Hadronic vacuum polarization in true muonium

Abstract

In order to reduce the theoretical uncertainty in the prediction, the leading-order hadronic vacuum polarization contribution to the hyperfine splitting of true muonium is reevaluated in two ways. A more complex pionic form factor and better estimates of the perturbative QCD contributions are used to study the model dependence of the previous calculation. The second, more accurate method directly integrates the Drell ratio $R(s)$ to obtain ${C}_{1,\mathrm{HVP}}=\ensuremath{-}0.04874(9)$. This corresponds to an energy shift in the hyperfine splitting (HFS) of $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{HFS},\mathrm{HVP}}^{\ensuremath{\mu}}=\ensuremath{-}8202(16)$ MHz and represents a factor-of-50 reduction in the theoretical uncertainty from hadronic sources. We also compute the contribution in positronium, which is too small at present to detect.