The $η/η' \rightarrow π^+ π^- γ$ Decays within BHLS$_2$ and the Muon HVP

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 $4.2 \sigma$ to $5.0 \sigma$, with respect to the White Paper consensus, it may indicate a hint for new physics. As the most delicate piece of $a_\mu$ is its leading order HVP part $a_\mu^{HVP-LO}$, methods to ascertain its theoretical value are crucial to interpret this discrepancy. We propose to examine the dipion spectra from the $\eta/\eta' \rightarrow \pi^+ \pi^- \gamma$ 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 $\eta/\eta'$ and $\tau$ decays and of the $e^+e^- \to \pi^+\pi^-$ annihilation can be simultaneously considered. A special focus is put to the high statistic dipion spectra from the $\eta$ decay collected by the KLOE/KLOE2 Collaboration and $\eta'$ 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 Omn`es representation real polynomial is requested, common to both the $\eta$ and $\eta'$ dipion spectra. Moreover, it is shown that fits involving the $\eta/\eta'/\tau$ 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 fairly agree with the usual dipion spectra collected in the $e^+e^- \to \pi^+\pi^-$ annihilation channel. Even if more precise $\eta/\eta'/\tau$ dipion spectra would help to be fully conclusive, this confirms the Dispersive Approach results for $a_\mu^{HVP-LO}$ and points towards a common non experiment-dependent origin to this tension with the now well accepted LQCD result.