Formal calculation of the pick-off annihilation rate for ortho- and parapositronium

The annihilation of a positronium atom in matter is often described as occurring through two alternative channels; self-annihilation (involving the electron in the atom itself) and pick-off annihilation (involving an external electron). For many years it has been debated if it is correct to assume the same pick-off annihilation rate for orthopositronium and for parapositronium. We address this problem with a formal calculation that takes into full account the spin variables of all the electrons involved in the process as well as the indistinguishability of the electrons. On the basis of our results, we make clear the conditions leading to a definition of the pick-off rate independent of the spin state of the positronium atom