I'm still trying to calculate the cross-section of the $e^- e^+ \rightarrow \mu^- \mu^+$ interaction in first order. This time I'm struggling with the phase space measure. Note that I have two questions, which are not exactly related but since they deal with the same equation I'll post them both at once.
1) So generally one has to integrate over the unknown momenta, i.e. the momenta of the outgoing muons, while keeping in mind that energy and momentum conservation needs to hold:
$$d\Phi_{12 \rightarrow 34} = [dp_3][dp_4] (2 \pi)^4 \delta(p_1 + p_2 - p_3 - p_4) = \frac{d^4p_3}{(2 \pi )^3}\delta(p_3^2 - m_{\mu}^2)\frac{d^4p_4}{(2\pi)^3}\delta(p_4^2 - m_{\mu}^2)(2\pi)^4\delta(p_1 + p_2 - p_3 - p_4)$$
In my script they jump from this point to this expression:
$$\frac{d^4p_3}{(2\pi)^2}\delta(p_3^2 - m_{\mu}^2)\delta((p_1 + p_2 - p_3)^2 - m_{\mu}^2)$$
Can somebody explain to me what happened here? Where did the integration over $p_4$ vanish to?
2) In the expression below:
$$\frac{|\vec{p_3}|^2 d|\vec{p_3}| d(\cos\theta) d\phi}{(2\pi)^2} \frac{1}{4E} \delta(E_3^2 - |\vec{p_3}|^2 - m_{\mu}^2) = \frac{|\vec{p_3}|d(\cos\theta)d\phi}{(2\pi)^2}\frac{1}{8E}$$
How was the Delta function manipulated here to give make the two terms equal? I have trouble seeing this.