Lattice QCD calculation of the $\pi^0$-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon

We develop a method to compute the pion transition form factor directly at arbitrary space-like photon momenta and use it to determine the $\pi^0$-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon. The calculation is performed using eight gauge ensembles generated with 2+1 flavor domain wall fermions, incorporating multiple pion masses, lattice spacings, and volumes. By introducing a pion structure function and performing a Gegenbauer expansion, we demonstrate that about 98\% of the $\pi^0$-pole contribution can be extracted in a model-independent manner, thereby ensuring that systematic effects are well controlled. After applying finite-volume corrections, as well as performing chiral and continuum extrapolations, we obtain the final result for the $\pi^0$-pole contribution to the hadronic light-by-light scattering in the muon's anomalous magnetic moment, $a_{\mu}^{\pi^0\mathrm{-pole}}=61.2(1.7)\times 10^{-11}$, and the $\pi^0$ decay width, $\Gamma_{\pi^0\to \gamma\gamma}=7.60(27)$ eV.