Positron annihilation in the MuPs system

The life-time of the four-body atomic system MuPs (${\mu }^{+}{e}_2^{-}{e}^{+}$ or muonium-positronium) against positron annihilation has been evaluated as $\tau = \frac{1}{\Gamma} \approx 4.076453 \cdot 10^{-10}$ $sec$. Various annihilation rates for MuPs are determined to a good numerical accuracy, e.g., $\Gamma_{2 \gamma} \approx$2.446485$\cdot 10^{9}$ $se{c}^{-1}$, ${\mathrm{\Gamma }}_{3\gamma }\approx$ 6.62798$\cdot {10}^6$ $se{c}^{-1}$, ${\mathrm{\Gamma }}_{4\gamma }\approx$ 3.61680$\cdot 10^{3}$ $se{c}^{-1}$, ${\mathrm{\Gamma }}_{5\gamma }\approx$ 6.32973 $se{c}^{-1}$. The hyperfine structure splitting for the ground state in the MuPs system has also been evaluated as $\mathrm{\Delta }$ = 23.078 $MHz$.