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Pseudoscalar-pole diagrams are an important component of estimates of thehadronic light-by-light (HLbL) contribution to the muon $g-2$. We report on ourcomputation of the transition form factors $\mathcal{F}_{P \rightarrow \gamma^*\gamma^*}$ for the neutral pseudoscalar mesons $P=\pi^0$ and $\eta$. Thecalculation is performed using twisted-mass lattice QCD with physical quarkmasses. On the lattice, we have access to a broad range of (space-like) photonfour-momenta and therefore produce form factor data complementary to theexperimentally accessible single-virtual direction, which directly leads to anestimate of the pion- and $\eta$-pole components of the muon $g-2$. For thepion, our result for the $g-2$ contribution in the continuum is comparable withprevious lattice and data-driven determinations, with combined relativeuncertainties below $10\%$. For the $\eta$ meson, we report on a preliminarydetermination from a single lattice spacing.

Pseudoscalar-pole contributions to the muon $g-2$ at the physical point