On the Four-Dimensional Formulation of Dimensionally Regulated Amplitudes

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. 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 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.

Four Dimensional Feynman Rules for gauge theories bare one-loop dimensionally regularized diagrams The external legs are treated as usual four dimensional states.
◮ The pure Yang-Mills (YM) loop propagators in Feynman-'t Hooft gauge The scalars come from a dimensional reduction of D = 4 − 2ǫ dimensional gluons vector fields.
In D = 4 − 2ǫ dimensions we perform the decomposition of the loop momentumk α in a 4-dimensional part k α and in its orthogonal complement the −2ǫ-dimensional fixed vector µ ᾱ k α = k α + µ α µ α µ α = −µ 2 where the A and B label the components of the complementary space of dimension D − 4. The metric G AB and the vector Q A needed to reformulate the Feynman rules satisfy ◮ Fermion propagator in a loop Dirac matrices have the following splittinḡ γ α = γ α +γ α and satisfy in D dimensions the Clifford algebra A possible 4-dimensional representation ofγ matrices is in terms of γ 5 by the replacement By imposing the rule Q A Γ A = 1 needed to recover / µ / µ = −µ 2 , ◮ Selection rules (−2ǫ SRs) or about the algebra in the D − 4 dimensional complementary space.
In the −2ǫ-dimensional vector space the following rules completely define our four dimensional formulation in agreement with the Four Dimensional Helicity scheme up to spurious terms as explicitly checked in reproducing the integrand numerator of QCD amplitudes of the following processes Generalized Internal legs ◮ Generalized subluminal Dirac equation. Given the ℓ four dimensional vector ◮ Solutions of the generalized Dirac equation ◮ Polarization sum of the solutions of the generalized Dirac equation The hatted are the shifted complex momenta and P * shows that the amplitude has been stripped of his external spinor wave function.

PROOF OF THE COMPLETNESS RELATIONS
Chirality projectors and we show that: and similarly In Arnowitt-Fickler gauge the helicity sum of the transverse D-dimensional polarization vectors is From the gauge invariance in D dimensions the choice of the fixed D-dimensional gauge vector ◮ Generalized Polarization Vectors Once again let us decompose the massive four-dimensional vector (ℓ 2 = µ 2 ) ℓ α = ℓ ♭ α +q α ℓ the µ-massive polarizations vectors are with the usual Proca's completness relation The numerator of cut propagator of the scalar can be expressed in terms of the (−2ǫ)-SRs: The factorĜ AB can be easily accounted for by defining the cut propagator as The previous Feynman rules and cuts prescriptions fully reconstruct the µ 2 dependence of the ǫ− dimensional numerator of scattering amplitudes of renormalized gauge theories.
Four point massless one-loop color ordered amplitudes A 4 From the reduction theorem a dimensionally regularized A 4 is decomposed in a cut-constructible part and in a rational part (R) expressed in terms of scalar integrals in D = 4 − 2ǫ dimensions. The coefficients c i are rational functions of the external momenta and polarizations.
By the separation and using polar coordinates in the −2ǫ dimensional Euclidean vector space, all the integrals in R can be computed. In particular We found a way of computing the rational part of scattering amplitudes by unitarity cuts with loop momenta in D = 4.
The all helicity-plus four gluons planar amplitude with a gluonic loop In order to reconstruct the full µ dependence and to obtain by cut construction the rational coefficients of the master integral decomposition, the following color-ordered trees amplitudes are needed. With all outgoing complex momenta The box coefficients are obtained by the following attaching procedure, with the external legs of the trees on the generalized mass-shell   .