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Homework Help: QFT: calculating decay rates from invariant matrix element M

  1. Feb 10, 2010 #1
    Hi!

    I am currently taking a first course in QFT with Peskin & Schroeder's book. I've got stuck with the equation that relates the differential decay rate of a particle A at rest into a set of final particles with the invariant matrix element M of the process. M can be found from the Feynman rules.
    The equation is:

    [tex]d\Gamma = \frac{1}{2m_A}\left(\prod_f \frac{d^3 p_f}{(2\pi)^3}\frac{1}{2E_f}\right) |M(m_A \rightarrow \{p_f\})|^2 (2\pi)^4 \delta^{(4)}(p_A-\sum p_f)[/tex]

    where

    [tex]\Gamma[/tex] is the decay rate
    [tex]m_A[/tex] is the rest energy of the initial particle A
    [tex]p_A[/tex] is the momentum of the initial particle A
    [tex]E_f[/tex] is the energy of one of the final particles
    [tex]p_f[/tex] is the momentum of one of the final particles

    In my problem, 1 particle decays into 2, so we get 2 factors with one integral each when integrating both sides to find [tex]\Gamma[/tex]. I feel kinda unsure on how to treat the delta function since it is four-dimensional and the integrals over p are three-dimensional. It's clear that the delta function imposes momentum conservation, but how to integrate?

    Any help is greatly appreciated.
     
  2. jcsd
  3. Feb 10, 2010 #2

    vela

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    I seem to recall you can convert the 3D integral into a 4D integral. It's been way too long since I took QFT to remember the details, and I'm too lazy to go dig out my books. But I found this on the web:

    http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/phase_space_integral [Broken]
     
    Last edited by a moderator: May 4, 2017
  4. Feb 11, 2010 #3
    Thanks a lot! That link was really helpful.
     
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