The lifetimes for each state of $l=0$ levels of the para-positronium

Abstract

We developed a model for a para-positronium system by taking into account the annihilation condition which requires $l=0$. A para-positronium system composed by an electron-antielectron pair is an unstable system that decays predominantly into two gamma photons via self-annihilation process. This is because of the conservation of (overall) charge conjugation parity. Our $2+1$-dimensional model was based on an exact solution of a two-body Dirac–Coulomb type equation (one-time). The resulting non-perturbative spectra were found to have a form of quasi-normal modes that naturally give proper and damping oscillations of the system. As a result, with a single equation, not only the binding energies and total annihilation energy transmitted by the annihilation photons but also the lifetimes, $\tau \left( n,\alpha \right) \propto n^{3}/\alpha ^{3}$, for each state of $l=0$ levels of the para-positronium were obtained. With this unique presentation of above-mentioned characteristic of the system, we also calculated transition frequencies and transition times between differing n level of $l=0$. This also gave us an opportunity to discuss the screening effect from electronic environment of the system. For this, the effective dielectric constant of the medium that surrounds the annihilation point was adapted into the calculated non-perturbative spectra. This may provide an important tool for the positron annihilation lifetime spectroscopy studies and for development of related techniques.