Annihilation of the electron-positron pairs in polyelectrons

Annihilation of the electron-positron pairs (or (e(-), e(+))-pairs, for short) in various polyelectrons e(n)(+)e(m)(-) = e(m)(-)e(n)(+) (where n >= 1 and m >= 1) is considered. In particular, we discuss the three- and four-photon annihilation of the ( e-, e+)pairs in the three- body Ps(-) ion and four-body bi-positronium system Ps(2). It is shown that the five-body e(2)(+)e(3)(-) ion is an unbound system. The closed-form expression is derived for the amplitude-square vertical bar M vertical bar(2) of the three- photon annihilation of (e(-), e(+))-pair at arbitrary energies of the colliding particles. Analogous amplitude-square vertical bar M vertical bar(2) for the four-photon annihilation is reduced to the form which is convenient for future analytical calculations. A method which can be used to produce macroscopic polyelectrons is briefly discussed.