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

3.2. The cut-constructible parts 70[124,165] (see section 2.6), and checked using Feynman diagrams. For simplicity,we focus our efforts on the four point amplitude A (1)4 (φ, 1− , 2 + , 3 − , 4 + ) which we areultimately interested in.The tree-level φ-MHV amplitude has the following form [95],A (0)n (φ, 1− , 2 + , . . ., m − , (m + 1) + , . . .,n + ) =〈1m〉 4∏ n−1(3.1)α=1〈α(α + 1)〉〈n1〉.Here we refer to the colour stripped primitive amplitude. The details of how toobtain colour dressed φ plus parton amplitudes are given in Appendix A. We willdecompose the loop amplitude into cut-constructible and rational pieces C n and R nrespectively, which are defined as follows,A (1)n (φ, 1− , 2 + , . . ., m − , . . ., n + ) = c Γ(C n (φ, 1 − , 2 + , . . .,m − , . . .,n + ))+R n (φ, 1 − , 2 + , . . .,m − , . . .,n + ) , (3.2)wherec Γ = Γ2 (1 − ǫ)Γ(1 + ǫ)(4π) 2−ǫ Γ(1 − 2ǫ) . (3.3)We will also use the following notation to define the cut-completed cut-constructiblepieces Ĉn and the remaining rational terms ̂R n (with the completion terms removed).A (1)n (φ, 1− , 2 + , . . ., m − , . . ., n + ) = c Γ(Ĉ n (φ, 1 − , 2 + , . . .,m − , . . .,n + ))+ ̂R n (φ, 1 − , 2 + , . . .,m − , . . .,n + ) . (3.4)In the following section we calculate the cut-constructible part Ĉn for all n, thenin section 4.3 we use the BCFW recursion relations to calculate the rational contribution̂R n , which we check against a Feynman diagram calculation. Finally insection 3.4 we justify the calculations by performing extensive collinear and softchecks on the amplitude.3.2 The cut-constructible partsIn this section we will use generalised unitarity methods to calculate C n whichappears in eq. (3.2). In general the cut-constructible pieces have the following basis