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

4.4. Higgs plus four gluon amplitudes 125+ 〈24〉〈34〉〈3|p H|1][41]3s 124 s 12 [42]− [12]2 〈23〉 2− 〈24〉(s )}23s 24 + s 23 s 34 + s 24 s 34 )s 14 [42] 2 3〈12〉〈14〉[23][34][42]{ }+ (2 ↔ 4) . (4.34)4.4 Higgs plus four gluon amplitudesIn this section we present complete expressions for the one-loop amplitudes neededto calculate the process 0 → Hgggg at NLO.The one-loop amplitudes presented here are computed in the four-dimensionalhelicity scheme and are not renormalised. To perform an MS renormalisation, oneshould subtract an MS counterterm (in the t’Hooft-Veltman scheme) from A (1)4 ,A (1)4 → A (1)4 − c Γ 2 β 0ǫ A(0) 4 . (4.35)The Wilson coefficient eq. (1.53) produces an additional finite contribution,A (1)4 → A (1)4 + 11(4π) 2 A(0) 4 . (4.36)We choose to split the un-renormalised amplitude into (completed) cut-constructiblepieces and rational terms. We also separate the infra-red divergent and finite partsof the amplitude. The basis functions for the finite part of the cut-constructablepieces are one-mass and two-mass boxes, three-mass triangles, and completed functionsˆL i (s, t) of eq. (4.30). We define the finite pieces of the box and three-masstriangle integrals in Appendix B.We express a generic helicity configuration in the following formA (1)4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4) = c Γ (C 4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4) + R 4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4)),(4.37)where C 4 represents the cut-constructible part of the amplitude and R 4 the rationalpieces. We further separate C 4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4) into divergent and finite pieces,C 4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4) = V 4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4) + F 4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4). (4.38)