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## Homework Statement

I am pretty sure it's been done many times before, but I can't seem to figure it out:

Consider the collision 1 + 2 -> 3 + 4 in the lab frame (2 at rest), with particles 3 and 4 massless. Derive the forumla for the differential cross section

## Homework Equations

We have Fermi's Golden Rule for scattering:

[tex] d\sigma = \left|M\right|^{2}\frac{\hbar^{2} S}{4\sqrt{\left(p_1.p_2\right)^{2}-\left(m_{1}m_{2} c^{2}\right)^{2}}} \left(\frac{cd^{3}p_{3}}{\left(2\pi\right)^{3}2E_{3}}\right) \left(\frac{cd^{3}p_{4}}{\left(2\pi\right)^{3}2E_{4}}\right) X \left(2\pi\right)^{4}\delta^{4}\left(p_1+p_2-p_3-p_4\right) [/tex]

(My god it took a while to type that out!)

## The Attempt at a Solution

I start by figuring out the dot product [itex] p_{1}.p_{2}[/itex]. We get [itex] m_2 \left|p_{1}\right| c[/itex]

So what we have is:

[tex]d\sigma = \left(\frac{\hbar}{8\pi}\right)^{2} \frac{S\left|M\right|^{2}}{m_2 \left|p_{1}\right| c} \frac{d^{3}p_{3}d^{3}p{4}}{\left|p_3\right|\left|p_4\right|} \delta\left(\frac{E_{1}}{c}+m_{2}c-\left|p_3\right|-\left|p_4\right|\right) \delta^{3}\left(p_{1}-p_{3}-p_{4}\right) [/tex]

From here on, I don't quite understand. In the textbook we use (Griffiths), it says to integrate [itex] p_{4}[/itex] which replaces it with [itex] p_{1}-p_{3} [/itex]. So the formula will look like:

[tex]d\sigma = \left(\frac{\hbar}{8\pi}\right)^{2} \frac{S\left|M\right|^{2}}{m_2 \left|p_{1}\right| c} \frac{\delta\left(\frac{E_{1}}{c}+m_{2}c-\left|p_3\right|-\left|p_{1}-p_{3}\right|\right) }{\left|p_3\right|\left|p_{1}-p_{3}\right|} d^{3}p_{3}[/tex]

Now we let:

[tex]d^{3}p_{3}=\left|p_{3}\right|^{2}d\left|p_{3}\right|d\Omega[/tex]

where [itex]d\Omega=sin\theta d\theta d\phi[/itex]

...And somehow we should get the right answer:

[tex]\frac{d\sigma}{d\Omega} = \left(\frac{\hbar}{8\pi}\right)^{2} \frac{S\left|M\right|^{2}\left|p_{3}\right|}{m_2 \left|p_{1}\right| \left(E_{1}+m_{2}c^{2}-\left|p_{1}\right|ccos\theta\right)}[/tex]

Can someone help me out? Thanks!

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