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 $\Delta \approx$ 23.064(5) $MHz$.