Pion decay

As you may or may not know, positive pions decay by
\[
\pi^+ \to \mu^+ + \nu_\mu
\]
Well, this is a little puzzling. Why does it decay to a muon instead of a positron? In general, decays to less massive particles are favorable because of something called phase space (I think), so it seems very strange that the positron decay is so suppressed: 0.01% of the decays, as compared to 99.98% for the muon decay (the remaining 0.01% of decays consist of around 6 other rare processes).

The solution, it turns out, comes because of the maximal parity violation in the weak force I've mentioned before. We've seen that neutrinos are always left-handed and anti-neutrinos are always right-handed. (Remember that the handedness of a particle is a measure of the projection of its spin onto its momentum vector; if the projection points in the same direction as the particle's momentum, it's called right-handed.) Well, it turns out that in general, particles (with positive lepton number) are preferentially left-handed, while antiparticles prefer to be right-handed. The critical bit is that the more massive the particle, the less it has a preference for handedness. That's why neutrinos, which have absolutely minuscule masses, are entirely polarized according to their lepton number.

Based on this, positrons "want" to be right-handed far more than positive muons. Now, recalling that the pion has spin zero, let's look at the its decay. If we start in the rest frame of the pion, then by conservation of momentum, the neutrino and positively charged muon that are produced have to be going in opposite directions with equal (magnitude) momenta. We also know that the neutrino has to be left-handed, that is, with a spin opposite (at least in projection) to its momentum. Since the original pion had a spin of zero, conservation of angular momentum tells us that the muon must also have a spin opposite to its momentum, as in the following sketch:

Pion decay in the pion's rest frame. The black arrows show
momenta, while the colored arrows show the direction of the
spin projected onto the particle's momentum (helicity).

Note that this makes the positive muon a left-handed particle, whereas we've seen that it 'wants' to be right-handed. Aha! The same logic would apply to an electron decay as well, and we've seen that less massive particles have much stronger helicity preferences than more massive ones. Thus the decay into a muon is more acceptable from a helicity perspective than to an electron, so pions tends to decay into muons. Magic!