Comparison of the hadronic vacuum polarization between hadronic $\tau$-decay data and lattice QCD

We compare the isospin-one, vector-current hadronic vacuum polarization (HVP) obtained from isospin-symmetric lattice QCD with that obtained from a dispersive representation employing inclusive hadronic $\tau$ decay data corrected for isospin breaking. We consider the subtracted HVP evaluated at squared Euclidean momenta ranging from $0.5$ GeV$^2$ to $12$ GeV$^2$, together with the light-quark-connected HVP contribution to the muon anomalous magnetic moment and the short-, intermediate- and long-distance RBC/UKQCD window components thereof. Dispersive contributions from the region of hadronic invariant masses above the $\tau$ mass are evaluated using perturbative QCD. We also consider dispersive determinations using $\tau$ data only for contributions from two-pion, or two-pion and four-pion, modes, and evaluating the remaining contributions using exclusive-mode $e^+e^-\to\mbox{hadrons}$ cross sections up to about 2 GeV, lessening the dependence on perturbation theory. We find generally good agreement between lattice and $\tau$-based results. However, a comparison of $\tau$-based window-quantity contributions for the two four-pion modes to expectations for those contributions based on the Pais relations and $e^+e^-$ four-pion cross sections, reveals significant differences for the $2\pi^-\pi^+\pi^0$ mode.