Abstract
We explain how one-loop amplitudes with massive fermions can be computed using only on-shell information. We first use the spinor-helicity formalism in six dimensions to perform generalised unitarity cuts in d dimensions. We then show that divergent wave-function cuts can be avoided, and the remaining ambiguities in the renormalised amplitudes can be fixed, by matching to universal infrared poles in 4 − 2ϵ dimensions and ultraviolet poles in 6 − 2ϵ dimensions. In the latter case we construct an effective Lagrangian in six dimensions and reduce the additional constraint to an on-shell tree-level computation.
Highlights
With three jets [4]
The obstacle is that the traditional approach to renormalisation requires the amputation of wavefunction graphs, and the addition of counterterm diagrams
Naive attempts to amputate wavefunction graphs in generalised unitarity are precluded by the presence of an on-shell propagator, leading to a factor 1/0: this is depicted explicitly in figure 1, where the on-shell tree amplitude appearing on the right hand side of a two-particle cut is expanded to reveal a divergent propagator inside
Summary
We review the basics of one-loop integrand parametrisations in d dimensions. The amplitudes we will consider in this paper are QCD amplitudes with one massive fermion flavour The amplitudes Bn(1) can be written in the usual integrand basis of irreducible scalar products including extra dimensional terms following the OPP [25]/EGKM [26, 27] constructions, Bn(1) = μ2R. The irreducible numerators k · wx;i1,...,is can be constructed using the spurious directions of van Neerven and Vermaseren [28] and vanish after integration. Μ2 = −k · kis the extra dimensional scalar product where k = k + k These give rise to dimension shifted integrals which in turn lead to rational terms in d = 4 − 2 dimensions. After elimination of vanishing integrals over the spurious directions, the d-dimensional representation of the amplitude is, Bn(1)(d, ds) =.
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