Lattice calculation of the pion transition form factor $\pi^0 \to \gamma^* \gamma^*$

We calculate the pion transition form factor ${\cal F}_{\pi^0\gamma^*\gamma^*}(q_1^2,q_2^2)$, which describe the interaction of an on-shell pion with two off-shell photons, using lattice QCD simulations with two degenerate flavors of dynamical quarks. This form factor is the main ingredient in the calculation of the pion-pole contribution to hadronic light-by-light scattering in the muon $g-2$, $a_\mu^{\mathrm{HLbL}; \pi^0}$. We focus our study on the spacelike region with photon virtualities up to $1.5~\mathrm{GeV}^2$, not yet measured experimentally. Several lattice spacings and pion masses are used to extrapolate the results to the physical point and a comparison with different phenomenological models is performed. Finally, we use our extrapolated form factor to provide a lattice determinaiton of $a_\mu^{\mathrm{HLbL}; \pi^0}$.