Rutherford cross-section from QED

[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

 

[Bill_K]

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.

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.

 

[Einj]

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

 

[Bill_K]

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

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.

 

[Einj]

Sounds good, thanks!