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

3.2. The cut-constructible parts 97with( (= A (0)n A φGn;1 (m, n) − 4 1 − N ) (fA φFn;14N (m, n) − 2 1 − N )fA φSn;1),c N (m, n) cA φGn;1(m, n) = − 1 2− 1 2n∑i=1n∑n+i−1∑i=1 j=i+3F 2me4 (s i+1,j , s i,j−1 ; s i,j , s i+1,j−1 )(3.101)F 1m4 (s i,i+1, s i+1,i+2 ; s i,i+2 ) + (F 1m3 (s i,n+i−2) − F 1m3 (s i,n+i−1)),(3.102)A φFn;1(m, n) =m−1∑n∑i=2 j=m+1m−1∑+n∑i=2 j=m+1m−1∑−n∑i=2 j=mm−1∑++1∑i=2 j=m+1m∑n∑i=2 j=m+1m−1∑−n∑i=1 j=m+1b ij1m F 2me4F (s i+1,j , s i,j−1 ; s i,j , s i+1,j−1 )b ij1m F2me 4F (s j+1,i, s j,i−1 ; s j,i , s j+1,i−1 )tr − (m, P (i,j) , i, 1)A ijs 2 m1 L 1(P (i+1,j) , P (i,j) )1mtr − (1, P (j,i) , i, m)A i(j−1)s 2 1m L 1 (P (j,i−1) , P (j,i) )1mtr − (m, P (i,j) , j, 1)A j(i−1)s 2 m1 L 1 (P (i,j−1) , P (i,j) )1mtr − (1, P (j,i) , j, m)A jis1mL 2 1 (P (j+1,i) , P (j,i) ). (3.103)1mHere we have introduced the following function, which will make the collinear behaviourof eq. (3.103) more apparent,( )A ij1m = tr− (1, i, j, m)− (j → j + 1)s ij. (3.104)The final piece of eq. (3.101) is the most complicated and has the following structure,m−1A φSn;1 (m, n) = − ∑−n∑i=2 j=m+1n∑m−1∑i=m+1 j=2m−1∑+n∑i=2 j=m[(b ij1m )2 F 2me4F (s i+1,j, s i,j−1 ; s i,j , s i+1,j−1 )(b ij1m) 2 F 2me4F (s i+1,j, s i,j−1 ; s i,j , s i+1,j−1 )− tr −(m, P (i,j) , i, 1) 3A ij3sm1L 4 3 (P (i+1,j) , P (i,j) )1m