Improved Standard-Model predictions for $η^{(\prime)}\to \ell^+ \ell^-$

TL;DR

The paper delivers a high-precision Standard-Model prediction for the rare decays $\eta^{(\prime)}\to \ell^+\ell^-$ by employing a dispersive reconstruction of the $P\gamma^*\gamma^*$ transition form factors, decomposing the low-energy contributions into isovector, isoscalar, and effective-pole components, and incorporating mass-corrected asymptotics. By robustly accounting for the imaginary parts from subleading channels beyond the two-photon cut, the authors obtain precise branching fractions and quantify uncertainties from the dispersive input, high-energy matching (Brodsky–Lepage), and asymptotics. The results show a mild tension with experiment for $\eta\to \mu^+\mu^-$ and reveal notable mass-correction effects for $\eta'$ decays, while providing a concrete framework to constrain Beyond-Standard-Model operators and light mediators. Overall, this work strengthens SM tests in pseudoscalar dilepton decays and offers quantitative benchmarks for future high-precision measurements and BSM searches.

Abstract

The rare decays $η^{(\prime)}\to\ell^+\ell^-$, $\ell\in\{e,μ\}$, are highly suppressed in the Standard Model, both by their chirality structure and the required loop attaching the lepton line to the $η^{(\prime)}\toγ^*γ^*$ matrix element. The latter is described by a single scalar function, the transition form factor, which has recently been studied in great detail for $η^{(\prime)}$ in the context of the pseudoscalar-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon. Based on these results, we evaluate the corresponding prediction for the $η^{(\prime)}$ dilepton decays, supplemented by an improved evaluation of the asymptotic contributions including pseudoscalar mass effects. In particular, the dispersive representation for the $η^{(\prime)}$ transition form factors allows us, for the first time, to perform a robust evaluation of the imaginary parts due to subleading channels besides the dominant two-photon cut. Our final results are $\text{Br}[η\to e^+e^-]=5.37(4)(2)[4]\times 10^{-9}$, $\text{Br}[η\to μ^+μ^-]=4.54(4)(2)[4]\times 10^{-6}$, $\text{Br}[η'\to e^+e^-]=1.80(2)(3)[3]\times 10^{-10}$, and $\text{Br}[η'\to μ^+μ^-]=1.22(2)(2)[3]\times 10^{-7}$, where the errors refer to the uncertainty in the normalized branching fraction, the one propagated from $\text{Br}[η^{(\prime)}\toγγ]$, and the total uncertainty, respectively. The branching fraction for $η\toμ^+μ^-$ exhibits a mild $1.6σ$ tension with experiment, and we explore the bounds that can be derived on physics beyond the Standard Model.

Figures (2)

• Figure 1: Dilepton decay $P\to\ell^+\ell^-$, $P=\pi^0,\eta,\eta^\prime$, $\ell=e,\mu$, in the SM, via two-photon exchange (left) and $Z$-boson exchange (right). The gray circle denotes the pseudoscalar TFF into two virtual photons.
• Figure 2: Normalized isovector double-spectral densities of Eq. \ref{['Eq:doubspecdens']} for $\eta$ (left) and $\eta'$ (right).