Many-body aspects of positron annihilation in the electron gas

We investigate positron annihilation in the electron gas as a case study for many-body theory, in particular, the Fermi-hypernetted-chain Euler-Lagrange (FHNC-EL) method. We examine several approximation schemes and show that one has to go up to the most sophisticated implementation of the theory available at the moment in order to get annihilation rates that agree reasonably well with experimental data. Even though there is basically just one number we look at, namely, the electron-positron pair-distribution function at zero distance, it is exactly this number that dictates how the full pair distribution behaves: in most cases, it falls off monotonously towards unity as the distance increases. Cases where the electron-positron pair distribution exhibits a dip are precursors to the formation of bound electron-positron pairs. The formation of electron-positron pairs is indicated by a divergence of the FHNC-EL equations; from this we can estimate the density regime where positrons must be localized. This occurs in our calculations in the range $9.4<~r_s<~10,$ where $r_s$ is the dimensionless density parameter of the electron liquid.