Positronium energy level in QED I'd like to know if it is possible to compute positronium mass and lifetime from a QED approach.
I'm searching for some literature on how to treat resonances in QED (or general QFT) ; most of the articles I've found are about corrections to these results, I'm searching for the beginning of these computations.
 A: 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...
A: Volume 4 of Landau & Lifshitz give an excellent account on both points. I am going to give a summary of it.
The following abreviations will be used (apart from well-known $c$ and $\hbar$ ) :
$m$ electron respectively positron mass
$e$ electron charge
$\alpha$ fine structure constant
First of all, one has to distinguish 2 positronium types according to their total spin $S=0$ and $S=1$.

*

*$S=0$: para-positronium (decays in $2\gamma$) singlet state


*$S=1$: ortho-positronium (decays in $3\gamma$) triplet state
A) rest mass:
The rest mass is the mass of both particles + (binding energy)$/c^2$ (negative). The latter can easily be found by using the formula for the binding energy of an electron in a hydrogen atom. The only thing that has to be changed is the reduced mass $M_{red} =\frac{m_1 m_2}{m_1+ m_2} = m/2$ of the bound state. In the hydrogen atom however, it is $M_{red} \approx m$ (proton mass is assumed to be very large).
Therefore the binding energy of the Positronium is (knowing the result from the hydrogen atom and modifying it according to the different reduced mass):
$\frac{E_b}{c^2} = - \frac{M_{red}}{2n^2}\alpha^2 = - \frac{m}{4n^2}\alpha^2$
So for the rest mass we get immediately (assuming the lowest energy state $n=1$):
$m_{rest} = 2m - \frac{m\alpha^2}{4}$
There are a couple of corrections to this value if further physical effects as spin-orbit correction, relativistic energy correction, spin coupling and what is called by Landau & Lifshitz "annihilation"-interaction of the positronium are considered.
However, these effects lead to corrections of the order $m\alpha^4$, i.e. they are very small. Also the difference of the binding energy between ortho-positronium and para-postronium is an effect of the order $m\alpha^4$ which can be easily neglected. In case of particular interest refer to Landau & Lifshitz.
B) Lifetime:
As Ali Moh already explained, the decay probability can be found by the computation of the product of decay cross section $\sigma$ (which is different for a $2\gamma$ and a $3\gamma$ decay) and "luminosity" also known as current density (at the origin of the center of mass system ). The cross section is computed with QED tools (Feyman diagrams) whereas the current density is a result of non-relativistic QM.
$w \sim \sigma \cdot v |\psi(0)|^2$
where $v$ is the relative velocity between both particles.
$\psi(0) = 1/\sqrt{\pi a^3}$ with $a^{positronium}_{bohr} \equiv a= \frac{2\hbar^2}{m e^2}$ twice larger than the Bohr-radius of the hydrogen atom.
The cross section is averaged of all spin directions of the particles. For the decay probability the statistical weight of the spin state that decays has to be considered. For the para-positronium it is g=1/4, whereas for the ortho-positronium it is g=3/4.
Putting everything together the lifetime for the para-positronium is (take the $2\gamma$ cross section from the post of Ali Moh (actually his value already contains the statistical weight for the spin state)):
$\tau_P = \frac{2 \hbar}{m c^2 \alpha^5} =1.23 \cdot 10^{-10}$ s
whereas for the ortho-positronium the spin-averaged decay cross section is
$\sigma_{3\gamma} = \frac{4 (\pi^2-9) \alpha }{3}\frac{c}{v} \left(\frac{e^2}{mc^2}\right)^2$
that leads to the lifetime of:
$ \tau_O = \frac{9\pi}{2(\pi^2-9)}\frac{\hbar}{mc^2 \alpha^6}= 1.4\cdot 10^{-7}$s
Again for more details consult Landau & Lifshitz volume 4.
By the way, if you are interested in the computation of bound states in QED, look for the Bothe-Salpeter equation (also Landau & Lifshitz vol. 4  gives a good account on this. )
