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Next to the π^0 pole, η and η^' 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⏛. 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 η^ (') →γγ decay widths, η^ (') →π^+π^-γ spectra, chiral symmetry for the η^ (') →2 (π^+π^-) amplitudes, vector-meson couplings, and asymptotic limits. Crucially, we investigate the role of the leading left-hand singularity generated by the exchange of the a₂ tensor meson, yielding, for the first time, an estimate of the associated factorization-breaking corrections. Our final results, a⏛^η pole=14. 7 (9) ×10^-11 and a⏛^η^{' pole}=13. 5 (7) ×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⏛^PS poles=91. 2-₂. ₄^+2. 9×10^-11.
Holz et al. (Tue,) studied this question.