Pseudoscalar-pole diagrams are an important component of estimates of the
hadronic light-by-light (HLbL) contribution to the muon $g-2$.
We report on our computation of the transition form factors $\mathcal{F}_{P \rightarrow \gamma^* \gamma^*}$ for the neutral pseudoscalar mesons $P=\pi^0$ and $\eta$. The calculation is performed using twisted-mass lattice QCD with physical quark masses.
On the lattice, we have access to a broad range of (space-like) photon four-momenta and therefore produce form factor data complementary to the experimentally accessible single-virtual direction, which directly leads to an estimate of the pion- and $\eta$-pole components of the muon $g-2$.
For the pion, our result for the $g-2$ contribution in the continuum is comparable with previous lattice and data-driven determinations, with combined relative uncertainties below $10\%$.
For the $\eta$ meson, we report on a preliminary determination from a single lattice spacing.