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.
Highlights
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
Integrand-reduction methods [6,20], instead, allow one to decompose the integrands of scattering amplitudes are into multi-particle poles, and the multi-particle residues are expressed in terms of irreducible scalar products formed by the loop momenta and either external momenta or polarization vectors constructed out of them
We elaborate on the four-dimensional helicity (FDH) scheme [28,40,41] and we propose a fourdimensional formulation (FDF) of the d-dimensional regularization scheme which allows for a purely four-dimensional regularization of the amplitudes
Summary
The recent development of novel methods for computing one-loop scattering amplitudes has been highly stimulated by a deeper understanding of their multi-channel factorization properties in special kinematic conditions enforced by on-shellness [1,2,3] and generalized unitarity [4,5], strengthened by the complementary classification of the mathematical structures present in the residues at the singular points [6,7,8,9,10,11]. Using basic principles of algebraic geometry [7,8,21,22,23], have shown that the structure of the multi-particle poles is determined by the zeros of the denominators involved in the corresponding multiple cut This new approach to integrand reduction methods allows for their systematization and for their all-loop extension. In many cases generalized unitarity in arbitrary non-integer dimensions is avoided and cut-constructible and rational terms are obtained in separate steps The former are computed by performing four-dimensional generalized cuts in the un-regularized amplitudes. The space-time dimensions have to admit an explicit representation of the Clifford algebra [32] This idea has been combined with the six-dimensional helicity formalism [39] for the analytic reconstruction of one-loop scattering amplitudes in QCD via generalized unitarity.
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