A dispersive analysis of ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ and ${\eta }^{\prime }\to {\ell }^{+}{\ell }^{-}\gamma$

Abstract

We present a dispersive representation of the ${\eta }^{\prime }$ 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 ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ to constrain the isovector part of the form factor, individually and in combination with data for the pion vector form factor. 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 ${\eta }^{\prime }\to {\ell }^{+}{\ell }^{-}\gamma$, in particular we find the slope parameter $b_{{\eta }^{\prime }}=1.431(23){\text{GeV}}^{-2}$. With forthcoming data on the latter process, our results establish the necessary framework to improve the evaluation of the ${\eta }^{\prime }$-pole contribution to the anomalous magnetic moment of the muon using experimental input from both ${\eta }^{\prime }$ decay channels.

18 pages, 2 figures, tension in ${\epsilon }_{\rho \omega }$ resolved by including higher orders in $e^2$

References in corpus (1)

• Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization

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