A dispersive analysis of η${}^{\prime }$ → π${}^{+}$π${}^{-}$γ and η${}^{\prime }$ → ${\ell }^{+}{\ell }^{-}$γ

We present a dispersive representation of the η${}^{\prime }$ transition form factor that allows one to account, in a consistent way, for the effects of ρ-ω mixing in both the isoscalar and the isovector contributions. Using this formalism, we analyze recent data on η${}^{\prime }$ → π${}^{+}$π${}^{-}$γ 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 ρ-ω 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 η${}^{\prime }$ → ${\ell }^{+}{\ell }^{-}$γ, in particular we find the slope parameter b${}_{{\eta }^{\prime }}$ = 1.455(24) GeV${}^{-2}$. With forthcoming data on the latter process, our results establish the necessary framework to improve the evaluation of the η${}^{\prime }$-pole contribution to the anomalous magnetic moment of the muon using experimental input from both η${}^{\prime }$ decay channels.