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A type of problem that often appears is to find the relation between cross sections of some processes. An example would be:

$$\pi _{- }+ p \rightarrow K_0 + \Sigma_0$$

$$\pi _{- }+ p \rightarrow K_+ + \Sigma_-$$

$$\pi _{+}+ p \rightarrow K_+ + \Sigma_+$$

To do this, I argue that

$$\sigma \, \alpha \, \Gamma \, \alpha \, M$$

with ##M## the transition matrix element.

In this case, the interactions are strong. I write the initial and final states for each process in the ##{I,i}## basis and I then write ##M=\langle i | H_s | f \rangle##.

I end up getting, for example for the third process: ##M=\langle 3/2,3/2 | H_s | 3/2, 3/2 \rangle##.

For the first process, I get one term of the form ##\langle 3/2,-1/2 | H_s | 3/2,-1/2 \rangle##.

My question: are there two expressions I just wrote equal? i.e. does ##M=\langle I,i | H_s | I,i' \rangle## depend only on the value of ##I##? Why?

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# I Transition matrix element and Isospin

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