Spectral-weight sum rules for the hadronic vacuum polarization

TL;DR

This paper introduces spectral-weight sum rules to compare spectral integrals and Euclidean two-point functions, aiding in resolving discrepancies in hadronic vacuum polarization data relevant to the muon g-2 anomaly.

Contribution

It develops sum rules with tailored spectral weights for detailed comparison of data-driven and lattice determinations of hadronic vacuum polarization.

Findings

Spectral weights can emphasize narrow energy regions for discrepancy analysis.

Sum rules enable precise comparison between different experimental and lattice data.

Method can help resolve the BaBar-KLOE discrepancy in muon g-2 calculations.

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 $a_\mu^{\rm HVP}$ 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.