PINNing the pion: conformal deep learning for Fπ(s) and the (g−2)μ hadronic contribution
Abstract
Extracting the pion electromagnetic form factor Fπ(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 Fπ(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, ⟨ rπ2 ⟩ = 0.435 ± 0.008stat ± 0.007cali fm2, the second-sheet pole parameters, mρpole = 761.72± 1.04 MeV and Γρpole = 135.99 ± 1.20 MeV, and the two-pion contribution to the muon anomalous magnetic moment, aμππ = (506.48 ± 2.02stat ± 1.70cali) × 10−10.
Related
- Spacelike and timelike kernels for the hadronic vacuum polarization contribution to the muon lifetime
- One-nucleon intermediate-state contribution to the pion-coupled four-quark condensate in the N--J/ψ interaction
- Probability-based Estimates of the Uncalculated N5LO QCD Contribution to the Hadronic W-Boson Decay Width
- The hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment in isospin-symmetric lattice QCD with twisted-mass fermions
- Understanding Energy Dependent Hadronic Calorimeter Response from a Machine Learning Perspective
BibTeX
@misc{goel2026pinning,
title = {{PINNing the pion: conformal deep learning for $F_π(s)$ and the $(g-2)_μ$ hadronic contribution}},
author = {Mayank Goel and Subhadip Mitra and Monalisa Patra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40008},
eprint = {2609.40008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40008},
}
RIS
TY - UNPB
TI - PINNing the pion: conformal deep learning for F_π(s) and the (g−2)_μ hadronic contribution
AU - Mayank Goel
AU - Subhadip Mitra
AU - Monalisa Patra
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40008
UR - https://arxiv.org/abs/2609.40008
AB - 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, ⟨ r_π^2 ⟩ = 0.435 ± 0.008_stat ± 0.007_cali fm^2, the second-sheet pole parameters, m_ρ^pole = 761.72± 1.04 MeV and Γ_ρ^pole = 135.99 ± 1.20 MeV, and the two-pion contribution to the muon anomalous magnetic moment, a_μ^ππ = (506.48 ± 2.02_stat ± 1.70_cali) × 10^−10.
ER -