Charged Pion Branching Ratio: Part I

Inherent symmetries in nature lead to conservation laws (perhaps someone will make a post about Noether's Theorem). When we think about the stability of a sub-atomic particle we must think in these terms. Take for example the charged pion, in free space at the tree level. The particle* has mass and charge. (Don't forget to boost into the pion's rest frame) There are no lighter charged hadrons therefore the final state must include a charged lepton of less mass (ie an electron or muon). Our situation looks as follows

Now we invoke conservation of lepton number and conclude that we cannot make the lepton without an associated anti-lepton; charge conservation rules out all particles save neutrinos. So now we have

This situation is the end of the story for the vast majority of pion decays (99,999,999/100,000,000 times in fact). However we can sneak in another particle! The neutral pion is less massive than the charged pion by more than the mass of the electron so we can add a pi-zero to the electron channel (beta decay). We have now arrived at the complete picture.

The next step is to calculate the different decay rates for the various processes and that is a topic for part II!

* I'm playing fast and loose with what is a particle and what is an anti-particle, deal with it.