch

Received: 16 September 2014
Accepted: 24 November 2014
Published online: 18 December 2014

Abstract

Elaborating on the four-dimensional helicity scheme, we propose a pure four-dimensional formulation (FDF) of the $$d$$-dimensional regularization of one-loop scattering amplitudes. In our formulation particles propagating inside the loop are represented by massive internal states regulating the divergences. The latter obey Feynman rules containing multiplicative selection rules which automatically account for the effects of the extra-dimensional regulating terms of the amplitude. We present explicit representations of the polarization and helicity states of the four-dimensional particles propagating in the loop. They allow for a complete, four-dimensional, unitarity-based construction of $$d$$-dimensional amplitudes. Generalized unitarity within the FDF does not require any higher-dimensional extension of the Clifford and the spinor algebra. Finally we show how the FDF allows for the recursive construction of $$d$$-dimensional one-loop integrands, generalizing the four-dimensional open-loop approach.