Contribution of neutral pseudoscalar mesons to $a_\mu^{HLbL}$ within a Schwinger-Dyson equations approach to QCD

A symmetry preserving formalism, based upon Schwinger-Dyson and Bethe-Salpeter equationsis employed to provide a tightly constrained prediction for the $\gamma^{(*)} \gamma^{(*)} \rightarrow M$ transition form factors (TFFs), where $M$ is any neutral pseudoscalar mesonand the process involves arbitrary virtualities of both photons. The numerical results are parametrized meticulously, ensuring the accurate reproduction of the known theoretical limits of the TFFs. We then evaluate the corresponding polecontribution to the hadronic light-by-light (HLbL) piece of the anomalous magnetic moment of the muon ($a_\mu$). As a result of this analysis, we obtain $a_{\mu}^{\pi^0-\textrm{ pole}} = (6.14 \pm 0.21) \times 10^{-10}$, $a_{\mu}^{\eta-\textrm{ pole}} = (1.47 \pm 0.19) \times 10^{-10}$, $a_{\mu}^{\eta'-\textrm{ pole}} = (1.36 \pm 0.08) \times 10^{-10}$, yielding a total value of $a_{\mu}^{{\pi^0+\eta+\eta'}-\textrm{ pole}} = (8.97 \pm 0.48) \times 10^{-10}$, which is compatible with recent determinations. Notably, we find that $a_{\mu}^{\eta_c-\textrm{ pole}} = (0.09 \pm 0.01) \times 10^{-10}$, which might not be negligible once the percent precision is reached in the computation of the light pseudoscalarcontributions.