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I took a look at the Particle Data Group, and the most likely modes are $\Omega ^{-}\rightarrow \Lambda K^{-}$ (68%) and $\Omega ^{-}\rightarrow\Xi ^{0}\pi ^{-}$ (24%). I have 2 questions:

1) The exercise only says that it can decay in 2 hadrons, but it doesn't say which ones. How can i know this without looking at books as i did?

2) Once i find the particle couples, how can i calculate the transition probability? Solution is given by Fermi's Golden Rule $\Gamma _{if}=2\pi |<\Psi _{i}|H_{Int}|\Psi _{f}>|^{2} \rho(E_{f})$ , but how can i calculate the density $\rho(E_{f})$ and the matrix element? enter image description here

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For 1) you can guess because of the available energy and the masses of the possible decay products, plus the need of various conservation laws

For 2) you cannot do this reliably. These are strong interaction decays,i.e.QCD, which is not easy to calculate as the Feynman diagram approach does not work for coupling constants of order one so the summation of diagrams cannot stop at the first few.

Here is a description of color force. What has been fairly successful in calculating strong interaction quantities is lattice QCD, but does not give the matrix elements you are talking about.

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