  A quantum mechanical approach is proposed for the formation of antihydrogen
in the ground and excited states (2s, 2p) via the mechanism of three body
recombination (TBR) inside a trapped plasma of anti proton and positron or in
the collision between the two beams of them. Variations of the differential
(DCS) as well as the total (TCS) formation cross sections are studied as a
function of the incident energies of both the active and the spectator
positrons. Significantly large cross sections are found at very low incident
energies in the TBR process as compared to other processes leading to
antihydrogen. The present formation cross section decreases with increasing
positron energy (temperature) but no simple power law could be predicted for it
covering the entire energy range, corroborating the experimental findings
qualitatively. The formation cross sections are found to be much higher for
unequal energies of the two positrons than for equal energies, as expected
physically.
