Hadronic production of a Higgs boson in association with two jets at ...
Hadronic production of a Higgs boson in association with two jets at ...
Hadronic production of a Higgs boson in association with two jets at ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
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 1mF<strong>in</strong>ally, <strong>in</strong> the i ‖ j coll<strong>in</strong>ear 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 not<strong>in</strong>g th<strong>at</strong> the comb<strong>in</strong><strong>at</strong>ion,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 th<strong>at</strong> all s<strong>in</strong>gularities cancel. The same arguments apply to the cut-constructiblepieces associ<strong>at</strong>ed <strong>with</strong> the scalar pieces.3.4.4 Coll<strong>in</strong>ear factoris<strong>at</strong>ion <strong>of</strong> the r<strong>at</strong>ional piecesThis section is devoted to the coll<strong>in</strong>ear factoris<strong>at</strong>ion <strong>of</strong> the r<strong>at</strong>ional pieces <strong>of</strong> thefour po<strong>in</strong>t amplitude. As a result <strong>of</strong> the symmetries <strong>of</strong> the amplitude there are <strong>two</strong><strong>in</strong>dependent limits 1 ‖ 2 and 2 ‖ 3. We first consider the coll<strong>in</strong>ear limit 2 ‖ 3. It isstraightforward to see th<strong>at</strong> 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 rema<strong>in</strong><strong>in</strong>g non-trivial coll<strong>in</strong>ear 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=±