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12 2.1. Kinematics and observablesFigure 2.1 – Schematic representation of an electron scattering from a nucleon at rest, producing a kaonand hyperon in the final state.2.1 Kinematics and observables2.1.1 Reference framesElectron-induced kaon production from a free nucleon assuming the OPE approximation is illustratedschematically in Figure 2.1. The incoming and outgoing electrons, with four-momenta k(E e , ⃗ k)and k ′ (E ′ e, ⃗ k ′ ) respectively, radiate a virtual photon with four-momentum p γ (E γ , ⃗p γ ) = k − k ′ andvirtuality Q 2 = −p γ ·p γ = |⃗p γ | 2 −E 2 γ. This photon interacts with a nucleon p N(E N , ⃗p N) and producestwo outgoing particles, i.e. the kaon and hyperon, which are labelled p K(E K , ⃗p K) and p Y(E Y , ⃗p Y)respectively. The three-vectors ⃗ k and ⃗ k ′ span the lepton plane, whereas the three-momenta of thephoton, nucleon, kaon and hyperon lie in the reaction plane. The relative inclination of both planesis given by the angle φ. For kaon photoproduction, this angle is arbitrary and the photon’s virtualityQ 2 vanishes.The N(γ (∗) , K)Y reaction can be readily described in the KY-CM frame. Three different referenceframes are commonly adopted in order to express the particles’ four-momenta and polarisations.They are indicated in Figure 2.2. All three choices have their y axis perpendicular to the reactionplane. For the unprimed reference frame, we take the z axis along the photon three-momentum. Theother reference frames have the z ′ (l) axis parallel to the kaon (hyperon). Explicitly we have,⃗z = ⃗p γ|⃗p γ | ,⃗y = ⃗p γ × ⃗p K, ⃗x = ⃗y × ⃗z , (2.1)|⃗p γ × ⃗p K|⃗z ′ = ⃗p K|⃗p K| , ⃗y ′ = ⃗y , ⃗x ′ = ⃗y ′ × ⃗z ′ , (2.2)⃗ l =⃗p Y|⃗p Y| , ⃗n = ⃗y , ⃗t = ⃗n × ⃗ l . (2.3)In addition, a fourth double-primed reference frame is defined in Figure 2.1, which has its y ′′ axisperpendicular to the lepton plane.⃗z ′′ = ⃗p γ|⃗p γ | , ⃗y ′′ = ⃗ k × ⃗ k ′| ⃗ k × ⃗ k ′ | , ⃗x ′′ = ⃗y ′′ × ⃗z ′′ . (2.4)Variables that are expressed in the KY-CM frame are labelled with an asterisk to make the distinctionwith those declared in the LAB frame where the target nucleon is at rest.