I try to calculate the scattering amplitude of a process $A + A \rightarrow B + B $ in lowest order contribution.
For one part of the amplitude $M$ i got: $$M=\frac{g^2}{(p_4-p_2)^2-m_C^2}$$
We are working in the center of mass frame and we set all masses equal and give the virtual particel zero mass: $$m_A=m_B=m\\ m_C=0$$
Then:$$M=\frac{g^2}{(p_4-p_2)^2}$$
$p_2$ and $p_4$ are four momentum vectors. So therefore $$(p_4-p_2)^2=p_4^2+p_2^2-2p_2\cdot p_4=2m^2-2\cdot E_2\cdot E_4+2\cdot \vec p_2 \cdot \vec p_4$$
My Question now:
How do we get from this result to $-2(\vec p_1)^2(1-\cos\theta)$
(where $\vec p_1$ ist the incident momentum of particle 1)?
I know, that $\vec p_1 = -\vec p_2$ and similarly for $p_3$ and $p_4$, but $E_2$ and $E_4$ are not the same, since $p_2\neq p_4$ an the cosine comes from the dot product of the space momentum.