Precision evaluation of the $\eta$- and $\eta'$-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon

TL;DR

This paper calculates the contributions of η and η' meson poles to the muon's anomalous magnetic moment using dispersion relations, incorporating various theoretical constraints and the effects of tensor meson exchanges.

Contribution

It provides the first estimate of factorization-breaking corrections from the $a_2$ tensor meson exchange in η and η' transition form factors.

Findings

η-pole contribution: 14.7(9)×10⁻¹¹

η'-pole contribution: 13.5(7)×10⁻¹¹

Total pseudoscalar-pole contribution: 91.2(+2.9/-2.4)×10⁻¹¹

Abstract

Next to the $\pi^0$ pole, $\eta$ and $\eta'$ intermediate states give rise to the leading singularities of the hadronic light-by-light tensor, resulting in sizable contributions to the anomalous magnetic moment of the muon $a_\mu$. The strength of the poles is determined by the respective transition form factors (TFFs) to two (virtual) photons. We present a calculation of these TFFs that implements a number of low- and high-energy constraints, including the $\eta^{(\prime)}\to\gamma\gamma$ decay widths, $\eta^{(\prime)}\to\pi^+\pi^-\gamma$ spectra, chiral symmetry for the $\eta^{(\prime)}\to 2(\pi^+\pi^-)$ amplitudes, vector-meson couplings, and asymptotic limits. Crucially, we investigate the role of the leading left-hand singularity generated by the exchange of the $a_2$ tensor meson, yielding, for the first time, an estimate of the associated factorization-breaking corrections. Our final results, $a_\mu^{\eta\text{-pole}}=14.7(9)\times 10^{-11}$ and $a_\mu^{\eta'\text{-pole}}=13.5(7)\times 10^{-11}$, conclude a dedicated effort to evaluate the pseudoscalar-pole contributions to hadronic light-by-light scattering using dispersion relations, amounting to a combined $a_\mu^{\text{PS-poles}}=91.2^{+2.9}_{-2.4}\times 10^{-11}$.