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

4.2. Cut-Constructible Contributions 121There are four three-mass triangles, which satisfy,C 3;φ|34|12 (φ, 1 + , 2 − , 3 − , 4 − ) = C 3;φ|12|34 (φ, 1 + , 2 − , 3 − , 4 − ) (4.19)C 3;φ|41|23 (φ, 1 + , 2 − , 3 − , 4 − ) = C 3;φ|23|41 (φ, 1 + , 2 − , 3 − , 4 − ). (4.20)The symmetry under the exchange of gluons with momenta p 2 and p 4 relates theremaining two coefficients,C 3;φ|23|41 (φ, 1 + , 2 − , 3 − , 4 − ) = C 3;φ|12|34 (φ, 1 + , 4 − , 3 − , 2 − ). (4.21)To compute C 3;φ|23|41 we use both Forde’s method [122] and the spinor integrationtechnique [128]. For a given triangle coefficient C 3;K1 |K 2 |K 3(φ, 1 + , 2 − , 3 − , 4 − ) withoff-shell momenta K 1 , K 2 and K 3 , we introduce the following massless projectionvectorsIn terms of these quantities we find,C 3;φ|12|34 (φ, 1 + , 2 − , 3 − , 4 − ) =K ♭µ1 = γ γKµ 1 − K1K 2 µ 2,γ 2 − K1 2K2 2K ♭µ2 = γ γKµ 2 − K2 2Kµ 1,γ 2 − K1K 2 22 √γ ± (K 1 , K 2 ) = K 1 · K 2 ± K 1 · K2 2 − K1K 2 2. 2 (4.22)∑γ=γ ± (p φ ,p 1 +p 2 )m 4 φ−〈K♭ 1 2〉3 〈34〉 32γ(γ + m 2 φ )〈K♭ 11〉〈K13〉〈K ♭ 14〉〈12〉 , ♭which, as expected, correctly vanishes in the soft Higgs limit (p φ → 0).(4.23)4.2.3 Bubble Integral CoefficientsThe non-vanishing bubble topologies for the φ-NMHV amplitude are shown inFig. 4.3. We find that the double-cuts associated with Fig. 4.3(a) contain onlycontributions from boxes and triangles, and therefore the coefficient of log(s 1234 ) iszero. In a similar fashion, the double cuts associated with diagram Fig. 4.3(c) withtwo external gluons with negative helicity emitted from the right hand vertex haveonly box and triangle contributions, so that the coefficients of log(s 23 ) and log(s 34 )are also zero.