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

5.1. Introduction 136H amplitude φ amplitude φ † amplitudeA(H, 1 −¯q , 2 + q , 3 + g , 4 + g ) A(φ, 1 −¯q , 2 + q , 3 + g , 4 + g ) [106,109] A(φ † , 1 −¯q , 2 + q , 3 + g , 4 + g )A(H, 1 −¯q , 2 + q , 3− g , 4− g ) A(φ, 1−¯q , 2 + q , 3− g , 4− g ) A(φ† , 1 −¯q , 2 + q , 3− g , 4− g ) [106,109]A(H, 1 −¯q , 2 + q , 3 + g , 4 − g ) A(φ, 1 −¯q , 2 + q , 3 + g , 4 − g ) [109] A(φ † , 1 −¯q , 2 + q , 3 + g , 4 − g ) [109]A(H, 1 −¯q , 2 + q , 3 − g , 4 + g ) A(φ, 1 −¯q , 2 + q , 3 − g , 4 + g ) [109] A(φ † , 1 −¯q , 2 + q , 3 − g , 4 + g ) [109]Table 5.1: φ and φ † amplitudes needed to construct a given one-loop H¯qqgg amplitude,together with the references where they can be obtained. In all cases the φ †amplitudes are constructed from the φ amplitudes given in the reference, using theparity operation. The cases where the gluons have the same helicity, which have noassociated references, are the subject of this chapter.− iA f 4 (φ, 1−¯q , 2+ q , 3+ g , 4+ g ) = 1 〈1 3〉 [3 4] 〈4 1〉3 〈1 2〉 〈3 4〉 2 , (5.14)whilst the subleading partial amplitude A 4;3 also requires the results,−iA L 4 (φ, 1−¯q , 2+ g , 3+ q , 4+ g ) = 〈1 3〉 2 [3 4]2 〈1 2〉 〈2 3〉 〈3 4〉 − 2〈1 |p φ| 4] 2〈1 2〉 〈2 3〉 s 123− 〈1 3〉 〈1 |p φ| 2]2 〈2 3〉 〈3 4〉 〈4 1〉 ,(5.15)A f 4 (φ, 1−¯q , 2+ , 3 + q , 4+ g ) = 0. (5.16)To obtain the form that is most useful for the calculation of A(H, 1 −¯q , 2 + q , 3− g , 4− g ),we relate the φ †¯qqgg amplitudes to the φ¯qqgg ones by using the relation in Eq. (5.10).Thus we obtain the required results by performing the transformation 1 ↔ 2, 3 ↔ 4,〈〉 ↔ [] and reversing the sign. The amplitudes contributing to A 4;1 are,− iA L 4 (φ † , 1 −¯q , 2 + q , 3 − g , 4 − g ) = 2i A (0)4 (φ, 1 −¯q , 2 + q , 3 − g , 4 − g )+ 1 [− 〈3 |p ]φ| 2] [2 1] 〈1 4〉 [2 4]+ − 1 [2 4] 〈3 4〉 [2 3]2 [1 4][3 4] [1 4][3 4][2 3] 3 [1 2][3 4] 2 . (5.17)−iA R 4 (φ† , 1 −¯q , 2+ q , 3− g , 4− g ) = 1 [ ]〈3 |pφ | 2] [2 1] 〈1 4〉[2 4]− , (5.18)2 [1 4][3 4] [1 4][3 4][2 3]−iA f 4 (φ† , 1 −¯q , 2+ q , 3− g , 4− g ) = 1 [2 4] 〈3 4〉 [2 3]3 [1 2][3 4] 2 , (5.19)while the additional subleading contributions become,[]−iA L 4 (φ† , 1 −¯q , 3− g , 2+ q , 4− g ) = 2 〈3 |p φ | 2] 2+ 1 [2 1] 〈4 |p φ | 2][2 4][4 1] s 124 2 [4 1][1 3][3 2] − 3 [1 2]2 〈1 3〉[2 4][4 1][1 3]