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

A.1. Spinor Helicity Formalism notation and conventions 172then its determinant vanishes, in this case we may write a light-like vector in thefollowing form,p aȧ = λ a˜λȧ,(A.1.12)which is used occasionally in this thesis.The following identities are useful and often used to simplify formulae in thisthesis [218]: Gordon identity and Fierz rearrangement,〈i|γ µ |i] = 2k µ i , 〈i|γµ |j]〈k|γ µ |l] = 2〈ik〉[lj] (A.1.13)The Schouten identity is also extremely useful,〈ij〉〈kl〉 = 〈ik〉〈jl〉 + 〈il〉〈kj〉(A.1.14)Finally polarisation vectors associated with external gluons have the following representation,ε + µ (k, q) = + 〈q|γ µ|k]√2〈qk〉ε − µ (k, q) = − [q|γ µ|k〉√2[qk](A.1.15)where q is a reference momenta which reflects the freedom of on-shell gauge transformations.A.1.2Colour ordering of φ plus parton amplitudes at treeleveland one-loopThe tree level amplitudes linking a φ with n gluons can be decomposed into colourordered amplitudes as [64,209],A (0)n (φ, {k i , λ i , a i }) = iCg n−2∑σ∈S n/Z ntr(T a σ(1)· · ·T a σ(n)) A (0)n (φ, σ(1 λ 1, .., n λn )).(A.1.16)Here S n /Z n is the group of non-cyclic permutations on n symbols, and j λ jlabels themomentum p j and helicity λ j of the j th gluon, which carries the adjoint representationindex a i . The T a iare fundamental representation SU(N c ) colour matrices, normalisedso that Tr(T a T b ) = δ ab . Tree-level amplitudes with a single quark-antiquark