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

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