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

3.2. The cut-constructible parts 79(3.34)We can use the Schouten identity to rewrite the numerator〈m|l 4 |n] 4 〈1|l 3 |2] 4 + 〈m|l 3 |2] 4 〈1|l 4 |n] 4 = 〈1m〉 4 [2|l 3 l 4 |n] 4+4〈1m〉 2 [2|l 3 l 4 |n] 2 〈m|l 4 |n]〈1|l 3 |2]〈m|l 3 |2] 4 〈1|l 4 |n]+2〈m|l 4 |n] 2 〈1|l 3 |2] 2 〈m|l 3 |2] 2 〈1|l 4 |n] 2 . (3.35)When we calculate the contributions which arise when a fermion propagates in theloop we findD (c)f= − 〈m|l 4|n] 3 〈1|l 3 |2] 3 〈m|l 3 |2]〈1|l 4 |n] + 〈m|l 3 |2] 3 〈1|l 4 |n] 3 〈m|l 4 |n]〈1|l 3 |2]D (c),denomg(3.36)where D (c),denomgis the denominator of eq. (3.34). We expand the fermionic numeratorinto the following,〈m|l 4 |n] 3 〈1|l 3 |2] 3 〈m|l 3 |2]〈1|l 4 |n] + 〈m|l 3 |2] 3 〈1|l 4 |n] 3 〈m|l 4 |n]〈1|l 3 |2] =〈1m〉 2 [2|l 3 l 4 |n] 2 〈m|l 4 |n]〈1|l 3 |2]〈m|l 3 |2] 4 〈1|l 4 |n]+2〈m|l 4 |n] 2 〈1|l 3 |2] 2 〈m|l 3 |2] 2 〈1|l 4 |n] 2 . (3.37)The fermion and gluon loops are hence made of similar contributions. This willoccur frequently in this chapter and we use the following notation to describe therelevant contributions,D (c)= D g(c) + N fDfc N c(= G 1m (1) + 4 1 − N ) (fF 1m (1) + 2 1 − N )fS 1m (1) (3.38)4N c N cHere the subscript indicates the cut integral we are referring to, and the bracketsdescribe the kinematic dependence (here indicating that p 1 is the massless leg ratherthan p m ). These integrals have the following form,G 1m (1) = −A (0)nF 1m (1) = A (0)n〈(n − 1)n〉〈n1〉〈12〉〈23〉[2|l 3 l 4 |n] 3〈l 1 l 2 〉〈l 1 (n − 1)〉〈l 2 3〉〈l 3 1〉〈l 4 1〉[l 3 l 2 ][l 4 l 1 ][nl 1 ][2l 2 ]〈l 3 l 4 〉〈l 3 m〉〈l 4 m〉〈(n − 1)n〉〈n1〉〈12〉〈23〉[nl 4 ] 3 [2l 3 ] 3〈l 1 l 2 〉〈l 1 (n − 1)〉〈l 2 3〉〈1m〉 2 [l 3 l 2 ][l 4 l 1 ][nl 1 ][2l 2 ](3.39)(3.40)