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32 3.1. Symmetry considerations at the strong-interaction vertexFigure 3.1 – The three tree-level diagrams considered in the RPR framework. Special emphasis is put onthe strong-interaction vertices.3.1 Symmetry considerations at the strong-interaction vertexThe relations among the coupling constants in the strong-interaction vertices of the various kaonproductionreactions, depicted in Figure 3.1, can be derived effortlessly. This is made possible bymeans of the isospin symmetry of the strong force. In the strong-interaction vertex, the hadroniccouplings are proportional to the Clebsch-Gordan coefficients〈g KΛN (∗) ∼ I K = 1 2 , M K I ; I Λ = 0, MΛ I = 0∣ I N (∗) = 1 〉2 , M I , (3.3)N (∗)〈g KΛ∆ ∗ ∼ I K = 1 2 , M K I ; I Λ = 0, MΛ I = 0∣ I ∆ ∗ = 3 〉2 , M ∆ I ∗ , (3.4)and〈g KΣN (∗) ∼ I K = 1 2 , M K I ; I Σ = 1, MΣI ∣ I N (∗) = 1 〉2 , M I , (3.5)N (∗)〈g KΣ∆ ∗ ∼ I K = 1 2 , M K I ; I Σ = 1, MΣI ∣ I ∆ ∗ = 3 〉2 , M ∆ I ∗ . (3.6)We adopt the following conventions for the isospin states of the N (∗) , ∆ ∗ , K (∗) and Σ particles,p, K (∗)+ , N ∗+ → ∣ ∣I = 1 2 , M I = + 1 2〉,n, K (∗)0 , N ∗0 → ∣ ∣I = 1 2 , M I = − 1 2〉,Λ → |I = 0, M I = 0 〉 ,Σ + → −|I = 1, M I = 1〉 ,(3.7)Σ 0 → |I = 1, M I = 0〉 ,Σ − → |I = 1, M I = − 1〉 ,∆ ∗+ → ∣ ∣I = 3 2 , M I = + 1 2〉,∆ ∗0 → ∣ I =32 , M I = − 1 〉2 .The phase of the Σ − state is taken to be positive. With this choice, the Condon-Shortley phaseconvention dictates a minus sign for the Σ + state. Using these isospin states to calculate theClebsch-Gordan coefficients of Eqs. (3.3)–(3.6), we easily find the conversion factors of interest.