The short life of orthopositronium
Positronium is a bound state of an electron and a positron, similar to a hydrogen-like atom. Its states are classified by the total spin:
parapositronium (spin 0) decays mainly into two photons;
orthopositronium (spin 1) decays into three.
In this project you will study the decay mechanism of orthopositronium, estimate its lifetime and see how QED describes the decay of composite particles.
- Consider the decay of orthopositronium into three photons, \(o\text{-Ps} \to 3\gamma\), assuming that the electron–positron system is at rest (the approximation \(v \ll c\)).
Draw all six tree-level diagrams of the process. Explain why there are exactly six. Write the full matrix element \(\mathcal{M}\), taking into account that the photons are identical.
Compute the squared matrix element \(|\mathcal{M}|^2\), averaged over the initial and summed over the final spins, in the approximation of fermions at rest (\(\boldsymbol{p}=\boldsymbol{p}'=0\)).
Find the differential decay rate in the energy of one of the photons, \[\frac{d\Gamma}{d\omega_1},\] where \(\omega_1\) is the energy of the first photon in the center-of-mass frame. Take the symmetry of the process into account.
Integrate and obtain the total decay width \(\Gamma(o\text{-Ps} \to 3\gamma)\). Using \[\tau = \frac{\hbar}{\Gamma},\] find the lifetime of orthopositronium in vacuum.
Modify the calculation assuming that the electron and the positron are in the ground state of a hydrogen-like atom (wave function \(\psi(0) \neq 0\)). How does this change the result?