A dispersive analysis of $\eta'\to\pi^+\pi^-\gamma$ and $\eta'\to \ell^+\ell^-\gamma$

TL;DR

This paper develops a dispersive approach to analyze the $ ext{eta'}$ transition form factor, accounting for $ ho$-$ ext{omega}$ mixing, and uses it to connect experimental data with predictions relevant for muon magnetic moment calculations.

Contribution

It introduces a dispersive formalism that incorporates $ ho$-$ ext{omega}$ mixing effects into the $ ext{eta'}$ transition form factor analysis, enabling improved predictions for related decay processes.

Findings

Constrained the isovector part of the form factor using recent decay data.

Predicted the $ ext{eta'} o ext{l}^+ ext{l}^- ext{gamma}$ spectrum and slope parameter.

Established a framework for refining the $ ext{eta'}$-pole contribution to muon g-2.

Abstract

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