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

3.2. The cut-constructible parts 84The vanishing of three-mass triangle coefficientsThe three-mass triangle integral is listed with the other basis integrals in Appendix Band is finite (i.e. it has no poles in the dimensional parameter ǫ) and as such thecoefficient of the three-mass triangle cannot be fixed by infra-red safety conditions.However, for any φ-MHV helicity configuration there can be no non-zero three-masstriangle coefficients. The general topologies are shown in Fig. 3.7 for each one thereis always at least one vertex which vanishes.One- and two-mass triangle coefficientsThe remaining triangle coefficients which are associated with one- and two-masstriangles can be calculated from infra-red safety conditions. To ensure correct infraredbehaviour the ǫ −2 pieces of the amplitude must have the following form,A (1)n= − c Γǫ 2 A(0) nn∑( µ2i=1−s i,i+1) ǫ+ O(ǫ 0 ). (3.54)In general we expect the coefficients of the various triangles to possess a similarstructure to the box coefficients, i.e. we expect to find the following sorts of termsin our amplitude,(−4(Cn;1 3−cut (φ, 1 − , 2 + , . . ., m − , . . ., n + ) = A (0)n(1 − N f4N c)A φF,3−cutn;1 (m, n) − 2A φG,3−cut1 − N fN c)A φS,3−cutn;1 (m, n)n;1 (m, n)). (3.55)In this decomposition it is clear that only A φG,3−cutn;1 (m, n) can contribute to eq. (3.54),further we can infer that since no infra-red poles are proportional to N f (since thereexists no (n+1) φ plus gluon tree amplitude which has an N f dependence) we knowthat the triangles which occur in these pieces must cancel the poles which arise fromthe box contributions. The case where m = 2 has been calculated [108] and thefollowing contributions were found,A φG,3−cutn;1 (2, n) =n∑i=1(F 1m3 (s i,n+i−2 ) − F 1m3 (s i,n+i−1 )) (3.56)A φF,3−cutn;1 (2, n) = A φS,3−cutn;1 (2, n) = 0 (3.57)