Inseparable positron annihilation and positronium formation in positron-atom collisions: Description in terms of an absorption potential

Pair annihilation in the elementary process of positron (e⁺) collisions with atoms is discussed in terms of an imaginary, absorption potential. This potential represents the QED effect of both direct annihilation during the collision and the indirect one via positronium (Ps) formation in a unified manner. These two mechanisms are inseparable from each other near the threshold for positronium formation, where the collision time becomes comparable to or even longer than the lifetime of the positronium. The theory is applied to the e⁺-H collisions. The dominant s-wave annihilation cross section follows the Baz’ threshold law near the thresholds E_th(1s^1,3S) for the formation of Ps(1s^1,3S). Simple relations between the singlet and triplet cross sections are derived. The spin-averaged annihilation cross section rises sharply but continuously, first across E_th(1s¹S), and then across E_th(1s³S), which lies at 0.841 meV above E_th(1s¹S). The cross section would diverge at E_th(1s¹S) and E_th(1s³S) in the conventional theory where Ps is assumed to have an infinite lifetime. The change of the annihilation probability as the Ps leaves H⁺, is studied by decomposing it into the contributions from direct and indirect annihilation and from the interference between them.