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

3.4. Cross Checks and Limits 111as these become collinear. In the following section we prove that this is indeed thecase. To be explicit, we consider the collinear limit i ‖ j with,i → zK,j → (1 − z)K. (3.179)Let us consider the cut-constructible pieces associated with the fermionic loopcontribution, A φFn;1(m, n) given in eq. (3.103). There are ten terms containing anexplicit pole in s ij which are given by,b ij1m F 2me4F (s i,j , s i+1,j−1 ; s i+1,j , s i,j−1 )+b ij1m F2me4F (s j,i, s j+1,i−1 ; s j+1,i , s j,i−1 )− tr −(m, P (i+1,j) , i, 1) tr − (m, i, j, 1)Ls 2 1 (P (i+1,j) , P (i,j) )1m s ij+ tr −(m, P (i+1,j−1) , i, 1) tr − (m, i, j, 1)s 2 1m− tr −(1, P (j,i−1) , i, m)s 2 1ms ijL 1 (P (i+1,j−1) , P (i,j−1) )tr − (1, i, j, m)s ijL 1 (P (j,i−1) , P (j,i) )+ tr −(1, P (j+1,i−1) , i, m) tr − (1, i, j, m)Ls 2 1 (P (j+1,i−1) , P (j+1,i) )1m s ij− tr −(m, P (i,j−1) , j, 1) tr − (m, j, i, 1)Ls 2 1 (P (i,j−1) , P (i,j) )1m s ij+ tr −(m, P (i+1,j−1) , j, 1) tr − (m, j, i, 1)Ls 2 1 (P (i+1,j−1) , P (i+1,j) )1m s ij− tr −(1, P (j+1,i) , j, m) tr − (1, j, i, m)s 2 1m+ tr −(1, P (j+1,i−1) , j, m)s 2 1mL 1 (P (j+1,i) , P (j,i) )s ijtr − (1, j, i, m)L 1 (P (j+1,i−1) , P (j,i−1) ). (3.180)s ijUsing P (i+1,j) = P (i+1,j−1) + p j , P (j,i−1) = P (j+1,i−1) + p j , P (i,j−1) = P (i+1,j−1) + p i andP (j+1,i) = P (j+1,i−1) + p i , as well as tr − (1, j, i, m) = − tr − (1, i, j, m) + O(s ij ) etc, wecan rewrite these terms asb ij (1m F2me4F (s i,j , s i+1,j−1 ; s i+1,j , s i,j−1 ) − s ij L 1 (P (i+1,j) , P (i,j) ) − s ij L 1 (P (i,j−1) , P (i,j) ) )(+b ij1m F2me4F (s j,i, s j+1,i−1 ; s j+1,i , s j,i−1 ) − s ij L 1 (P (j,i−1) , P (j,i) ) − s ij L 1 (P (j+1,i) , P (j,i) ) )− tr −(m, P (i+1,j−1) , i, 1) tr − (m, i, j, 1) (L1 (Ps 2 (i+1,j) , P (i,j) ) − L 1 (P (i+1,j−1) , P (i,j−1) ) )1m s ij+ tr −(m, P (i+1,j−1) , j, 1)s 2 1mtr − (m, i, j, 1)s ij(L1 (P (i,j−1) , P (i,j) ) − L 1 (P (i+1,j−1) , P (i+1,j) ) )