The η /η ^ → π ⁺ π ⁻ γ decays within BHLS₂ and the muon HVP

The departure of the latest FNAL experimental average for the muon anomalous magnetic moment a_μ =(g_μ -2)/2 a μ = ( g μ - 2 ) / 2 measurements, having increased from 4.2 σ 4.2 σ (Abi et al. in Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, arXiv:2104.03281 , 2021) to 5.0 σ 5.0 σ (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_μ a μ is its leading-order hadronic vacuum polarization (HVP) part a_μ HVP-LO a μ H V P - L O , 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 η /η ^ → π ⁺ π ⁻ γ η / η ′ → π + π - γ decays in the hidden local symmetry (HLS) context using its BHLS ₂ 2 broken variant. We thus have at our disposal a framework where the close relationship of the dipion spectra from the η /η ^ η / η ′ and τ τ decays and of the e⁺e⁻ → π ⁺π ⁻ e + e - → π + π - annihilation can be simultaneously considered. A special focus is given to the high-statistics dipion spectra from the η η decay collected by the KLOE/KLOE2 Collaboration and η ^ η ′ decay collected by the BESIII Collaboration, and it is shown that the BHLS ₂ 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 η η and η ^ η ′ dipion spectra. Moreover, it is shown that fits involving the η /η ^ /τ η / η ′ / τ dipion spectra, and excluding the e⁺e⁻ → π ⁺π ⁻ e + e - → π + π - annihilation data, allow for a prediction of the pion vector form factor data F_π (s) F π ( s ) which agrees fairly well with the usual dipion spectra collected in the e⁺e⁻ → π ⁺π ⁻ e + e - → π + π - annihilation channel. Even if more precise η /η ^ /τ η / η ′ / τ dipion spectra would help to be fully conclusive, this confirms the dispersive approach results for a_μ HVP-LO a μ H V P - L O and points toward a common non-experimentally dependent origin to this tension with the now well-accepted LQCD result.