Pion Transition Form Factor from Twisted-Mass Lattice QCD and the Hadronic Light-by-Light ${\pi }^0$-pole Contribution to the Muon $g-2$

The neutral pion generates the leading pole contribution to the hadronic light-by-light tensor, which is given in terms of the nonperturbative transition form factor ${\mathcal{F}}_{{\pi }^0\gamma \gamma }(q_1^2,q_2^2)$. Here we present an ab-initio lattice calculation of this quantity in the continuum and at the physical point using twisted-mass lattice QCD. We report our results for the transition form factor parameterized using a model-independent conformal expansion valid for arbitrary space-like kinematics and compare it with experimental measurements of the single-virtual form factor, the two-photon decay width, and the slope parameter. We then use the transition form factors to compute the pion-pole contribution to the hadronic light-by-light scattering in the muon $g-2$, finding $a_{\mu }^{{\pi }^0\text{-pole}}=56.7(3.2)\times {10}^{-11}$.