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

A.2. Tree-level amplitudes 173pair can be decomposed into colour-ordered amplitudes as follows,A (0)n (φ, {p i, λ i , a i }, {p j , λ j , i j })∑= iCg n−2(A.1.17)σ∈S n−2(T a σ(2)· · ·T a σ(n−1)) i1 i nA n (φ, 1 λ , σ(2 λ 2, . . .,(n − 1) λ n−1), n −λ ) ,where S n−2 is the set of permutations of (n−2) gluons. Quarks are characterised withfundamental colour label i j and helicity λ j for j = 1, n. By current conservation, thequark and antiquark helicities are related such that λ 1 = −λ n ≡ λ where λ = ± 1 2 .The one-loop amplitudes which are the main subject of this paper follow thesame colour ordering as the pure QCD amplitudes [110] and can be decomposedas [106–108],[n/2]+1A (1)n (φ, {k ∑i, λ i , a i }) = iCg nwherec=1∑σ∈S n/S n;cG n;c (σ)A (1)n (φ, σ(1λ 1, . . .,n λn )) (A.1.18)G n;1 (1) = N c tr(T a1 · · ·T an )G n;c (1) = tr(T a1 · · ·T a c−1) tr(T ac · · ·T an ) , c > 2.(A.1.19)(A.1.20)The sub-leading terms can be computed by summing over various permutations ofthe leading colour amplitudes [110].A.2 Tree-level amplitudesIn this section we list the tree level amplitudes which have been used as ingredientsin the construction of Higgs plus four parton one-loop amplitudes. We also recallthe following relations for constructing Higgs amplitudes,A (l)n (H; {p k}) = A (l)n (φ, {p k}) + A (l)n (φ† , {p k }).(A.2.21)We can also generate pseudo-scalar amplitudes from the difference of φ and φ †components,A n (l) (A; {p k}) = 1 (A(l)ni(φ, {p k}) − A n (l) (φ† , {p k }) ) .(A.2.22)