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

In summary, the phase space factor for the decay of \pi\rightarrow \mu + \upsilon is \rho \propto p^2 dp/dE, where E is the total energy. The expression for p^2 is ({m_\pi}^2 - {m_\mu}^2)^2/4{m_\pi}^2, and this was calculated for the muon (or neutrino) in the center of mass frame. The phase space factor is the number of final states per initial energy and can be written as a product of integrals over each particle's momentum, with a delta function introduced to account for the dependencies.
  • #1
ncs22
6
0
Show that the phase space factor [tex]\rho \propto p^2 dp/dE [/tex] for the decay [tex]\pi\rightarrow \mu + \upsilon[/tex] is

[tex]\rho \propto \frac{({m_\pi}^2 - {m_\mu}^2)^2}{{m_\pi}^3}E_\mu [/tex]

where E is the total energy.I can show that [tex]p^2 = ({m_\pi}^2 - {m_\mu}^2)^2/4{m_\pi}^2[/tex]

but then I get stuck, I don't know how to evaluate dp/dE and I'm not sure what p here is referring 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.
 
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  • #2
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
Thanks for the reply.

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 [tex]\rho[/tex]. 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.
 

1. How is the phase space factor calculated for pion to muon decay?

The phase space factor for pion to muon decay is calculated by using the conservation laws of energy and momentum. This involves considering the masses and momenta of all the particles involved in the decay process and using mathematical equations to determine the allowed phase space for the decay to occur.

2. Why is the phase space factor important in particle physics?

The phase space factor is important in particle physics because it helps us understand the probability of a decay process occurring. By calculating the phase space factor, we can determine the rate at which particles decay and make predictions about the behavior of subatomic particles.

3. What factors affect the phase space factor for pion to muon decay?

The phase space factor for pion to muon decay is affected by the masses of the particles involved, the energy of the initial state, and any external forces or interactions present. It can also be affected by the spin and angular momentum of the particles.

4. How does the phase space factor differ for different types of particle decays?

The phase space factor can differ for different types of particle decays depending on the specific conservation laws and interactions involved. For example, the phase space factor for a two-body decay will be different from a three-body decay due to the varying number of particles and their properties.

5. Can the phase space factor be experimentally measured?

Yes, the phase space factor can be experimentally measured by analyzing the decay products of a particle and comparing it to theoretical predictions. However, this can be challenging as it requires precise measurements and understanding of the decay process and its underlying physics.

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