Physics-informed neural network models pion form factor precisely
PINNing the pion: conformal deep learning for $F_π(s)$ and the $(g-2)_μ$ hadronic contribution
Machine Learning
Summary
Measuring a property called the pion electromagnetic form factor often relies on fitting models that can introduce errors or inconsistencies. The authors designed a neural network guided by fundamental physics principles to directly determine this form factor more reliably across different energy regions. Their approach respects key rules from particle physics and incorporates experimental data, producing precise estimates of important physical quantities related to the pion and its role in particle interactions. This helps improve understanding of the pion’s structure and its contribution to the muon’s magnetic properties.
What this means in practice
- •For particle physicists: Provide a physics-based machine learning tool to extract pion form factors free of model bias and consistent with fundamental principles.
- •For computational physicists: Use the conformal PINN framework to improve numerical stability and accuracy in modeling complex analytic functions in physics.
Authors
Mayank Goel, Subhadip Mitra, Monalisa Patra
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, $a_μ^{ππ} = (506.48 \pm 2.02_{\text{stat}} \pm 1.70_{\text{cali}}) \times 10^{-10}$.