Hyperfine Splitting in True Muonium to $\mathcal{O}(m_{\mu }{\alpha }^6)$: Two Photon Annihilation Contribution from Other Flavors

The two-photon-annihilation contribution to the true muonium hyperfine splitting arising from $e$ and $\tau$ loops is obtained analytically at order $m_{\mu }{\alpha }^6$. The contribution to the hyperfine splitting is $-2.031092873 m_\mu\alpha^6/n^3\pi^2=-793.926988/n^3$ MHz. The contribution to the triplet true muonium decay rate has also been obtained and was found to be $9.825708266 m_\mu\alpha^6/n^3\pi^2=3840.737698/n^3$ MHz. Additional results have been computed for other purely leptonic bound states.