Spectral-weight sum rules for the hadronic vacuum polarization
Diogo Boito, Maarten Golterman, Kim Maltman, Santiago Peris
Abstract
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 ${a}_{\ensuremath{\mu}}^{\mathrm{HVP}}$ 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 ${a}_{\ensuremath{\mu}}^{\mathrm{HVP}}$. Alternatively, for a narrow region around the $\ensuremath{\rho}$ mass, they may allow for a comparison of the dispersive determination of ${a}_{\ensuremath{\mu}}^{\mathrm{HVP}}$ with lattice determinations 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-$\ensuremath{\tau}$-decay-based data and ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\text{hadrons}(\ensuremath{\gamma})$-based data, where lattice computations can provide the necessary isospin-breaking corrections.