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

3.3. The rational pieces 1053.3.5 Combined rational piecesCombining contributions, the full four-point amplitude is given by,A (1)4 (φ, 1 − , 2 + , 3 − , 4 + ) = C 4 (φ, 1 − , 2 + , 3 − , 4 + ) + CR 4 (φ, 1 − , 2 + , 3 − , 4 + )+ ˆR 4 (φ, 1 − , 2 + , 3 − , 4 + ), (3.145)withˆR(φ, 1 − , 2 + , 3 − , 4 + ) = O 4 (φ, 1 − , 2 + , 3 − , 4 + ) + R 4 (φ, 1 − , 2 + , 3 − , 4 + )− InfCR 4 (φ, 1 − , 2 + , 3 − , 4 + ). (3.146)After some algebra, the combination of overlapping and recursive terms can bewritten in the following form, free of spurious singularities 1 ,ˆR 4 (φ, 1 − , 2 + , 3 − , 4 + ) = −2A (0) (A, 1 − , 2 + , 3 − , 4 + )+ N P [2 4] 4 (6 [1 2][2 3][3 4][4 1]− s 23s 34+ 3 s 23s 34− s 12s 41+ 3 s 12s 41s 24 s 412 s 2 24 s 24 s 234 s 2 24where A (0) (A, 1 − , 2 + , 3 − , 4 + ) is the difference of φ and φ †),(3.147)amplitudes. We havechecked this amplitude against a Feynman diagram calculation and found agreement.Finally the full Higgs amplitude is given by the sum of φ and φ † amplitudesA (1)4 (H, 1 − , 2 + , 3 − , 4 + ) = A (1)4 (φ, 1 − , 2 + , 3 − , 4 + ) + A (1)4 (φ † , 1 − , 2 + , 3 − , 4 + ),(3.148)with,A (1)4 (φ † , 1 − , 2 + , 3 − , 4 + ) = A (1)4 (φ, 2 − , 3 + , 4 − , 1 + ) 〈i j〉↔[i j] . (3.149)We note that the rational terms not proportional to N P in eq. (3.147) cancel whenforming the Higgs amplitude, just as for the A (1)4 (H, 1 − , 2 − , 3 + , 4 + ) amplitude ofref. [108].1 Which we have checked with the aid of the package S@M [204]