Neutral Pseudoscalar Mesons, Hadron Light-by-Light Scattering, and the Muon Anomalous Magnetic Moment

Continuum Schwinger function methods\linebreak (CSM) are used to provide estimates of pseudoscalar meson pole contributions - $\mathsf{P}={\pi }^0$, $\eta$, ${\eta }^{\mathrm{\prime }}$, ${\eta }_c$, ${\eta }_b$ - to the muon anomalous magnetic moment. The analysis framework is benchmarked via successful applications to an array of other meson and nucleon properties. It uses the leading order CSM truncation, augmented by a parametrisation of non-Abelian anomaly effects, to deliver an internally consistent treatment of all the pseudoscalar–two-photon transition form factors required to complete such a calculation. Employing those form factor predictions, the following results are obtained (units ${10}^{-11}$): $a_{\mu }^{{\pi }^0}+a_{\mu }^{\eta }+a_{\mu }^{{\eta }^{\mathrm{\prime }}}=84.3(0.8)$ - a value lower than some other contemporary estimates; $a_{\mu }^{{\eta }_c}=0.82(0.04)$; and a value of $a_{\mu }^{{\eta }_b}$ that is negligible. The total contribution is ${\sum }_{\mathsf{P}}{a}_{\mu }^{\mathsf{P}}=85.1(0.8)\mathrm{/}{10}^{11}$. Our analysis suggests that this pole contribution to $a_{\mu }$ is still not known with precision better than 5%.