# Kaon reaction rates

I have a big difficulty in grasping weak interactions.

For example, how would I go on about determining the following: $\frac{\Gamma(K^{+} \to \pi^{+}K^{0})}{\Gamma(K^{+} \to \pi^{0}K^{+})}$

I don't understand much of the reaction at all. I can find some standard decay reactions of Kaons, am I supposed to combine these from them? How?

And for those reactions, "Cabibbo-angles" are defined. Are they related to some known properties, or defined experimentally?

In weak interactions there is a concept called isospin. Using this one can determine the ratio of decay widths you asked for just in terms of the Wigner-Eckart-Theorem.

The Cabibbo angle (and there is just one of them, but also have a look at the CKM matrix) defines how probable it is that a strange quark turns into a down quark in weak interactions. In other words it describes the mismatch between the interaction eigenstates and the mass eigenstates (which we would call proper particles).

• I'll look into it. Thank you. I'm pretty sure this is the method I'm supposed to use as I know that $I_{3}$ is either $1/2$ or $3/2$. [I accept your answer once I figure out how to do it] – swirld May 20 '14 at 7:37
• Realize that you can write $\Gamma$ as a matrix element of an operator with definite isospin. Then you can use the WET and the generalized matrix element will cancel, leaving you with a ratio of Clebsch-Gordan coefficients that can be computed directly. – Neuneck May 20 '14 at 8:36

These are the Feynman diagrams, as far as I know. When worked out they are quite simple. What exactly are you calculating? Can you be more specific?

• I would not call these diagrams "simple" if you work them out. These processes take place at lower energies than the W mass, and therefore one needs to use techniques like operator product expansion to properly calculate what's going on from scratch. Also, the hadronic matrix elements can so far not be determined in direct computations (although lattice simulations exist), and are often extracted from experiment. – Neuneck May 20 '14 at 7:30
• Thank you. I'm pretty sure this is too complicated approach for me (this is an introductory course, and I am unable to calculate out Feymann diagrams.) – swirld May 20 '14 at 7:39
• Isn't that why we have computers? For them to solve this after it has already been set up will be a piece of cake. You don't have anything that strange occurring; so what?, a gluon is emitted and breaks into an anti quark pair that allows the decay to occur. – Gödel May 20 '14 at 7:40
• Watch Suskind's lectures free on youtube and there won't be a Feynman diagram you can't calculate :) . Good luck in A level! – Gödel May 20 '14 at 7:43