Rutherford cross-section from QED

1. May 20, 2014

Einj

Hi everyone. I have a question about the calculation of Rutherford cross section in the context of QED. I know how to compute it using the usual four potential:
$$A_\mu(q)=(\frac{e}{q^2},0,0,0)$$
and taking the matrix element to be:
$$\mathcal{M}=\bar u_{s'}(p')\gamma_\mu u_s(p)A^{\mu}(q).$$

I was wondering if it is possible to compute it also by considering it as the process e+γ→e, i.e. taking the matrix element to be:
$$\mathcal{M}=e\bar u_{s'}(p')\gamma_\mu u_s(p)\epsilon^\mu,$$
where $\epsilon^\mu$ is the photon polarization.

In this case I got (if I did everything correctly):
$$\frac{1}{2}\sum_{spin}|\mathcal{M}|^2=2e^2q^2.$$

My question is: how should I now integrate this to obtain the cross section? In order words, what is the phase space for this weird 2→1 process?

Thanks

2. May 20, 2014

Bill_K

No, but you can compute it from Møller scattering, that is, e + e → e + e, with exchange of a photon, in which M is something like

$$\mathcal{M}=e^2 \frac{\bar u(p_1')\gamma^\mu u(p_1) \bar u(p_2')\gamma_\mu u(p_2)}{(p_1' - p_1)^2}$$

and then take the nonrelativistic limit.

3. May 20, 2014

Einj

Does it mean just to take the mass of one of the two electrons to be infinite?

4. May 20, 2014

Bill_K

I don't think you need to, I think you can just go to the CM system and regard it as one of the electrons scattering off its mirror image.

5. May 20, 2014

Einj

Sounds good, thanks!