PINNing the pion: conformal deep learning for $F_π(s)$ and the $(g-2)_μ$ hadronic contribution

2026-09-30

AI Summary

This study addresses the challenges of model dependence, unphysical artifacts, and kinematic inconsistencies in extracting the pion electromagnetic form factor by proposing a physics-informed neural network (PINN) embedded within the conformal z-plane. Methodologically, a conformal mapping transforms the cut complex plane into the unit disk to optimize training stability, while the loss function is constructed from S-matrix principles and perturbative QCD asymptotic behavior. This framework directly generates the form factor from first principles, natively isolating the pure isovector component. The proposed approach achieves model-independent, high-precision estimates of key physical observables, including the pion charge radius, the ρ(770) pole parameters, and the two-pion contribution to the muon anomalous magnetic moment.

Abstract

Extracting the pion electromagnetic form factor $F_π(s)$ through phenomenological curve-fitting models introduces model dependence, unphysical artefacts, and kinematic inconsistencies. We introduce a Physics-Informed Neural Network (PINN) embedded in a conformal $z$-plane that constructs $F_π(s)$ directly from first principles across spacelike and timelike domains: charge normalisation and Schwarz reflection are enforced by construction, while Cauchy-Riemann analyticity, dispersion relations, Watson's theorem, and perturbative QCD asymptotics enter through the loss functional. Thus, the fundamental S-matrix principles dictate the form factor's behaviour while data act as constraints. Mapping the cut complex plane onto the unit disk bounds the Hessian norm and prevents Neural Tangent Kernel spectral starvation, two known failure modes of deep-learning optimisation. Besides $e^+e^-$ scattering data, we also incorporate $τ$-decay data through a switch that isolates the pure isovector form factor natively, bypassing model-dependent isospin-breaking pre-corrections. The network organically yields an interior zero-free form factor, while the framework tests experimental tensions around the $ρ(770)$ peak against analyticity and dispersion constraints. We obtain model-independent estimates of the pion charge radius, $\langle r_π^2 \rangle = 0.435 \pm 0.008_{\text{stat}} \pm 0.007_{\text{cali}}$ fm$^2$, the second-sheet pole parameters, $m_ρ^{\text{pole}} = 761.72\pm 1.04$ MeV and $Γ_ρ^{\text{pole}} = 135.99 \pm 1.20$ MeV, and the two-pion contribution to the muon anomalous magnetic moment, $a_μ^{ππ} = (506.48 \pm 2.02_{\text{stat}} \pm 1.70_{\text{cali}}) \times 10^{-10}$.

Problem

• pion electromagnetic form factor
• model dependence
• Physics-Informed Neural Network
• muon anomalous magnetic moment
• hadronic contribution

Innovation

• Physics-Informed Neural Network
• Conformal Mapping
• Pion Form Factor
• Neural Tangent Kernel
• Dispersion Relations