Muonium decay

Modifications of the ${\mu }^{+}$ lifetime in matter due to muonium $(M={\mu }^{+}{e}^{-})$ formation and other medium effects are examined. Muonium and free ${\mu }^{+}$ decay spectra are found to differ at $\mathcal{O}\left(\alpha m_e{/m}_{\mu }\right)$ from Doppler broadening and $\mathcal{O}\left({\alpha }^2{m}_e{/m}_{\mu }\right)$ from the Coulomb bound state potential. However, both types of corrections are shown to cancel in the total decay rate due to Lorentz and gauge invariance respectively, leaving a very small time dilation lifetime difference, $\left({\tau }_M-{\tau }_{{\mu }^{+}}\right)/{\tau }_{{\mu }^{+}}={\alpha }^2{m}_e^2{/2m}_{\mu }^2\simeq 6\times {10}^{-10},$ as the dominant bound state effect. It is argued that other medium effects on the stopped ${\mu }^{+}$ lifetime are similarly suppressed.