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

2.6. The unitarity-bootstrap 572.6.1 BCFW at one-loopSince the rational pieces of one-loop amplitudes contain no discontinuities they havethe same kinematic structure as tree-level amplitudes. This lead to the realisationthat they could be calculated, in principle, using BCFW recursion relations [161].In the previous section the proof of BCFW recursion relations was sketched, herewe consider the integral over the shifted amplitude A(z),I = 1 ∮A(z)2πi z . (2.96)CThe contour C is taken around the circle at infinity, if, as was required in the proof,A(z) → 0 as z → ∞, we find the following result,0 = A(0) + ∑poles αRes z=zαA(z)z . (2.97)If, on the other hand there is a contribution from A(z) as z → ∞ (equal to C ∞ ),the relation would beA(0) = C ∞ − ∑poles αRes z=zαA(z)z . (2.98)For one-loop amplitudes many of the properties of the shifted amplitude A(z) changefrom those at tree-level. For example the proof in the previous section required thatthe amplitude contained only simple poles which arose from internal propagatorsgoing on-shell. At one-loop the situation changes and logarithms appear whichpossess discontinuities and are defined on the complex plane with a branch cut. Thedifferences between the complex planes for tree-level and one-loop amplitudes areshown schematically in Fig. 2.4 and Fig. 2.5. The BCFW recursion relations mustbe altered to work at one-loop [161–163]. We begin by assuming once again thatfor a specific shift A(z) → 0 as z → ∞. We then consider the following vanishingintegral which is taken along the circle at ∞ and deformed such that it moves aroundthe branch cuts.0 = 1 ∮A(z)2πi C z . (2.99)The integral, although it still vanishes, is no longer equal to the sum of residuesof the simple poles. We must also include the contribution from the line integrals