  We investigate positron annihilation in electron liquid as a case study for
many-body theory, in particular the optimized Fermi Hypernetted Chain (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 to look at, 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.
