On the four-dimensional formulation of dimensionally regulated amplitudes

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 fourdimensional open-loop approach.

To $${d}$$ d , or not to $${d}$$ d : recent developments and comparisons of regularization schemes

“…Its aim is to achieve the ddimensional regularization of one-loop scattering amplitudes in a purely four-dimensional framework [32]. The starting point for the formulation of the scheme is the structure of the quasi-d s -dimensional fdh space, Eq.…”

“…Using the Feynman rules of Ref. [32] together with the (−2 )-SRs, we obtain for the case of massless fermions…”

BCJ identities and d-dimensional generalized unitarity

“…One of most striking implications of the C/K duality is the existence of relations between color-ordered tree-level amplitudes [1] which, together with U(1) symmetry and Kleiss-Kuijf relations [6], can be used to further reduce the number of independent partial amplitudes to be considered in tree-level calculations. In [7], by adopting the FourDimensional-Formulation (FDF) [8] variant of the Four-Dimensional-Helicity (FDH) [9][10][11] regularization scheme, we studied the C/K-duality for tree-level amplitudes in d dimensions and we derived a set of BCJ identities, for four-and five-point amplitudes, which take into account the explicit dependence on the regulating parameter, together with a general strategy for the determination of analogous relations between higher-multiplicity amplitudes.…”

γ5 in the four-dimensional helicity scheme

“…FDF is a novel regularization approach that was introduced to reproduce FDH results at the one-loop level [21].…”

“…[16]. Among the considered dimensional schemes are the 't Hooft-Veltman scheme (HV) [1], conventional dimensional regularization (CDR) [17], dimensional reduction (DRED) [18], the four-dimensional helicity scheme (FDH) [19,20], and its recently proposed four-dimensional formulation (FDF) [21] at one loop.…”