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

Abstract

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 ${\gamma }^{\ast }{\gamma }^{\ast }\to \{{\pi }^0,\eta ,{\eta }^{\prime },{\eta }_c,{\eta }_b\}$ 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: $a_{\mu }^{{\pi }^0-\text{pole}}=(6.14\pm 0.21)\times 10^{-10}$, $a_{\mu }^{\eta -\text{pole}}=(1.47\pm 0.19)\times 10^{-10}$, $a_{\mu }^{{\eta }^{\prime }-\text{pole}}=(1.36\pm 0.08)\times 10^{-10}$, yielding a total value of $a_{\mu }^{{\pi }^0+\eta +{\eta }^{\prime }-\text{pole}}=(8.97\pm 0.48)\times 10^{-10}$, 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.

Comments: 13 pages, 4 figures, 3 tables