PINNing the pion: conformal deep learning for F_pi(s) and the (g-2)_mu hadronic contribution

arXiv:2609.40008v1 Announce Type: cross Abstract: Extracting the pion electromagnetic form factor F_{pi}(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_{pi}(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 au-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 rho(770) peak against analyticity and dispersion constraints. We obtain model-independent estimates of the pion charge radius, langle r_{pi}^2 rangle = 0.435 pm 0.008_{ext{stat}} pm 0.007_{ext{cali}} fm^2, the second-sheet pole parameters, m_{rho}^{ext{pole}} = 761.72pm 1.04 MeV and Gamma_{rho}^{ext{pole}} = 135.99 pm 1.20 MeV, and the two-pion contribution to the muon anomalous magnetic moment, a_{mu}^{pipi} = (506.48 pm 2.02_{ext{stat}} pm 1.70_{ext{cali}}) imes 10^{-10}.

Source: arXiv cs.LG | 2026-10-01