Production of parapositronium and orthopositronium at relativistic heavy ion colliders

We consider the orthopositronium and parapositronium production in the process $A\overrightarrow{A}AA+$positronium where A is a nucleus with the charge number Z. The inclusive cross section and the energy distribution of the relativistic positronium are calculated which are of primary interest from the experimental point of view. The accuracy of the corresponding cross sections is given by omitting terms $\sim (Z\alpha {)}^2{/L}^2$ for the parapositronium and $\sim (Z\alpha {)}^2/L$ for the orthopositronium production where $L=\ln {\gamma }^2\approx 9$ and 16 for the RHIC and the LHC. Within this accuracy the multiphoton (Coulomb) corrections are taken into account. We show that the RHIC and the LHC will be positronium factories with a production rate of about ${10}^5–{10}^8$ relativistic positronium per day. The fraction of the orthopositronium is expected to be of the same order as that of the parapositronium for Au-Au and Pb-Pb collisions.