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

3.4. Cross Checks and Limits 109the momentum P (i,j) always contains both m − 1 and m, while P (j,i) never includeseither m − 1 or m, and we find relations such as,tr − (m, P (i,j) , i, 1)s 2 1mA ij m1tr − (1, P (j,i) , i, m)A i(j−1)s 2 1m1mWe note that for the special case i = m − 1,m−1‖m tr −(K, P (i,j) , i, 1)−−−−−→ A ijs 2 K1 ,1Km−1‖m tr −(1, P (j,i) , i, K)−−−−−→ A i(j−1)s 2 1K . (3.169)1KA m−1,jm1 = tr −(m, j, m − 1, 1)s m−1,j− tr −(m, j, m, 1)s m,jA m−1,j1mA i,m−1m1m−1‖m−−−−−→ 0,m−1‖m−−−−−→ 0,m−1‖m−−−−−→ 0. (3.170)Similar relations hold for the terms involving K and I. Therefore, all terms in the n-gluon version of A φFn;1 and AφS n;1 therefore either collapse onto similar terms, or vanishin such a way that the reduced summation precisely matches onto the correspondingA φFn−1;1 and AφS n−1;1 .Two positive collinear limitNext we consider the limit when two positive helicity gluons become collinear. Wefocus on the specific example where l −1 ‖ l with 3 ≤ l ≤ m −1. As in the previoussubsection, let first consider the ranges 2 ≤ i ≤ m − 1 and m ≤ j ≤ n. We notethat,b l−1j1mb lj1ml−1‖l−−−→l−1‖l−−−→b Kj1m ,b Kj1m. (3.171)The collinear factorisation of box functions has been well studied [110,111,166] andin this case, the relation,(b l−1j1m) n (F2me4F (s l−1,j , s l,j−1 ; s l,j , s l−1,j−1 ) +(l−1‖l−−−→b lj1mb Kj1m) nF2me4F (s l,j , s l+1,j−1 ; s l+1,j , s l,j−1 )) nF2me4F (s K,j, s l+1,j−1 ; s K,j , s l+1,j−1 )ensures the box terms correctly factorise onto the lower point amplitude.(3.172)