The / ^ ^+ ^- decays within BHLS₂ and the muon HVP

Abstract The departure of the latest FNAL experimental average for the muon anomalous magnetic moment a_ = (g_ -2) /2 a μ = (g μ - 2) / 2 measurements, having increased from 4. 2 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 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_ a μ is its leading-order hadronic vacuum polarization (HVP) part a_ ^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 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 ′ and τ decays and of the e^+e^- ^+ ^- e + e - → π + π - annihilation can be simultaneously considered. A special focus is given to the high-statistics 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 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 and ^ η ′ dipion spectra. Moreover, it is shown that fits involving the / ^ / η / η ′ / τ dipion spectra, and excluding the e^+e^- ^+ ^- e + e - → π + π - annihilation data, allow for a prediction of the pion vector form factor data F_ (s) F π (s) which agrees fairly well with the usual dipion spectra collected in the e^+e^- ^+ ^- e + e - → 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_ ^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.