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In this paper we present a first ab-initio calculation of the $\pi^0$, $\eta$and $\eta^{\prime}$ transition form factors performed with physical light-quarkmasses. We provide a complete parametrization of the form factors that includesboth single and double-virtual kinematics. Our results are compared withexperimental measurements of the form factors in the space-like region and withthe measured two-photon decay widths. In a second step, our parametrizations ofthe transition form factors are used to compute the dominant pseudoscalar-polecontributions to the hadronic light-by-light scattering in the muon $g-2$. Ourfinal result reads $a_{\mu}^{\rm hlbl, ps-pole} = (85.1 \pm 5.2) \times10^{-11}$. Although the pion-pole is dominant, we confirm that, together, the$\eta$ and $\eta^{\prime}$ provide roughly half of its contribution.

Lattice calculation of the $\pi^0$, $\eta$ and $\eta^{\prime}$  transition form factors and the hadronic light-by-light contribution to the  muon $g-2$