Hadronic Vacuum Polarization in True Muonium

Abstract: The leading-order hadronic vacuum polarization contribution to the hyperfine splitting of true muonium is reevaluated in two ways. The first considers a more complex pionic form factor and better estimates of the perturbative QCD contributions. The second, more accurate method directly integrates the Drell ratio $R(s)$ to obtain $C_{1,\rm hvp}=-0.0489(3)$. This corresponds to an energy shift in the hyperfine splitting of $\Delta E^\mu_{hfs,\rm hvp}=276196(51)$ MHz.