Golden Rule for Decays in Griffiths' Introduction to Elementary Particles ?

[TMFKAN64]

Golden Rule for Decays in Griffiths' "Introduction to Elementary Particles"?

Hello, I'm getting a bit bogged down in constant factors while reading Griffiths' "Introduction to Elementary Particles", 1st edition. In particular, I'm having problems with equation 6.15:
$$d\Gamma = |M|^{2}\frac{S}{2\hbar m_{1}}\left[ \left(\frac{c\, d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\right) \left(\frac{c\, d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\right)\cdots \left(\frac{c\, d^{3}p_{n}}{(2\pi)^{3}2E_{n}}\right)\right]\times (2\pi)^{4}\delta^{4}(p_{1}-p_{2}-p_{3}\cdots-p_{n})$$

I've found some slides online at http://www-pnp.physics.ox.ac.uk/~libby/Teaching/Lecture4.pdf that discuss normalizing the wave function and phase space so it is Lorentz invariant, but this only explains part of my problem. What I still don't understand is:

• Where do the $$c$$ in the numerators of the momentum terms come from?
• Where does the $$(2\pi)^{4}$$ term before the delta function come from?
• Where did the $$\hbar^{3n}$$ in the density of states go? (The $$\frac{1}{(2\pi\hbar)^{3n}}$$ is responsible for the $$2\pi$$ factors in the momentum terms, right?)
• Why doesn't the first fraction have $$m_{1}c^{2}$$ in the denominator?

Any assistance would be greatly appreciated.

 

[LightHero]

Thanks! The "Golden Rule" for decays in Griffiths' "Introduction to Elementary Particles" is the following: 1. All decay products must be on-shell (have real energies). 2. The total four-momentum of all decay products must equal the four-momentum of the original particle (momentum conservation). 3. The transition amplitude must be Lorentz-invariant. 4. The phase space integral must be Lorentz-invariant. 5. The decay rate is given by the square of the transition amplitude multiplied by the phase space integral. 6. The c in the numerators of the momentum terms come from the fact that the momentum is measured in units of energy. 7. The (2π)4 term before the delta function comes from the fact that the integration over phase space is performed in four dimensions. 8. The 1/(2πℏ)3n in the density of states is responsible for the 2π factors in the momentum terms. 9. The first fraction does not have m1c2 in the denominator because the energy and momentum are related by E=pc, so m1c2 is already included in the momentum terms.