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AbstractThe electric field surrounding a single positron in a metal is screened by an increase in the local electron density which, in the case of nearly free-electron metals (like Al, Na, etc.), has a radial distribution similar to that of the electron in positronium (Ps). In such metals, a singlet pair of positrons would experience an attractive interaction and at low enough electron densities could possibly form a bound state that is held together by exchange and correlation energies, thus forming structures analogous to that of the positronium molecule (Ps$$_2$$ ${}_2$), with binding energies of a few tenths of an eV. Such di-positrons could be prevalent at positron densities of around 10$$^{18}$$ ${}^{18}$cm$$^{-3}$$ ${}^{-3}$and, if so, would be evident from an apparent broadening of the sharp step at the Fermi surface in measurements of the electron momentum distribution by the angular correlation of the 2$$\gamma $$ $\gamma$annihilation radiation. Even if di-positrons are not directly formed in a metal, optical spectroscopy of Ps$$_2$$ ${}_2$formed in vacuum via pairs of positrons simultaneously being emitted from the surface could be applied to the direct measurement of the momentum distribution of Cooper pairs. If they exist, di-positrons in metals would yield interesting information about electron and positron interactions and at very high densities might allow the study of a di-positron Bose–Einstein condensate immersed in an electron gas. Graphic Abstract