Phase space element calculation

[kaksmet]

Can anyone see what's not right?

In the phase space calculation of a 2 to 2 process I get to
$$I=\int dp_1d\Omega \frac{1}{(2\pi)^2}\frac{p_1^2}{2E_12E_2}\delta(E_1+E_2-E)$$

then I use
$$p_1=\sqrt{E_1^2+m_1^2} \Rightarrow dp_1=\frac{E_1}{\sqrt{E_1^2+m_1^2}}dE_1$$

thus
$$I = \int dE_1d\Omega \frac{1}{(2\pi)^2}\frac{E_1^2-m_1^2}{2E_12E_2}\frac{E_1}{\sqrt{E_1^2-m_1^2}}\delta(E_1+E_2-E)$$
$$=\int d\Omega \frac{\sqrt{E_1^2-m_1^2}}{16\pi^2E_2}$$

$$=\int d\Omega \frac{|p_1|}{16\pi^2E_2}$$

However, this is the right answer, which should be
$$\int d\Omega \frac{|p_1|}{16\pi^2E_{CM}}$$All ideas greatly appreciated

 

[kaksmet]

Problem sovled. Misstake was that E_2 is a function of E_1 so I cannot directly use the delta function. Taking this into account the correct answer is obtained.