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

3.2. The cut-constructible parts 87We will presently show that A φG,3−cut is helicity blind and as such A φG,3−cutn;1 (2, n) =A φ,G,3−cutn;1 (m, n) = A φ,G,3−cutn;1 (n). It was also shown in [108] that the combinationof A φG,3−cut and A φG,4−cut correctly generates the pole structure of eq. (3.54). Thegeneral structure of one- and two-mass triangles are shown in Fig. 3.9 of these (a)(c) − (f) and (h) represent pieces which only contribute to A φG . As an example weconsider (a) in detail and show that the dependence on p m factors into the tree-levelprefactor. The product of tree amplitudes has the following form,D (a) = A (03 (φ, l − 1 , l − 2 )A (0) (l + 2 , i + , l − 3 )A (0) (l − 3 , (i + 1) + , 1 − , . . ., m − , (i − 1) + , l + 1 )= −〈l 1 l 2 〉 2 [l 2 i] 3〈1m〉 4[il 3 ][l 3 l 2 ] 〈l 3 (i + 1)〉 ∏ i−2α=i+1 〈α(α + 1)〉〈(i − 1)l 1〉〈l 1 l 3 〉= −A (0 〈l 1 l 2 〉 2 [l 2 i] 3 〈i(i + 1)〉〈(i − 1)i〉n[l 3 l 2 ][il 3 ]〈l 3 (i + 1)〉〈(i − 1)l 1 〉〈l 1 l 3 〉 . (3.58)It is trivial to show that diagrams (c) − (f) and (h) factorise in the same manner.As such we know that these integrals will be identical to the adjacent minus φ-MHVcase [108]. This leaves us with the task of determining the coefficients representedby Fig. 3.9(b) which allow both fermions and gluons to propagate in the loops.We follow the same procedure as we did for the box diagrams and decompose thediagram into constituent pieces,(D (b) = G 2,1m (a, i, j) + 4 1 − N )(fF 2,1m (a, i, j) + 2 1 − N )fS 2,1m (a, i, j)4N c N c(3.59)As before a ∈ {1, m} indicates which of the negative helicity gluons is not pairedwith φ at a vertex. Here we do not distinguish explicitly between one- and two-masstriangles (i.e. we consider (b) knowing we can obtain (g) in the soft limit), in theapproach we will use [122] we choose two momenta K 1 and K 2 (which are externalmomenta) and parameterise the loop momentum in terms of massless projectionsof these vectors. In these calculations we can always set K 2 1 = 0 K2 2 ≠ 0 regardlessof whether the triangle has one or two massive legs. A schematic representation ofthe kinematics we will use for the calculation is shown in Fig. 3.8. The masslessprojections of K 1 and K 2 which we will use to construct our basis in which the loopmomentum is decomposed have the following form,K ♭,µ1 = p µ i , (3.60)