The $\eta /{\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ Decays within BHLS${}_2$ and the Muon HVP

Abstract

The departure of the latest FNAL experimental average for the muon anomalous magnetic moment $a_{\mu }=(g_{\mu }-2)/2$ measurements having increased from $4.2\sigma$ to $5.0\sigma$, with respect to the White Paper consensus, it may indicate a hint for new physics. As the most delicate piece of $a_{\mu }$ is its leading order HVP part $a_{\mu }^{HVP-LO}$, methods to ascertain its theoretical value are crucial to interpret this discrepancy. We propose to examine the dipion spectra from the $\eta /{\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ 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 $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ annihilation can be simultaneously considered. A special focus is put to the high statistic 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 requested, 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 $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ annihilation data, allow for a prediction of the pion vector form factor data $F_{\pi }(s)$ which fairly agree with the usual dipion spectra collected in the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ 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 towards a common non experiment-dependent origin to this tension with the now well accepted LQCD result.

Comments: Revision 1