A dispersive analysis of η ' → π + π - γ and η ' → ℓ + ℓ - γ

Abstract

We present a dispersive representation of the ${\eta }^{\text{'}}$ transition form factor that allows one to account, in a consistent way, for the effects of $\rho$ - $\omega$ mixing in both the isoscalar and the isovector contributions. Using this formalism, we analyze recent data on ${\eta }^{\text{'}}\to {\pi }^{+}{\pi }^{-}\gamma$ to constrain the isovector part of the form factor, individually and in combination with data for the pion vector form factor, which suggests a tension in the $\rho$ - $\omega$ mixing parameter. As a first application, we use our results, in combination with the most recent input for the isoscalar part of the form factor, to predict the corresponding spectrum of ${\eta }^{\text{'}}\to {\ell }^{+}{\ell }^{-}\gamma$ , in particular we find the slope parameter $b_{{\eta }^{\text{'}}}=1.455\left(24\right){\text{GeV}}^{-2}$ . With forthcoming data on the latter process, our results establish the necessary framework to improve the evaluation of the ${\eta }^{\text{'}}$ -pole contribution to the anomalous magnetic moment of the muon using experimental input from both ${\eta }^{\text{'}}$ decay channels.