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Ratio of cross sections in terms of the isospin amplitudes

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Express the ratio of cross sections of K^−p→pi^+ Σ^−,K^−p→pi^0 Σ^0, K^−pi→pi6− Σ^+ in terms of the isospin amplitudes A0, A1, A2.

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FOR BEST VIEW OF SOLUTION, SEE ATTACHED FILE "539663-Revised solution.docx"

Here, three interactions are taking place,
(1) K^- p →π^+ ∑^-
(2) K^- p →π^0 ∑^0
(3) K^- p →π^- ∑^+

We know that isospin values for K^-, p, π^+, π^0, π^-, ∑^-, ∑^0, and ∑^+ are as follow,
K^-= |1/2┤ ,├ -1/2〉 , p =|1/2┤ ,├ +1/2〉 π^+= |1┤ ,├ +1〉, π^0= |1┤ ,├ 0〉, π^-=|1┤ ,├ -1〉, ∑^-= |1┤ ,├ -1〉, ∑^0= |1┤ ,├ 0〉 and ∑^+= |1┤ ,├ +1〉.
Both sides of reactions there are two single particle isospin states are associated, now we should decompose the two single-particle isospin states into total isospin states. This we can achieve by using Clebsch-Gordan coefficients table which is available in its standard form in literature. In the following discussion I have explained how to calculate for finding total isospin for the both sides of reaction process.
Therefore, For reaction (1), let us calculate total isospin states for left side. First write the isospin sates in bra-ket notation as shown below,
K^- p= |1/2┤ ,├ -1/2〉|1/2┤ ,├ +1/2〉
Note that the second value in both bra-ket is -1/2 and +1/2 respectively for K^- and p . Now on decomposition of isospin the total isospin state can be computed by using Clebsch-Gordan coefficients ...

Solution Summary

Step-by-step explanation is given for all interactions. The solution is thoroughly explained with necessary formulas. As solution is provided in attached docx file, for the best view of solutions please see attached file "539663-Revised solution.docx".