Rutherford cross-section from QED

In summary, the question concerns the calculation of Rutherford cross section in the context of QED using both the usual four potential and the process e+γ→e. The matrix elements for both cases are given, and the resulting cross section for the e+γ→e process is stated. The question then shifts to how to integrate this cross section and what the phase space for this process is. It is suggested to compute it from Møller scattering and take the nonrelativistic limit. However, it is also mentioned that it may be possible to simply go to the CM system and consider one of the electrons scattering off its mirror image.
  • #1
Einj
470
59
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 [itex]\epsilon^\mu[/itex] 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
 
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  • #2
Einj said:
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 [itex]\epsilon^\mu[/itex] 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.
 
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  • #3
Does it mean just to take the mass of one of the two electrons to be infinite?
 
  • #4
Einj said:
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.
 
  • #5
Sounds good, thanks!
 

What is the Rutherford cross-section from QED?

The Rutherford cross-section from QED (Quantum Electrodynamics) is a mathematical formula that describes the probability of a photon scattering off of a charged particle, such as an electron or proton, in an interaction known as Compton scattering.

How is the Rutherford cross-section from QED calculated?

The Rutherford cross-section from QED is calculated using a combination of classical and quantum mechanical principles. It takes into account the charge of the interacting particles, the energy and momentum of the photon, and the effects of relativity.

What is the significance of the Rutherford cross-section from QED?

The Rutherford cross-section from QED is significant because it helps us understand the behavior of light and matter at the subatomic level. It has been extensively tested and validated by experiments, and is an essential tool for theoretical predictions in particle physics.

How does the Rutherford cross-section from QED relate to the Rutherford scattering experiment?

The Rutherford cross-section from QED is a theoretical calculation that describes the probability of scattering, while the Rutherford scattering experiment was a physical experiment conducted by Ernest Rutherford in 1911 that demonstrated the existence of the atomic nucleus.

What are some real-world applications of the Rutherford cross-section from QED?

The Rutherford cross-section from QED has many practical applications, such as in medical imaging techniques like X-rays and PET scans, as well as in nuclear energy and particle accelerator technologies. It is also used in the development of new materials and technologies based on quantum mechanics.

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