# Particle physics: calculating the phase space factor for pion to muon decay

1. Nov 17, 2009

### ncs22

Show that the phase space factor $$\rho \propto p^2 dp/dE$$ for the decay $$\pi\rightarrow \mu + \upsilon$$ is

$$\rho \propto \frac{({m_\pi}^2 - {m_\mu}^2)^2}{{m_\pi}^3}E_\mu$$

where E is the total energy.

I can show that $$p^2 = ({m_\pi}^2 - {m_\mu}^2)^2/4{m_\pi}^2$$

but then I get stuck, I don't know how to evaluate dp/dE and I'm not sure what p here is refering to i.e. which particle and in which frame. I worked out the above expression for p2 taking it to be the energy for the muon (or neutrino) in the center of mass frame.

2. Nov 17, 2009

### turin

I'm not sure what exactly is meant by phase-space factor, but I would assume that a Dirac delta (for 4-momentum conservation) has already been factored out, which means that you can treat the momentum as the final-state momentum of the particle of your choice. Usually, you choose the visible one (e.g. the muon).

3. Nov 19, 2009

### ncs22

ok cool at least that means the first bit is probably right :-)

The phase space factor is the number of final states per initial energy, for example the term in Fermi's Golden rule usually denoted by a $$\rho$$. Yes I know you can write the phase space as a product of integrals over every particles momentum in which case a delta function has to be introduced to account for the fact that not every momentum is independent.

When I find the answer I will post it here.