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

A continuum approach to Quantum Chromodynamics (QCD), based upon Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations, is employed to provide a tightly constrained prediction for the $\gamma^{} \gamma^{} \rightarrow { \pi^0, \eta, \eta’, \eta_c, \eta_b }$ transition form factors (TFFs) and their corresponding pole contribution to the hadronic light-by-light (HLbL) piece of the anomalous magnetic moment of the muon ($a_{\mu }$). This work relies on a practical and well-tested quark-photon vertex Ansatz approach to evaluate the TFFs for arbitrary space-like photon virtualities, in the impulse approximation. The numerical results are parametrized meticulously, ensuring a reliable evaluation of the HLbL contributions to $a_{\mu }$. We obtain: $a_{\mu }^{{\pi }^0-\text{pole}}=(6.14\pm 0.21)\times {10}^{-10}$, $a_{\mu }^{\eta -\text{pole}}=(1.47\pm 0.19)\times {10}^{-10}$, $a_{\mu }^{{\eta }^{\mathrm{\prime }}-\text{pole}}=(1.36\pm 0.08)\times {10}^{-10}$, yielding a total value of $a_{\mu}^{\pi^0+\eta+\eta’-$\text{pole}$} = (8.97 \pm 0.48) \times 10^{-10}$, compatible with contemporary determinations. Notably, we find that $a_{\mu}^{\eta_c+\eta_b-$\text{pole}$} \approx a_{\mu}^{\eta_c-$\text{pole}$} = (0.09 \pm 0.01) \times 10^{-10}$, which might not be negligible once the percent precision in the computation of the light pseudoscalars is reached.