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

5.1. Introduction 133increased number of primitive amplitudes and the expressions for the amplitudes.5.1.1 Definition of colour ordered amplitudesThe colour decomposition of the H¯qqgg amplitudes is exactly the same as for thecase ¯qqgg which was written down in ref. [117]. For the tree-level case there are twocolour stripped amplitudes,A (0)4 (φ, 1¯q , 2 q , 3 g , 4 g ) = Cg 2 ∑ σ∈S 2(T a σ(3)T a σ(4)) ī1i 2A (0)4 (φ, 1¯q , 2 q , σ(3), σ(4)) . (5.1)At one-loop level the colour decomposition is,[A (1)4 (φ, 1¯q, ∑2 q , 3 g , 4 g ) = Cg 4 c Γ N c (T a σ(3)T a σ(4)) ī1i 2A 4;1 (φ, 1¯q , 2 q , σ(3), σ(4))σ∈S 2+ δ a 3a 4δ ī1i 2A 4;3 (φ, 1¯q , 2 q ; 3 g , 4 g )]. (5.2)In these equations g is the strong coupling constant and c Γ is the ubiquitous one-loopfactor,c Γ ≡1 Γ(1 + ǫ)Γ 2 (1 − ǫ). (5.3)(4π) 2−ǫ Γ(1 − 2ǫ)The colour stripped amplitudes A 4;1 and A 4;3 can further be decomposed into primitiveamplitudes,and,A 4;1 (φ, 1¯q , 2 q , 3 g , 4 g ) = A L 4 (φ, 1¯q, 2 q , 3 g , 4 g ) − 1 A RNc2 4 (φ, 1¯q, 2 q , 3 g , 4 g )+ N fN cA f 4 (φ, 1¯q, 2 q , 3 g , 4 g ) , (5.4)A 4;3 (φ, 1¯q , 2 q ; 3 g , 4 g ) = A L 4 (φ, 1¯q , 2 q , 3 g , 4 g ) + A R 4 (φ, 1¯q , 2 q , 3 g , 4 g )+ A L 4 (φ, 1¯q, 3 g , 2 q , 4 g ) + A L 4 (φ, 1¯q, 2 q , 4 g , 3 g )+ A R 4 (φ, 1¯q, 2 q , 4 g , 3 g ) + A L 4 (φ, 1¯q, 4 g , 2 q , 3 g ) . (5.5)All of these colour decomposition equations, namely eqs. (5.1, 5.2, 5.4, 5.5) areequally valid if φ is replaced by a φ † or a Higgs boson H. Sample diagrams contributingto each of the primitive amplitudes are shown in Figure 5.1.