The $η/η' \rightarrow π^+ π^- γ$ Decays within BHLS$_2$ and the Muon HVP
The departure of the latest FNAL experimental average for the muon anomalous magnetic moment $a_μ=(g_μ-2)/2$ measurements having increased from $4.2 σ$ to $5.0 σ$, with respect to the White Paper consensus, it may indicate a hint for new physics. As the most delicate piece of $a_μ$ is its leading order HVP part $a_μ^{HVP-LO}$, methods to ascertain its theoretical value are crucial to interpret this discrepancy. We propose to examine the dipion spectra from the $η/η' \rightarrow π^+ π^- γ$ 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 $η/η'$ and $τ$ decays and of the $e^+e^- \to π^+π^-$ annihilation can be simultaneously considered. A special focus is put to the high statistic dipion spectra from the $η$ decay collected by the KLOE/KLOE2 Collaboration and $η'$ 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 requested, common to both the $η$ and $η'$ dipion spectra. Moreover, it is shown that fits involving the $η/η'/τ$ dipion spectra, and excluding the $e^+e^- \to π^+π^-$ annihilation data, allow for a prediction of the pion vector form factor data $F_π(s)$ which fairly agree with the usual dipion spectra collected in the $e^+e^- \to π^+π^-$ annihilation channel. Even if more precise $η/η'/τ$ dipion spectra would help to be fully conclusive, this confirms the Dispersive Approach results for $a_μ^{HVP-LO}$ and points towards a common non experiment-dependent origin to this tension with the now well accepted LQCD result.