Hadronic vacuum polarization: Comparing lattice QCD and data-driven results in systematically improvable ways

The precision with which hadronic vacuum polarization (HVP) is obtained determines how accurately important observables, such as the muon anomalous magnetic moment $a_{\mu }$ or the low-energy running of the electromagnetic coupling $\alpha$, are predicted. The two most precise approaches for determining HVP are dispersive relations combined with $e^{+}{e}^{-}\to \text{hadrons}$ cross section data and lattice QCD. However, the results obtained in these two approaches display significant tensions, whose origins are not understood. Here we present a framework that sheds light on this issue and—if the two approaches can be reconciled—allows them to be combined. Via this framework, we test the hypothesis that the tensions can be explained by modifying the R-ratio in different intervals of center-of-mass energy $\sqrt{s}$. As ingredients, we consider observables that have been precisely determined in both approaches. These are the leading hadronic contributions to $a_{\mu }$, to the so-called intermediate window observable, and to the running of $\alpha$ between spacelike virtualities 1 and $10{\mathrm{GeV}}^2$ (for which only a preliminary lattice result exists). Our tests take into account all uncertainties and correlations, as well as uncertainties on uncertainties in the lattice results. For instance, using this framework we show that results obtained in the two approaches can be made to agree, for all three observables, by modifying the $\rho$ peak in the experimental spectrum. More specifically, we show that this requires a common $\sim 5\%$ increase in the contributions of the peak to each of the three observables. This result is robust against the presence or absence of the running of $\alpha$ in the comparison. However, such an increase is much larger than the uncertainties on the measured R-ratio. We also discuss a variety of generalizations of the methods used here, as well as the limits in the information that can be extracted from the R-ratio via a finite set of observables.