The paper reports a feasibility study that combines theoretical and experimental inputs to improve knowledge of the pion transition form factor. The pion transition form factor (TFF) describes how a neutral pion interacts with two photons. That interaction is a key ingredient in the hadronic light‑by‑light (HLbL) contribution to the muon’s anomalous magnetic moment, often called muon g‑2. Better knowledge of the TFF reduces an important source of uncertainty in the Standard Model prediction for muon g‑2.

The authors bring together lattice QCD (LQCD) calculations and e+e− scattering data in a single, global fit. LQCD produces robust predictions when both photons are virtual (so‑called doubly‑virtual kinematics), while scattering experiments give high‑precision measurements when one photon is nearly real and the other carries momentum (singly‑virtual kinematics) up to large momentum transfer Q2 (Q2 is the squared momentum carried by a photon). To combine the datasets in a statistically consistent way, the team used a modified z‑expansion (a flexible functional form), a synthetic jackknife procedure to handle uncertainties, and a normalized χ2 weighting scheme. Their LQCD input comes from twelve Nf=2+1 CLS ensembles with O(a)‑improved Wilson fermions, several lattice spacings (about 0.050, 0.064, 0.076 and 0.086 fm) and pion masses down to about 200 MeV; the lattice data cover spacelike photon virtualities up to roughly 5.5 GeV2 in the doubly‑virtual case. The fit also includes recent world experimental data for the spacelike TFF and the PrimEx‑II measurement of the pion decay width.

The combined analysis tightens the TFF constraints. When experimental data are included, the uncertainty in the singly‑virtual limit can shrink by up to a factor of three. For the pion‑pole piece of HLbL—which is the single largest HLbL contribution and amounts to roughly (60–64)×10−11 in the muon g‑2—the uncertainty improves by about a factor of 1.5. The authors point out that the more modest numerical gain for the muon g‑2 reflects how the g‑2 integral is dominated by low‑Q2 contributions, which are already constrained by the physical normalization from the pion’s decay into two real photons.