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

4.2. Cut-Constructible Contributions 117R 4 (φ, 1 + , 2 − , 3 − , 4 − ) is derived in section 4.3. Throughout this chapter we use thefollowing expression for the φ-NMHV tree amplitudeA (0)n (φ, 1+ , 2 − , 3 − , 4 − ) =−m 4 φ 〈24〉4s 124 〈12〉〈14〉〈2|p φ |3]〈4|p φ |3] + 〈4|p φ |1] 3s 123 〈4|p φ |3][12][23] − 〈2|p φ |1] 3s 134 〈2|p φ |3][14][34] . (4.3)This compact form can be derived using the BCFW recursion relations [140, 141]and agrees numerically with the previously known expression derived from MHVrules [95]. It clearly possesses the correct symmetry properties under the exchange{2 ↔ 4}, and factors onto the correct gluon tree amplitude (which is zero) in thelimit of vanishing p φ . Other tree amplitudes needed in this chapter can be found inAppendix A.4.2 Cut-Constructible ContributionsAs in chapter 3, we employ the generalised unitarity method [120,122,123,127,128,132] to calculate the cut-constructible parts of the one-loop amplitude. We canfurther decompose C 4 in (4.1) into a sum over constituent basis integrals,C 4 (φ, 1 + , 2 − , 3 − , 4 − ) = ∑ iC 4;i I 4;i + ∑ iC 3;i I 3;i + ∑ iC 2;i I 2;i . (4.4)As usual I j;i represents a j-point scalar basis integral, with a coefficient C j;i . Thesum over i represents the sum over the partitions of the external momenta over thej legs of the basis integral.As in previous chapters we use the methods of generalised unitarity to extractthe various coefficients of the basis integrals, four-cuts for boxes [120], triple cutsfor triangles [122] and double cuts for bubbles [132].4.2.1 Box Integral CoefficientsWe begin our calculation of the φ-NMHV amplitude by computing the coefficients ofthe scalar boxes using generalised unitarity with complex momenta [120]. In general