Vacuum polarization corrections to $\mu^+\mu^-$ spectrum

The energy levels of the $n=1$ and $n=2$ bound states of the $\mu^+\mu^-$ atom (true muonium) are calculated starting from a previously derived potential that correctly describes positronium to order $\alpha^5$ supplemented by the known $\alpha^6$ contributions. All electron vacuum polarization corrections on the true muonium levels are computed to order $\alpha^6$ order. The second order perturbative contributions from all of the $\alpha^4$ and $\alpha^5$ terms in the potential are also included. Additional order $\alpha^6$ contributions from the two photon and three photon annihilation processes are included in the evaluation of the ground state ($1^3S_1-1^1S_0$) hyperfine splitting.