[QUE/QFT-15006] Relativistic Coulomb scattering of Dirac Particle

Write Dirac equation in external electromagentic potential  \(A_\mu(\vec{x})\). Assume the potentials for Coulomb interactions with a nucleus of charge \(Ze\) to be
\[ \vec{A}=0, \quad A_0 = \frac{Ze}{4\pi |\vec{x}|}.  \]
Show that the differential cross section for electron scattering from nucleus is
given by
\[\frac{d\sigma}{d\Omega}= \left(\frac{d\sigma}{d\Omega}\right)_\text{R} \Big(1-v^2\sin^2(\theta/2)\Big)\] where
\[\left(\frac{d\sigma}{d\Omega}\right)_\text{R} = \frac{Ze^2}{64\pi m^2v^4 \sin^4(\theta/2)}\]
is the Rutherford cross section for nonrelativistic Coulomb scattering.