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14 2.1. Kinematics and observablesFor circularly polarised photons, one hasɛ λγ=±1 = ∓ 1 √2(0, 1, ±i, 0) , (2.12)whereas virtual photons can also have a longitudinal polarisationɛ λγ=0 = (0, 0, 0, 1) . (2.13)The transition current operator can be constructed from a set of relevant Feynman diagrams. InSections 2.2 and 2.3, the structure of this operator in the RPR formalism will be discussed in depth.The transition matrix elements fulfil the following useful property [100]T -λγ-λ N -λ Y= (−1) 1+λγ−λ N −λ YT λγλ N λ Y, for λ γ = 0, ±1 , (2.14)owing to the reaction’s invariance under space inversions and rotations. This relation reduces thenumber of independent amplitudes by a factor of two. In the transversity basis for real photonreactions, one chooses the following set of independent amplitudesb 1 = T y ,+ 1 2 ,+ 1 2b 2 = T y ,− 1 2 ,− 1 2b 3 = T x − 1 2 ,+ , 12(2.15)b 4 = T x + 1 2 ,− , 12where photons have a linear polarisation and the baryon polarisations are quantised along the y axis,i.e. normal to the reaction plane.2.1.4 Photoproduction observablesThe differential cross section for the most general scattering reaction is given in Eq. (D.1). Applyingthis to the kaon photoproduction reaction, one findsdσ =1 ∑√|T |2 d 3 ⃗p K d 3 ⃗p Y2 λ(s KY , 0, m 2 N ) 2E K (2π) 3 2E Y (2π) 3 (2π)4 δ (4) (p γ + p N − p K − p Y ) . (2.16)Here, the symbol ∑ implies that one sums (averages) over the polarisations of the outgoing (incoming)particles. After integrating out the energy-momentum conserving delta function, one commonlyexpresses the differential cross section asd 2 σdΩ ∗ K= 1 |⃗pK ∗ | ∑|T | 264π 2 EγW ∗ KY2 . (2.17)At higher energies, where the reaction becomes diffractive, the cross section is often reported in thefollowing Lorentz-invariant formdσdt = 1 1 ∑|T | 264π Eγ ∗2 WKY2 . (2.18)