Question

Consider the reactions at the same energy 

π⁺ + p → Σ⁺ + K + 

π⁻ + p → Σ⁻ + K ⁺ 

π⁻ + p → Σ⁰ + K⁰ 

Assuming that the isospin amplitude a_1/2 << a_3/2, show that the cross sections for the reactions will be in the ratio 9:1:2

Answer 1

The analysis is identical with that for πp reactions. The Σ’s are isospin triplet (T = 1) with the third components T3 = +1, 0, −1 for Σ⁺, Σ⁰, Σ⁻ similar to π⁺, π⁰, π⁻. Further, the K⁺ and K⁰ form a doublet (T = 1/2) with T₃ = +1/2 and −1/2, analogous to p and n. 

Therefore the result on the ratio of cross-section will be identical with that for pion – nucleon reactions as