Hadronic Vacuum Polarization in True Muonium

TL;DR

This paper reevaluates the hadronic vacuum polarization contribution to true muonium's hyperfine splitting using improved form factors and direct integration of experimental data, resulting in a precise energy shift estimate.

Contribution

It introduces two refined methods for calculating the hadronic vacuum polarization effect, enhancing accuracy over previous estimates.

Findings

Energy shift in hyperfine splitting: 276196(51) MHz

More accurate estimate of hadronic vacuum polarization contribution

Direct integration of Drell ratio improves precision

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.