Some radiative corrections to the hyperfine splitting of true muonium: two-photon-exchange contributions

We consider a set of radiative contributions from one-loop lepton vacuum polarization to the hyperfine splitting of true muonium. Improving previous results, we obtain values for the electron-loop coefficients and extract their leading dependence upon $\beta=m_\mu/m_e$. The coefficients are $C^{\mu}_{20,1{\rm VPCT}}=0.734931603\beta$, $C^{\mu}_{20,1{\rm VPCA}}=0.551198702\beta$, and $C^\mu_{20,1{\rm VPT}}=0.419149715\beta$. The mass-independent contribution from three one-loop vacuum polarization is $C_{40,{\rm VPX}}=\frac{1}{127575}\left(-5255-2016\pi^2+51840\zeta(3)\right)$. Contributions from $\tau$ up to $\mathcal{O}(m_\mu\alpha^8)$ are calculated.