Eur. Phys. J. C (2024) 84: 295
https://doi.org/10.1140/epjc/s10052-024-12608-w

Regular Article - Theoretical Physics

The $\eta /{\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ decays within BHLS${}_2$ and the muon HVP

¹ LPNHE des Universités Paris VI et Paris VII, IN2P3/CNRS, 75252, Paris, France
² Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489, Berlin, Germany

^b luigi.delbuono@lpnhe.in2p3.fr

Received: 28 September 2023
Accepted: 25 February 2024
Published online: 21 March 2024

Abstract

The departure of the latest FNAL experimental average for the muon anomalous magnetic moment $a_{\mu }=(g_{\mu }-2)/2$ measurements, having increased from $4.2\sigma$ (Abi et al. in Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, arXiv:2104.03281, 2021) to $5.0\sigma$ (Muon g-2, D.P. Aguillard et al. in Measurement of the positive muon anomalous magnetic moment to 0.20 ppm, arXiv:2308.06230, 2023) with respect to the white paper (WP) consensus (Aoyama et al. in Phys Rep 887:1, arXiv:2006.04822, 2020), may indicate a hint for new physics. As the most delicate piece of $a_{\mu }$ is its leading-order hadronic vacuum polarization (HVP) part $a_{\mu }^{HVP-LO}$, methods to ascertain its theoretical value are crucial to appropriately interpreting this departure from the measurement. We, therefore, propose to closely examine the dipion spectra from the $\eta /{\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ decays in the hidden local symmetry (HLS) context using its BHLS${}_2$ broken variant. We thus have at our disposal a framework where the close relationship of the dipion spectra from the $\eta /{\eta }^{\prime }$ and $\tau$ decays and of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ annihilation can be simultaneously considered. A special focus is given to the high-statistics dipion spectra from the $\eta$ decay collected by the KLOE/KLOE2 Collaboration and ${\eta }^{\prime }$ decay collected by the BESIII Collaboration, and it is shown that the BHLS${}_2$ framework provides a fair account of their dipion spectra. More precisely, it is first proven that a single Omnès representation real polynomial is required, common to both the $\eta$ and ${\eta }^{\prime }$ dipion spectra. Moreover, it is shown that fits involving the $\eta /{\eta }^{\prime }/\tau$ dipion spectra, and excluding the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ annihilation data, allow for a prediction of the pion vector form factor data $F_{\pi }(s)$ which agrees fairly well with the usual dipion spectra collected in the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ annihilation channel. Even if more precise $\eta /{\eta }^{\prime }/\tau$ dipion spectra would help to be fully conclusive, this confirms the dispersive approach results for $a_{\mu }^{HVP-LO}$ and points toward a common non-experimentally dependent origin to this tension with the now well-accepted LQCD result.