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

A.1. Spinor Helicity Formalism notation and conventions 171invariants through the following relation,s ij = 〈ij〉[ji].(A.1.5)Chains of spinor products are written as〈i|j|k] = 〈ij〉[jk] 〈i|jk|l〉 = 〈ij〉[jk]〈kl〉, etc. (A.1.6)For example, using momentum conservation we have,4∑〈i|p φ |k] = − 〈ij〉[jk].j=1(A.1.7)For purely partonic amplitudes (i.e. with no Higgs, φ or φ † present) momentumconservation is represented by the following equation0 =n∑p µ i =⇒ 0 =i=1n∑〈ij〉[jk]i=1(A.1.8)A good overview of the spinor helicity formalism and colour ordering (which isdescribed in the following section) can be found in [218].Intermittently in this thesis we have used the following notation to define afour-vector in terms of spinor indices [131]p aȧ = σ µ aȧ p µ(A.1.9)we use σ µ = (1, −→ σ ) and −→ σ is a representation of the Pauli spin matrices. Thismeans that we can expand p aȧp aȧ = p 0 + −→ σ · −→ p(A.1.10)This last equation implies thatp µ p µ = det(p aȧ )(A.1.11)Such that a vector is light-like if and only if the determinant of p aȧ vanishes, ageneral 2 × 2 matrix has at most rank two so can be expanded in terms of spinorsλ, µ as follows p aȧ = λ a˜λȧ + µ a˜µȧ. If the rank of a 2 × 2 matrix is less than two