Three-Photon Annihilation of an Electron-Positron Pair

Annihilation of an electron-positron pair accompanied by the emission of three photons is discussed for the case of small relative velocity of the two particles. The energy spectrum of the photons is derived and the cross section for the process is calculated. The ratio of this cross section to that of ordinary two-photon annihilation is found to be 1:370. When the result is applied to the ${}_{}{}^3{}_{}{}^{}S$ ground state of the positronium atom, for which two-photon annihilation is forbidden, one finds a lifetime 1.4×${10}^{-7\mathrm{}}$ sec.