Spectral-weight sum rules for the hadronic vacuum polarization

We develop a number of sum rules comparing spectral integrals involving judiciously chosen weights to integrals over the corresponding Euclidean two-point function. The applications we have in mind are to the hadronic vacuum polarization that determines the most important hadronic correction $a_\mu^{\rm HVP}$ to the muon anomalous magnetic moment. First, we point out how spectral weights may be chosen that emphasize narrow regions in $\sqrt{s}$, providing a tool to investigate emerging discrepancies between data-driven and lattice determinations of $a_\mu^{\rm HVP}$. Alternatively, for a narrow region around the $\rho$ mass, they may allow for a comparison of the dispersive determination of $a_\mu^{\rm HVP}$ with lattice deteruminations zooming in on the region of the well-known BaBar-KLOE discrepancy. Second, we show how such sum rules can in principle be used for carrying out precision comparisons of hadronic-$\tau$-decay-based data and $e^+e^-\to\mbox{hadrons}(\gamma)$-based data, where lattice computations can provide the necessary isospin-breaking corrections.