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

Abstract

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 $C_{20,1\mathrm{VPCT}}^{\mu }=0.734931603\beta$, $C_{20,1\mathrm{VPCA}}^{\mu }=0.551198702\beta$, and $C_{20,1\mathrm{VPT}}^{\mu }=0.419149715\beta$. The mass-independent contribution from three one-loop vacuum polarization is $C_{40,\mathrm{VPX}}=\frac{1}{127575}\left(-5255-2016{\pi }^2+51840\zeta (3)\right)$. Contributions from $\tau$ up to $\mathcal{O}(m_{\mu }{\alpha }^8)$ are calculated.

Comments: 5 pages, 1 figure, 2 tables