Using a continuum approach to the hadron bound-state problem, we calculate γ * γ → η, η transition form factors on the entire domain of spacelike momenta, for comparison with existing experiments and in anticipation of new precision data from next-generation e + e − colliders. One novel feature is a model for the contribution to the Bethe-Salpeter kernel deriving from the non-Abelian anomaly, an element which is crucial for any computation of η, η properties. The study also delivers predictions for the amplitudes that describe the light-and strange-quark distributions within the η, η . Our results compare favourably with available data. Important to this at large-Q 2 is a sound understanding of QCD evolution, which has a visible impact on the η in particular. Our analysis also provides some insights into the properties of η, η mesons and associated observable manifestations of the non-Abelian anomaly. * leichang@nankai.edu.cn † cdroberts@anl.gov the bound-state's total momentum. The complete transition form factor is obtained as a sum over the various qq subcomponent contributions:where ψ q M is a flavour weighting factor originating in the meson's wave function.Notably [3-5] (τ 2 := Λ 2 QCD /Q 2 ):i.e. the DA acquires its asymptotic profile and henceConsequently, on τ 0 the γ * γ → M transition form factor exhibits simple scaling; and the anomalous dimension, characteristic of gauge field theories quantised in four dimensions, is "hidden" in the manner of approach to the τ = 0 limit. (N.B. As will become clear, owing to the non-Abelian axial anomaly in QCD, Eq. (5) is amended when M = η, η [6, 7].)An array of experiments have been performed with a view to testing Eqs. (1), (5) for the neutral pion [8][9][10][11]. Such measurements are difficult, typically involving the study of e + -e − collisions, in which one of the outgoing fermions is detected after a large-angle scattering whilst the other is scattered through a small angle and, hence, undetected. The detected fermion is assumed to have emitted a highly-virtual photon, the undetected fermion, a soft-photon; and these photons are supposed to fuse and produce the final-state pseudoscalar meson. There are many possible background processes and loss mechanisms in this passage of events, and thus ample room for systematic error, especially as Q 2 increases [12].