Resolve the CMD-3 versus BaBar and tau-decay form-factor tension

Resolve the tension between the CMD-3 measurement and the BaBar and tau-decay data in the pion electromagnetic form factor, which is not resolved by the conformal physics-informed neural-network framework alone.

Background

The paper’s default reconstruction excludes CMD-3 because its cross sections are less compatible with the global holomorphy and dispersion constraints than the BaBar and tau-decay measurements. Separate dataset fits show that CMD-3 produces a higher form-factor normalization near the rho(770) resonance and shifts the inferred two-pion contribution to the muon anomalous magnetic moment. The authors therefore identify the experimental discrepancy as an unresolved issue requiring further investigation beyond the framework presented.

References

First, our default result excludes CMD-3: the network's global holomorphy and dispersion constraints fit CMD-3 less comfortably than BaBar and $\tau$-decay data, evidence bearing on the ongoing tension between these measurements, though the tension is not resolved by this framework alone.

— PINNing the pion: conformal deep learning for $F_π(s)$ and the $(g-2)_μ$ hadronic contribution  (2609.40008 - Goel et al., 30 Sep 2026) in Section 7, Discussion and future directions