Question

Consider the following decays of Σ hyperons 

Σ⁺ → n + π⁺ 

Σ⁻ → n + π⁻ 

Σ⁺ → p + π⁰ 

Using the ΔI = 1/2 rule, show that the triangle relation for the amplitudes is a⁺ + √2a₀ = a⁻

Answer 1

Let us introduce a fictitious particle called spurion of I = 1/2, I₃ = −1/2, and neutrally charged and add it on the left hand side so that the weak decay is converted into a strong interaction in which I is conserved. The reactions with Σ⁺ can proceed in I = 1/2 and 3/2 channels while that with Σ⁻ through pure I = 3/2 channel. We can then write down the amplitudes for the initial state by referring to the C.G. coefficients for 1 × 1/2: