Spectral-weight sum rules for the hadronic vacuum polarization

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 $a_{\mu }^{\mathrm{HVP}}$ 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 $a_{\mu }^{\mathrm{HVP}}$. Alternatively, for a narrow region around the $\rho$ mass, they may allow for a comparison of the dispersive determination of $a_{\mu }^{\mathrm{HVP}}$ 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 $e^+e^-\to\mbox{hadrons}(γ)$-based data, where lattice computations can provide the necessary isospin-breaking corrections.

33 pages, 5 figures; expanded discussions, version to be published, no changes in results

References in corpus (10)

• Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm
• The $g-2$ of charged leptons, $\alpha (M_Z^2)$ and the hyperfine splitting of muonium
• Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order
• Window observable for the hadronic vacuum polarization contribution to the muon $g-2$ from lattice QCD
• Data-driven evaluations of Euclidean windows to scrutinize hadronic vacuum polarization
• On the extraction of spectral densities from lattice correlators
• The muon anomalous magnetic moment with staggered fermions: is the lattice spacing small enough?
• The strong coupling from an improved $\tau$ vector isovector spectral function
• $F_K/F_{\pi }$ from Möbius domain-wall fermions solved on gradient-flowed HISQ ensembles
• Remarks about weighted energy integrals over Minkowski spectral functions from Euclidean lattice data

Cited by in corpus (5)

• Probing the energy-smeared R-ratio on the lattice
• Data-based determination of the isospin-limit light-quark-connected contribution to the anomalous magnetic moment of the muon
• Muon $g-2$: Lattice calculations of the hadronic vacuum polarization
• Renormalization group summation and analytic continuation from spacelike to timeline regions
• Model-independent extrapolation of MUonE data with Padé and D-Log approximants