Positronium energy level in QED

To find the lifetime you can use a mixture between QED and non relativistic QM. You use QED to find the cross-section of $e^+ e^- \rightarrow \gamma\gamma$ which is to first order in perturbation theory for the singlet state $\frac{4\pi}{cv}\left(\frac{\alpha}{m}\right)^2$. Then you must multiply this by the electron luminosity, this is where you use the QM result that gives you the relation between the radius of the positronium atom in terms of electron mass, as well as the wave function of the electron. From this you find the luminosity, which is the product of velocity and charge density (wave function modulus) and multiply by the cross section to get the decay rate...