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

3.4. Cross Checks and Limits 1133.4.5 Soft limit of A (1)4 (φ, 1− , 2 + , 3 − , 4 + )The final test is to take the limit as the φ momentum becomes soft, this limit occurswhen p φ → 0 such that m 2 φ → 0. Our naive expectation is that in this limit, the φfield is essentially constant so thatCφtrG SD µν G µ,νSD → trG SD µνG µ,νSD . (3.185)In other words, the amplitude should collapse onto the gluon-only amplitude. Inref [106], it was postulated that the amplitude should factorise in following form,A (1)n (φ, n − g − , n + g + ) p φ→0→ n − A (1)n (n − g − , n + g + ), (3.186)whileA (1)n (φ† , n − g − , n + g + ) p† φ →0→ n + A (1)n (n −g − , n + g + ). (3.187)We first consider the cut constructible contributions. These factorise onto thefour gluon amplitude in rather trivial manner since in our construction we separatedgluon-only like diagrams and those which require a non-vanishing φ-momentum. Inthe soft limit, the one and two mass easy box and triangle functions have smoothlimits so that,( µ2−m 2 φ) ǫ pφ →0→ 0, (3.188)( ) µ2 ǫp φ →0→ 0. (3.189)−s φiFurthermore, in the soft limit the L k functions become the massless bubble functions,L k (s 234 , s 23 ) = Bub(s 234) − Bub(s 23 ) p φ →0→(s 234 − s 23 ) kAltogether, we find that( )(−1) k µ2 ǫ. (3.190)s k 23ǫ(1 − 2ǫ) −s 23C 4 (φ, 1 − , 2 + , 3 − , 4 + ) p φ→0→ 2C 4 (1 − , 2 + , 3 − , 4 + ), (3.191)where C 4 (1 − , 2 + , 3 − , 4 + ) is the cut-constructible pieces of the four-gluon amplitude.This confirms that the cut-constructible terms of the amplitude do follow the naivefactorisation of eq. (3.186)