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 1133.4.5 S<strong>of</strong>t limit <strong>of</strong> A (1)4 (φ, 1− , 2 + , 3 − , 4 + )The f<strong>in</strong>al test is to take the limit as the φ momentum becomes s<strong>of</strong>t, this limit occurswhen p φ → 0 such th<strong>at</strong> m 2 φ → 0. Our naive expect<strong>at</strong>ion is th<strong>at</strong> <strong>in</strong> this limit, the φfield is essentially constant so th<strong>at</strong>CφtrG 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 postul<strong>at</strong>ed th<strong>at</strong> the amplitude should factorise <strong>in</strong> follow<strong>in</strong>g 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 <strong>in</strong> r<strong>at</strong>her trivial manner s<strong>in</strong>ce <strong>in</strong> our construction we separ<strong>at</strong>edgluon-only like diagrams and those which require a non-vanish<strong>in</strong>g φ-momentum. Inthe s<strong>of</strong>t limit, the one and <strong>two</strong> mass easy box and triangle functions have smoothlimits so th<strong>at</strong>,( µ2−m 2 φ) ǫ pφ →0→ 0, (3.188)( ) µ2 ǫp φ →0→ 0. (3.189)−s φiFurthermore, <strong>in</strong> the s<strong>of</strong>t 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 f<strong>in</strong>d th<strong>at</strong>( )(−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 <strong>of</strong> the four-gluon amplitude.This confirms th<strong>at</strong> the cut-constructible terms <strong>of</strong> the amplitude do follow the naivefactoris<strong>at</strong>ion <strong>of</strong> eq. (3.186)