Internal structure and positron annihilation in the four-body MuPs system

A large number of bound state properties of the four-body muonium-positronium system MuPs (or ${\mu }^{+}{e}_2^{-}{e}^{+}$) are determined to high accuracy. Based on these expectation values we predict that the weakly-bound four-body MuPs system has the `two-body’ cluster structure Mu + Ps. The two neutral clusters Mu (${\mu }^{+}{e}^{-}$) and Ps ($e^{+}{e}^{-}$) interact with each other by the attractive van der Waals forces. By using our expectation values of the electron-positron delta-functions we evaluated the half-life ${\tau }_a$ of the MuPs system against annihilation of the electron-positron pair: $\tau_a = \frac{1}{\Gamma} \approx 4.071509 \cdot 10^{-10}$ $sec$. The hyperfine structure splitting of the ground state in the MuPs system evaluated with our expectation values is $\mathrm{\Delta }\approx$ 23.064(5) $MHz$.