system
Given the following post, respond as if you're commenting on Reddit:
human
Why is the lifetime of a Muon (~2 microseconds) so much smaller than the lifetime of a Neutron (~15 minutes), even though they both decay due to the weak interaction?
gpt
Great question. If you look at the Feynman diagrams for muon decay and neutron decay you'll find they're essentially the same. Hence the *dynamics* of the decays are not any different, but they have a different lifetime due to their *kinematics*.

The muon is roughly 200 times heavier than the electron to which it decays. In computing the lifetime, we can therefore regard the decay products--an electron, a muon neutrino, and an electron antineutrino--as being effectively massless. On the other hand, the neutron is just slightly more massive than its decay products, because the proton is almost as heavy as the neutron.

In computing the lifetime of a particle, we integrate over the "phase space" of the decay products. Essentially, this means we have to sum over all the different possible configurations of the momenta and energy of the outgoing particles. In the decay of a neutron, most of the energy (i.e. mass) of the neutron is needed to supply the rest mass of the proton, so you only have a little bit of leftover energy to distribute among the decay products. In a muon decay, you have a lot more kinetic energy (about a hundred times more) to distribute among the decay products. There are many more possible configurations of the momenta of the outgoing particles and hence the lifetime is shorter.

To clarify this argument further, take a limiting case. Suppose the proton was heavier than the neutron. Then, by conservation of energy, the neutron would not be able to decay. So you know that if the neutron is only slightly heavier than the proton, the decay is suppressed and the lifetime is long.

Ninja edit: you can read more about this in a particle physics text, such as Griffiths' Introduction to Elementary Particles, chapters 9.2 and 9.3.