Combined Analysis of Lattice QCD and Experimental Data on the Pion Transition Form Factor
The evaluation of the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment requires precise knowledge of the pion transition form factor (TFF). In this work, we present a feasibility study for a combined analysis of lattice QCD (LQCD) and experimental data. Our methodology is driven by the goal of combining complementary datasets to leverage their respective kinematic advantages: while LQCD provides robust predictions for the doubly-virtual TFF, $e^+e^-$ scattering experiments offer high-precision singly-virtual measurements up to large momentum transfers. To ensure a statistically rigorous combination, we implement a global one-stage fitting approach based on the modified $z$-expansion, utilizing synthetic jackknife replicate sampling and a normalized $χ^2$ weighting scheme. We demonstrate that the inclusion of experimental data substantially tightens the constraints on the pion TFF, yielding up to a factor of three reduction in uncertainty in the singly-virtual limit. In contrast, the uncertainty of the resulting pion-pole contribution to the muon $g-2$ improves by a factor of $1.5$. This more modest improvement reflects the fact that the $g-2$ integral is heavily dominated by the low-$Q^2$ region, which is already well constrained by physical normalization constraints.