Positron annihilation in $e^+ -$H collision above the Positronium formation threshold

A long-standing problem to account for the electron-positron annihilation in positron Hydrogen scattering above the Positronium formation threshold has been resolved by the use of the three-body Faddeev formalism. The multichannel three-body theory for scattering states in presence of a complex absorbing potential is developed in order to compute the direct $e^+ e^-$ annihilation amplitude, the amplitude of Positronium formation and respective cross sections. A number of $e^+ e^-$ direct annihilation cross sections and Positronium formation cross sections in the energy gap between Ps$(1s)$ and H$(n=2)$ thresholds are reported for both the positron-Hydrogen incoming channel as well as the proton-Positronium incoming channel.