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

3.2. The cut-constructible parts 80S 1m (1) = A (0) 〈l 3 m〉 2 〈l 4 m〉 2 〈l 3 1〉〈l 4 1〉〈(n − 1)n〉3〈n1〉〈12〉〈23〉[nl 4 ] 3 [2l 3 ] 3n〈l 1 l 2 〉〈l 1 (n − 1)〉〈l 2 3〉〈1m〉 4 〈l 3 l 4 〉[l 3 l 2 ][l 4 l 1 ][nl 1 ][2l 2 ](3.41)The procedure to calculate these quantities is straightforward, remove l 1 , l 2 and l 4in terms of l 3 and substitute in the solution l (2) . This yields the followingG 1m (1) = A(0) n2 s 12s 1n , (3.42)( (0) ) AF 1m n 〈mn〉〈2m〉〈1n〉〈12〉(1) =2 s 12s 1n , (3.43)〈1m〉 2 〈2n〉 2( (0) )( ) 2 AS 1m n 〈mn〉〈2m〉〈1n〉〈12〉(1) = −2 s 12s 1n . (3.44)〈1m〉 2 〈2n〉 2This completes the analysis for one-mass boxes, we will observe in the next sectionthat the calculation of the two-mass box coefficients proceeds in an identical fashion.Two-mass boxesWhen calculating the coefficients of one-mass boxes which appear in φ-MHV amplitudeswe noted that there were two sorts of contributions. Diagrams in whichthe loop particle was constrained to be a gluon had a coefficient of the followingform, A (0)n s 1 s 2 where s 1 and s 2 are the invariants associated with pairs of (adjacent)massless legs. The diagrams which allowed fermions to propagate in the loop havea more complicated structure, they also contain a piece of the form A (0)n s 1 s 2 but inaddition contain pieces proportional to N f , the number of light flavours.We find an identical situation with the two-mass boxes which are depicted inFig. 3.6. Diagrams (a), (b), (d) and (e) which contain a gluon loop alone havecoefficients of the form A (0)n (s 1 s 2 − P 2 Q 2 ) where P and Q are the momenta of themassive legs. An example of this type of term was given in the previous chapter.We also find that the diagrams which allow a fermion to propagate in the loop havethe same breakdown as the one-mass boxes,(D (c) = G 2m (a, i, j) + 4 1 − N ) (fF 2m (a, i, j) + 2 1 − N )fS 2m (a, i, j).4N c N c(3.45)