# How to determine the decay mode probability of a pion

1. ### rwooduk

450
Hi,

Given the initial energy of a pion is it possible to calculate the probability and mode of its decay?

Ive looked everywhere for a relevent formula but everywhere simply states the probability and mode....

Wiki "The primary decay mode of a pion, with probability 0.999877, is a purely leptonic decay into an anti-muon and a muon neutrino"... "The second most common decay mode of a pion, with probability 0.000123, is also a leptonic decay into an electron and the corresponding electron antineutrino."

I considered doing some sort of conservation of energy / momentum, but how am i supposed to do that if i dont know what it decays into?

ive spent at least two hours trying to find an equation, i found something called the "branching ratio" is that at all related?

any direction on this would be appreciated.

2. ### Einj

444
There is no rule to determine the decay modes. In principle, a particle can decay in every final state allowed by conservation laws. First of all the total mass of the final state must be lighter than the decay particle. Then the right quantum numbers must be conserved (total angular momentum, lepton number and so forth).

Once you chose what particular final state you want to study you can compute the decay width for that particular final state using the Quantum Field Theory methods. A good reference on how to actually do this kind of calculation can be:

De Wit - Field Theory in Particule Physics (Chapter 3)

Is a very practical book that shows you how to compute cross sections and decay widths.

I hope this is useful

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3. ### rwooduk

450
That's very very useful! Thank you!!

4. ### dauto

The decay is truly random. There is no way to tell ahead of time which of the possible outcomes will actually happen. All you can tell are the probabilities of each possible outcome.

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### Staff: Mentor

The energy of the pion does not matter - in particular, you can always consider it in its rest frame.

Branching ratios are exactly the probabilities of the decay modes you are looking for.

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6. ### rwooduk

450
if the energy of the pion is completely independent of its decay mode then that again that answers my question because the only information given in the question is the pions total energy. So finding the main decay mode calculatively would not be possible.

Thanks all!

### Staff: Mentor

Calculating the main decay modes is possible with theoretical physics (quantum field theory). And those decay modes are completely independent of the energy of the pion in our lab frame (the pion just does not care about our lab).

8. ### rwooduk

450
hmm i'm not quite at quantum field theory level yet, but in essence i think we agree that decay mode is independent of energy :-)

9. ### ChrisVer

2,212
Even in its rest frame, the pion will have mass...