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 ...
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3.4. Cross Checks and Limits 109the momentum P (i,j) always conta<strong>in</strong>s both m − 1 and m, while P (j,i) never <strong>in</strong>cludeseither m − 1 or m, and we f<strong>in</strong>d rel<strong>at</strong>ions 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 th<strong>at</strong> 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 rel<strong>at</strong>ions hold for the terms <strong>in</strong>volv<strong>in</strong>g K and I. Therefore, all terms <strong>in</strong> the n-gluon version <strong>of</strong> A φFn;1 and AφS n;1 therefore either collapse onto similar terms, or vanish<strong>in</strong> such a way th<strong>at</strong> the reduced summ<strong>at</strong>ion precisely m<strong>at</strong>ches onto the correspond<strong>in</strong>gA φFn−1;1 and AφS n−1;1 .Two positive coll<strong>in</strong>ear limitNext we consider the limit when <strong>two</strong> positive helicity gluons become coll<strong>in</strong>ear. Wefocus on the specific example where l −1 ‖ l <strong>with</strong> 3 ≤ l ≤ m −1. As <strong>in</strong> the previoussubsection, let first consider the ranges 2 ≤ i ≤ m − 1 and m ≤ j ≤ n. We noteth<strong>at</strong>,b l−1j1mb lj1ml−1‖l−−−→l−1‖l−−−→b Kj1m ,b Kj1m. (3.171)The coll<strong>in</strong>ear factoris<strong>at</strong>ion <strong>of</strong> box functions has been well studied [110,111,166] and<strong>in</strong> this case, the rel<strong>at</strong>ion,(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 po<strong>in</strong>t amplitude.(3.172)