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 $a_\mu^{ \text{ PS-poles}}=91.2^{ +2.9}_{ -2.4}\times 10^{ -11}$.
Categories: hep-ph hep-ex hep-lat nucl-th