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

3.2. The cut-constructible parts 750000000001111111110100000000011111111101000000000111111111010000000001111111110100000 1111100000 1111100000 1111100000 11111m 1l − M 23l00000 1111100000 1111100000 1111100000 11111ml − m 2 00000 1111100000 11111m 2Figure 3.4: A one-mass box with arbitrary massless legs m 1 , m 2 and m 3on-shell constraint fixes ρ,0 = s m2 m 3− ρ〈m 2 |m 3 |m 1 ] =⇒ ρ = [m 2m 3 ][m 3 m 1 ] . (3.12)If on the other hand we had taken ρ = 0 we would have found that0 = s m2 m 3− δ〈m 1 |m 3 |m 2 ] =⇒ δ = 〈m 2m 3 〉〈m 1 m 3 〉 , (3.13)so that the two solutions to the on-shell conditions are,l µ (1) = [m 2m 3 ]2[m 3 m 1 ] 〈m 2|γ µ |m 1 ] and l µ (2) = 〈m 2m 3 〉2〈m 1 m 3 〉 〈m 1|γ µ |m 2 ]. (3.14)We will average over the solutions, but in general one is always zero due to thehelicities of m 1 and m 2 . With the general solution in hand we now proceed todetermine the types of one-mass box which can appear in the φ-MHV amplitude.Since there is only one-mass, φ must always be present at the massive vertex. We arethen free to assign the two negative helicity gluons throughout the various vertices,for which the general topologies are shown in Fig. 3.5.Diagrams 3.5(a), (b), (d) and (e) only allow gluons to propagate in the loop, andwe consider these first, diagram 3.5(a) has the following integrand,D (a) = A (0)n−1(φ, l + 1 , i+ , . . .,1 − , . . .,m − , . . .,j + , l + 2 )A(0) 3 (l − 2 , (j + 1)+ , l + 3 )×A (0)3 (l− 3 , (j + 2)+ , l − 4 )A(0) 3 (l+ 4 , (i − 1)+ , l − 1 ) (3.15)