[QUE/QFT-15001] Pair creation probability per unit volume

Prove that the probability per unit volume per unit time  that the external
potential
\[ \vec{A}= (0,0,a\cos\omega t), A_0=0\]
creates an electron-positron pair in the vacuum is given by
\[ R = \frac{2}{3} \frac{e^2}{4\pi} \Big(\frac{|a|^2}{2} \Big) \omega^2 \Big(
   1+ \frac{2m^2}{\omega^2}\Big) \sqrt{1- \frac{4m^2}{\omega^2}}  \]

Source :Sakurai