Precision evaluation of the $\eta$- and ${\eta }^{\mathrm{\prime }}$-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon

Next to the ${\pi }^0$ pole, $\eta$ and ${\eta }^{\mathrm{\prime }}$ 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 ${\eta }^{(\mathrm{\prime })}\to \gamma \gamma$ decay widths, ${\eta }^{(\mathrm{\prime })}\to {\pi }^{+}{\pi }^{-}\gamma$ spectra, chiral symmetry for the ${\eta }^{(\mathrm{\prime })}\to 2({\pi }^{+}{\pi }^{-})$ 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 $a_{\mu }^{{\eta }^{\mathrm{\prime }}\text{-pole}}=13.5(7)\times {10}^{-11}$, conclude a dedicated effort to evaluate the pseudoscalar-pole contributions to hadronic light-by-light scattering using dispersion relations, amounting to a combined $a_{\mu }^{\text{PS-poles}}=91.2_{-2.4}^{+2.9}\times {10}^{-11}$.