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

We calculate the $\pi^0\to \gamma^*\gamma^*$ transition form factor ${\cal F}_{\pi^0\gamma^*\gamma^*}(q_1^2,q_2^2)$ in lattice QCD with two flavors of quarks. Our main motivation is to provide the input to calculate the $\pi^0$-pole contribution to hadronic light-by-light scattering in the muon $(g-2)$, $a_\mu^{\rm HLbL;\pi^0}$. We therefore focus on the region where both photons are spacelike up to virtualities of about $1.5~$GeV$^2$, which has so far not been experimentally accessible. Results are obtained in the continuum at the physical pion mass by a combined extrapolation. We reproduce the prediction of the chiral anomaly for real photons with an accuracy of about $8-9\%$. We also compare to various recently proposed models and find reasonable agreement for the parameters of some of these models with their phenomenological values. Finally, we use the parametrization of our lattice data by these models to calculate $a_\mu^{\rm HLbL;\pi^0}$.