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In Weinberg QFT volume 2 p.273, the equation (20.6.4) says that the Mott scattering cross section for an electron by a point spinless particle is given by: $$(\frac{d\sigma}{d\Omega})_{Mott}=\frac{e^4}{4E_e^2} \frac{cos^2(\theta/2)}{\sin^4(\theta/2)}$$.

Here, $\pi-\theta$ is the scattering angle of the electron in the center of mass frame.

However, all other textbooks say the Mott scattering cross section is :$$\frac{\alpha^2}{4E_e^2} \frac{cos^2(\theta/2)}{\sin^4(\theta/2)}$$ where $\alpha =\frac{e^2}{4\pi}$ is the fine structure constant.

Is the Weinberg book wrong? I am quite confused...

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Weinberg is probably using electrostatic CGS units, where

$$\alpha=\frac{e^2}{\hbar c}.$$

See Wikipedia regarding what the fine-structure constant looks like in various non-SI unit systems.

Your definition of $\alpha$ is for natural units in which $\epsilon_0$ rather than $4\pi\epsilon_0$ is 1.

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  • $\begingroup$ Weinberg states at the start of his book.that he uses the natural unit. That is why I am confused.. $\endgroup$
    – Keith
    Apr 28, 2020 at 2:18
  • $\begingroup$ I don’t think everyone agrees on what “natural units” means. You have to look at his equations, not at his words. $\endgroup$
    – G. Smith
    Apr 28, 2020 at 4:03
  • $\begingroup$ Then natural unit he says is of course setting the speed of light and reduced Planck constant to be 1. He has been using this unit throughout the book without any further comment. There seems to be no reason for the equation (20.6.4) to be written in a different unit. $\endgroup$
    – Keith
    Apr 28, 2020 at 4:35

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