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

3.2. The cut-constructible parts 83We note that A φG,4−cutn;1 (m, n) is independent of the position of the two negativehelicity gluons.A φF,4−cutn;1 (m, n) =m−1∑n∑i=2 j=m+1m−1∑+n∑i=2 j=m+1b ij1m F 2me4 (s i+1,j , s i,j−1 ; s i,j , s i+1,j−1 )b ij1m F2me 4 (s j+1,i , s j,i−1 ; s j,i , s j+1,i−1 ) (3.52)andm−1∑A φS,4−cutn;1 (m, n) = −−n∑i=2 j=m+1n∑m−1∑i=m+1 j=2(b ij1m) 2 F 2me4 (s i+1,j , s i,j−1 ; s i,j , s i+1,j−1 )(b ij1m) 2 F 2me4 (s i+1,j , s i,j−1 ; s i,j , s i+1,j−1 ). (3.53)We leave the explicit definitions of the basis functions F 2me4 and F 1m4 to AppendixB, noting that the definitions we use are related to the basis integral I 4by a kinematic factor (which cancels those appearing in the coefficients given inthis chapter). The summations in A φF,4−cutn;1 and A φS,4−cutn;1 do not explicitly refer toone-mass boxes, this is because these terms arise naturally from the two-mass boxeswhen one of the massive legs becomes soft.3.2.2 Triangle CoefficientsIn this section we determine the coefficients associated with the triangle basis integrals.For the general φ-MHV configuration we show that there can never be anynon-zero three-mass triangle coefficients. Further, we show that the remaining coefficientsattributed to the one- and two-mass triangles can be split into two pieces.The first of these pieces is helicity blind whilst the second contains pieces proportionalto N f , and depends on the position of the two negative helicity gluons. Sincethe three-mass triangles do not contribute, knowledge of the triangle coefficients canalso be determined by infra-red safety conditions.