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We propose a pure four-dimensional formulation (FDF) of the ddimensional regularization of one and two loop scattering amplitudes in quantum chromodynamics. In our formulation particles propagating inside the loops are represented by massive internal states regulating the divergences. The equivalence between the FDF and the Four Dimensional Helicity scheme is discussed. We present explicit representations of the polarization and helicity states of the four-dimensional particles propagating in the loops. They allow for a complete, four-dimensional, unitarity-based construction of d-dimensional amplitudes. Generalized unitarity within the FDF does not require any higherdimensional extension of the Cli↵ord and the spinor algebra.

Two loop four dimensional formulation of dimensionally regularized amplitudes in quantum chromodynamics