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

3.2. The cut-constructible parts 93It is trivial to see that diagrams 3.11(c), 3.11(e) and 3.11(f) have exactly the sameintegrand (with the relevant values of i and j). Therefore to consider the puregluediagrams we merely need perform the double cut integration of the followingfunction,I ij = 〈l 1l 2 〉 2 〈(i − 1)i〉〈j(j + 1)〉〈l 1 (j + 1)〉〈(i − 1)l 2 〉〈l 2 i〉〈l 1 j〉 . (3.80)The Schouten identity can be used to relate I ij to simpler functions G ij ,I ij = −G ij + G (i−1)j + G i(j+1) − G (i−1)(j+1) (3.81)withG ij = 〈il 1〉〈jl 2 〉〈il 2 〉〈jl 1 〉 . (3.82)We will now proceed to integrate G ij using the method of [132]. First remove l 1 infavour of l 2 = l 1 + P and replace l 2 = tλ, t is then fixed by the δ function, leavingthe following integrand,∫∫G ij =dλs P i,j〈jλ〉〈i|P |λ]〈iλ〉〈j|P |λ]〈λ|P |λ] 2. (3.83)Next we replace |λ〉 with |p〉+z|η〉 and integrate in z removing the pieces proportionalto logarithms. It is interesting to note that if we had started with I ij and integratedwe would have found no pieces which are not proportional to logarithms and hencewould have concluded that I ij ∝ boxes and triangles. However, when we work withG ij we find a non-zero piece which has a non-zero residue at z = 0. In the previouschapter we described how these pieces arise from the integrand of a cut-bubble. Thisimplies that G ij contains bubbles whilst I ij does not. For both these statements tobe correct implies that G ij does not depend on i or j, indeed we find that∫G ij | 2−point = 1, (3.84)which ensures that I ij | 2−point = 0 as expected. In conclusion there are no pieces ofdiagrams 3.11(a), 3.11(c), 3.11(e) and 3.11(f), which are not proportional to boxesand triangles (and hence already known).This leaves diagrams 3.11(b) and 3.11(d) which are related to each other ((d)can be obtained from (b)), here the integrands are more complicated since there are