Hadronic production of a Higgs boson in association with two jets at ...

2.6. The unitarity-bootstrap 59deformed around the discontinuity (here referred to as B),∮1 A(z)+ 1 ∮A(z)2πi z 2πi z . (2.100)B ↑ +iǫB ↓ −iǫSince A(z) has a non-vanishing discontinuity along B,one may write2πi Disc B A(z) = A(z + iǫ) − A(z − iǫ) (2.101)0 = A(0) + ∑poles αRes z=zαA(z)z+ ∑ Disc B∫Bdzz Disc BA(z), (2.102)this is the structure of a one-loop amplitude under a BCFW shift. Of course, onemust take special care if a pole is located along a branch cut, here must one movethe pole away from the branch cut by a small amount δ calculate the branch cutterms separately and take the limit δ goes to zero at the end of the calculation.2.6.2 Cut-constructible, cut-completion, rational and overlaptermsEq. (2.102) describes the properties of one-loop amplitudes under a generic BCFWshift (which vanishes at ∞). However, as been described in detail in this chapter,we have simple and generic methods to extract the cut-constructible pieces of oneloopamplitudes. Ideally we would wish to only extract the rational pieces from arecursion relation, which are missed by our four-dimensional methods. Since thesepieces contain no discontinuities they should have simpler recursive properties thanthe whole one-loop amplitude.There is one complication however since rational and cut-constructible piecesneed to communicate with each other to ensure correct factorisation. Specificallywe will use the following basis functions which arise from the reduction of tensortriangles,L i (s, t) =1log (s/t). (2.103)(s − t)iThese terms become singular for i > 1 as s → t, since this is a non-physical singularityit must be cancelled by some part of the rational piece. To aid in the stability