The departure of the latest FNAL experimental average for the muon anomalous magnetic moment $$a_\mu =(g_\mu -2)/2$$ measurements, having increased from $$4.2 \sigma $$ (Abi et al. in Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, arXiv:2104.03281 , 2021) to $$5.0 \sigma $$ (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 $$a_\mu $$ is its leading-order hadronic vacuum polarization (HVP) part $$a_\mu ^{HVP-LO}$$ , 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 $$\eta /{\eta ^\prime }\rightarrow \pi ^+ \pi ^- \gamma $$ 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 $$\eta /{\eta ^\prime }$$ and $$\tau $$ decays and of the $$e^+e^- \rightarrow \pi ^+\pi ^-$$ annihilation can be simultaneously considered. A special focus is given to the high-statistics dipion spectra from the $$\eta $$ decay collected by the KLOE/KLOE2 Collaboration and $${\eta ^\prime }$$ 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 required, common to both the $$\eta $$ and $${\eta ^\prime }$$ dipion spectra. Moreover, it is shown that fits involving the $$\eta /{\eta ^\prime }/\tau $$ dipion spectra, and excluding the $$e^+e^- \rightarrow \pi ^+\pi ^-$$ annihilation data, allow for a prediction of the pion vector form factor data $$F_\pi (s)$$ which agrees fairly well with the usual dipion spectra collected in the $$e^+e^- \rightarrow \pi ^+\pi ^-$$ annihilation channel. Even if more precise $$\eta /{\eta ^\prime }/\tau $$ dipion spectra would help to be fully conclusive, this confirms the dispersive approach results for $$a_\mu ^{HVP-LO}$$ and points toward a common non-experimentally dependent origin to this tension with the now well-accepted LQCD result.
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"title": "The $$\\eta /{\\eta ^\\prime }\\rightarrow \\pi ^+ \\pi ^- \\gamma $$ <math> <mrow> <mi>\u03b7</mi> <mo>/</mo> <msup> <mi>\u03b7</mi> <mo>\u2032</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03c0</mi> <mo>+</mo> </msup> <msup> <mi>\u03c0</mi> <mo>-</mo> </msup> <mi>\u03b3</mi> </mrow> </math> decays within BHLS $$_2$$ <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math> and the muon HVP"
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"value": "The departure of the latest FNAL experimental average for the muon anomalous magnetic moment $$a_\\mu =(g_\\mu -2)/2$$ <math> <mrow> <msub> <mi>a</mi> <mi>\u03bc</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>g</mi> <mi>\u03bc</mi> </msub> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> </mrow> </math> measurements, having increased from $$4.2 \\sigma $$ <math> <mrow> <mn>4.2</mn> <mi>\u03c3</mi> </mrow> </math> (Abi et al. in Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, arXiv:2104.03281 , 2021) to $$5.0 \\sigma $$ <math> <mrow> <mn>5.0</mn> <mi>\u03c3</mi> </mrow> </math> (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 $$a_\\mu $$ <math> <msub> <mi>a</mi> <mi>\u03bc</mi> </msub> </math> is its leading-order hadronic vacuum polarization (HVP) part $$a_\\mu ^{HVP-LO}$$ <math> <msubsup> <mi>a</mi> <mi>\u03bc</mi> <mrow> <mi>H</mi> <mi>V</mi> <mi>P</mi> <mo>-</mo> <mi>L</mi> <mi>O</mi> </mrow> </msubsup> </math> , 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 $$\\eta /{\\eta ^\\prime }\\rightarrow \\pi ^+ \\pi ^- \\gamma $$ <math> <mrow> <mi>\u03b7</mi> <mo>/</mo> <msup> <mi>\u03b7</mi> <mo>\u2032</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03c0</mi> <mo>+</mo> </msup> <msup> <mi>\u03c0</mi> <mo>-</mo> </msup> <mi>\u03b3</mi> </mrow> </math> decays in the hidden local symmetry (HLS) context using its BHLS $$_2$$ <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math> broken variant. We thus have at our disposal a framework where the close relationship of the dipion spectra from the $$\\eta /{\\eta ^\\prime }$$ <math> <mrow> <mi>\u03b7</mi> <mo>/</mo> <msup> <mi>\u03b7</mi> <mo>\u2032</mo> </msup> </mrow> </math> and $$\\tau $$ <math> <mi>\u03c4</mi> </math> decays and of the $$e^+e^- \\rightarrow \\pi ^+\\pi ^-$$ <math> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03c0</mi> <mo>+</mo> </msup> <msup> <mi>\u03c0</mi> <mo>-</mo> </msup> </mrow> </math> annihilation can be simultaneously considered. A special focus is given to the high-statistics dipion spectra from the $$\\eta $$ <math> <mi>\u03b7</mi> </math> decay collected by the KLOE/KLOE2 Collaboration and $${\\eta ^\\prime }$$ <math> <msup> <mi>\u03b7</mi> <mo>\u2032</mo> </msup> </math> decay collected by the BESIII Collaboration, and it is shown that the BHLS $$_2$$ <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math> framework provides a fair account of their dipion spectra. More precisely, it is first proven that a single Omn\u00e8s representation real polynomial is required, common to both the $$\\eta $$ <math> <mi>\u03b7</mi> </math> and $${\\eta ^\\prime }$$ <math> <msup> <mi>\u03b7</mi> <mo>\u2032</mo> </msup> </math> dipion spectra. Moreover, it is shown that fits involving the $$\\eta /{\\eta ^\\prime }/\\tau $$ <math> <mrow> <mi>\u03b7</mi> <mo>/</mo> <msup> <mi>\u03b7</mi> <mo>\u2032</mo> </msup> <mo>/</mo> <mi>\u03c4</mi> </mrow> </math> dipion spectra, and excluding the $$e^+e^- \\rightarrow \\pi ^+\\pi ^-$$ <math> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03c0</mi> <mo>+</mo> </msup> <msup> <mi>\u03c0</mi> <mo>-</mo> </msup> </mrow> </math> annihilation data, allow for a prediction of the pion vector form factor data $$F_\\pi (s)$$ <math> <mrow> <msub> <mi>F</mi> <mi>\u03c0</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </math> which agrees fairly well with the usual dipion spectra collected in the $$e^+e^- \\rightarrow \\pi ^+\\pi ^-$$ <math> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03c0</mi> <mo>+</mo> </msup> <msup> <mi>\u03c0</mi> <mo>-</mo> </msup> </mrow> </math> annihilation channel. Even if more precise $$\\eta /{\\eta ^\\prime }/\\tau $$ <math> <mrow> <mi>\u03b7</mi> <mo>/</mo> <msup> <mi>\u03b7</mi> <mo>\u2032</mo> </msup> <mo>/</mo> <mi>\u03c4</mi> </mrow> </math> dipion spectra would help to be fully conclusive, this confirms the dispersive approach results for $$a_\\mu ^{HVP-LO}$$ <math> <msubsup> <mi>a</mi> <mi>\u03bc</mi> <mrow> <mi>H</mi> <mi>V</mi> <mi>P</mi> <mo>-</mo> <mi>L</mi> <mi>O</mi> </mrow> </msubsup> </math> and points toward a common non-experimentally dependent origin to this tension with the now well-accepted LQCD result."
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