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

3.4. Cross Checks and Limits 112− tr −(1, P (j+1,i−1) , i, m) tr − (1, i, j, m)s 2 1m+ tr −(1, P (j+1,i−1) , j, m)s 2 1mFinally, in the i ‖ j collinear limit,(L1 (P (j,i−1) , P (j,i) ) − L 1 (P (j+1,i−1) , P (j+1,i) ) )s ijtr − (1, i, j, m) (L1 (P (j+1,i) , P (j,i) ) − L 1 (P (j+1,i−1) , P (j,i−1) ) ) .s ijtr − (m, P (i+1,j−1) , i, 1) ( L 1 (P (i+1,j) , P (i,j) ) − L 1 (P (i+1,j−1) , P (i,j−1) ) )(3.181)→ tr − (m, P (i+1,j−1) , j, 1) ( L 1 (P (i,j−1) , P (i,j) ) − L 1 (P (i+1,j−1) , P (i+1,j) ) ) (3.182)and noting that the combination,F 2me4F (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) ) → O(s 2 ij ),we see that all singularities cancel. The same arguments apply to the cut-constructiblepieces associated with the scalar pieces.3.4.4 Collinear factorisation of the rational piecesThis section is devoted to the collinear factorisation of the rational pieces of thefour point amplitude. As a result of the symmetries of the amplitude there are twoindependent limits 1 ‖ 2 and 2 ‖ 3. We first consider the collinear limit 2 ‖ 3. It isstraightforward to see that the amplitude correctly factorises onto:ˆR 4 (φ, 1 − , 2 + , 3 − , 4 + ) + CR 4 (φ, 1 − , 2 + , 3 − , 4 + ) 2 −→ ‖ 3∑R 3 (φ, 1 − , K i , 4 + )Split (0) (−K −i , 2 + , 3 − ) (3.183)i=±In a similar fashion the remaining non-trivial collinear limit takes the form,ˆR 4 (φ, 1 − , 2 + , 3 − , 4 + ) + CR 4 (φ, 1 − , 2 + , 3 − , 4 + ) 1 −→ ‖ 2∑R 3 (φ, K i , 3 − , 4 + )Split (0) (−K −i , 1 − , 2 + ) (3.184)i=±