Review of hadronic vacuum polarization calculations via $e^+e^-$ measurements

Abstract

The discrepancy on the muon anomalous magnetic moment values obtained via a direct measurement and via a data-driven theory determination that uses the experimentally measured hadronic cross section, is among the long standing and most significant deviations from the Standard Model predictions. The recently presented final result of the direct measurement performed at the experiment at Fermilab, with an impressive accuracy of 127 parts-per-billion, further stresses the need for a theory estimate of comparable accuracy. The $e^+e^-$ hadronic cross section is the experimental input to the dispersive integral for the calculation of the hadronic contribution to the $g-2$, which is largely dominated by the $e^+e^-\to π^+π^-$ channel. Precise measurements of the $e^+e^-\to π^+π^-$ cross section with a sub-percent accuracy, in the energy region of the $ρ$-resonance peak, have been performed by several experiments, but the results differ way more than the published accuracies limiting the comparison with the Fermilab direct measurement. New results are expected to become available in the near future from KLOE, CMD-3, SND and BESIII experiments, while a new measurement from the BABAR collaboration is presented today and its impact on the present situation is discussed. This measurement makes use of the entire BABAR data set and adopts a different analysis strategy, with results (still preliminary) largely independent of the results published in the 2009 BABAR publication.

Figures (8)

• Figure 1: Relative contributions of the QED, hadronic (had) and EW predictions (blue bars) and their uncertainties (red bars) in comparison with the current measurement precision (dashed vertical line).
• Figure 2: Left: the total hadronic $e^+e^-$ annihilation rate $R(s)$ as a function of center-of-mass energy $\sqrt{s}$. Inclusive measurements from BES and KEDR are shown as data points, while the narrow blue bands correspond to the sum of exclusive channels, obtained using HVPTools Davier:2010rnx. Also shown for the purpose of illustration is the prediction from massless perturbative QCD (solid red line). Figure taken from Ref. Davier:2019can. Right: Contributions to the total hadronic $R$-ratio from different final states at low energies. The full $R$-ratio is shown in light blue. Each final state is included as a new layer on top in decreasing order of the size of its contribution to $a_\mu^\text{HVP LO}$. Figure taken from Ref. Keshavarzi:2018mgv.
• Figure 3: Relative differences in terms of pion form factors between CMD3 and previous scan-based measurements (left) and ISR photon-based measurements (right). The green band corresponds to the systematic uncertainty of the CMD-3 measurement. Figures taken from Ref. CMD-3:2023alj.
• Figure 4: The $\pi\pi(\gamma)$ contribution to the LO HVP from the energy range $0.6 <\sqrt{s} < 0.88$ GeV obtained from the CMD3 data and the results of the other experiments. Figure (Table) taken (adapted) from Ref. CMD-3:2023alj.
• Figure 5: Illustration of the kinematic (template) fit for the signals ($\pi\pi$ or $\mu\mu$) and background ($KK$ and $ee\gamma$) separation for two selected mass intervals at $0.30<m_{\pi\pi}<0.31$ GeV/c$^2$ (left) and $0.770<m_{\pi\pi}<0.772$ GeV/c$^2$ (right).