The $$\eta /{\eta ^\prime }\rightarrow \pi ^+ \pi ^- \gamma $$ η / η ′ → π + π - γ decays within BHLS $$_2$$ 2 and the muon HVP

Abstract

Abstract The departure of the latest FNAL experimental average for the muon anomalous magnetic moment $$a_\mu =(g_\mu -2)/2$$ a μ = ( g μ - 2 ) / 2 measurements, having increased from $$4.2 \sigma $$ 4.2 σ (Abi et al. in Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, arXiv:2104.03281 , 2021) to $$5.0 \sigma $$ 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 $$a_\mu $$ a μ is its leading-order hadronic vacuum polarization (HVP) part $$a_\mu ^{HVP-LO}$$ 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 $$\eta /{\eta ^\prime }\rightarrow \pi ^+ \pi ^- \gamma $$ 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 $$\eta /{\eta ^\prime }$$ η / η ′ and $$\tau $$ τ decays and of the $$e^+e^- \rightarrow \pi ^+\pi ^-$$ e + e - → π + π - annihilation can be simultaneously considered. A special focus is given to the high-statistics dipion spectra from the $$\eta $$ η decay collected by the KLOE/KLOE2 Collaboration and $${\eta ^\prime }$$ 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 $$\eta $$ η and $${\eta ^\prime }$$ η ′ dipion spectra. Moreover, it is shown that fits involving the $$\eta /{\eta ^\prime }/\tau $$ η / η ′ / τ dipion spectra, and excluding the $$e^+e^- \rightarrow \pi ^+\pi ^-$$ e + e - → π + π - annihilation data, allow for a prediction of the pion vector form factor data $$F_\pi (s)$$ F π ( s ) which agrees fairly well with the usual dipion spectra collected in the $$e^+e^- \rightarrow \pi ^+\pi ^-$$ e + e - → π + π - annihilation channel. Even if more precise $$\eta /{\eta ^\prime }/\tau $$ η / η ′ / τ dipion spectra would help to be fully conclusive, this confirms the dispersive approach results for $$a_\mu ^{HVP-LO}$$ 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.