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...