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

  1. Nov 17, 2009 #1
    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 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. jcsd
  3. Nov 17, 2009 #2


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    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).
  4. Nov 19, 2009 #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.
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