QFT: calculating decay rates from invariant matrix element M

[FredMadison]

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:

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

where

$$\Gamma$$ is the decay rate
$$m_A$$ is the rest energy of the initial particle A
$$p_A$$ is the momentum of the initial particle A
$$E_f$$ is the energy of one of the final particles
$$p_f$$ 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 $$\Gamma$$. 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.

 

[vela]

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

 

[FredMadison]

Thanks a lot! That link was really helpful.