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

eplacemen3.2. The cut-constructible parts 72++−−±∓±−−++−−++0000 1111∓±∓+−+−(a) (b) (c)Figure 3.1: Configurations where the coefficient of the four-mass box vanishes, externalparticles are either; positive helicity gluons (solid black lines) of which therecan be an arbitrary number at any vertex, negative helicity gluons (shown in green)of which there can be only two and one φ (dashed black line). In each configuration(a) − (c) we find one zero tree-level amplitude, these are highlighted by the redboxes.One-mass boxesThe one-mass box is simplest and we begin by classifying the two solutions forthe loop momenta l with the general kinematics shown in Fig.3.4. The on-shellconditions for the various loop momenta are as follows,(l + m 1 ) 2 = 0 (3.7)l 2 = 0 (3.8)(l − m 2 ) 2 = 0 (3.9)(l − M 23 ) 2 = 0 (3.10)where we have used the shorthand notation M ij = m i +m j . We choose to expand l interms of two massless basis vectors m 1 and m 2 with four free parameters {α, β, ρ, δ}which are fixed by the on-shell constraints.l µ = αm µ 1 + βmµ 2 + ρ 2 〈m 2|γ µ |m 1 ] + δ 2 〈m 1|γ µ |m 2 ]. (3.11)The first and third on-shell constraint require that, α = β = 0. Then setting l onshell requires that either ρ or δ = 0. We begin by choosing δ = 0. Then the final