PINNing the pion: conformal deep learning for $F_\pi(s)$ and the $(g-2)_\mu$ hadronic contribution

I. INTRODUCTION

• Chiral perturbation theory is precise near q^2 = 0 but breaks down near the rho resonance, while vector-meson dominance violates the required high-energy 1/q^2 scaling.Dispersive Omnès approaches depend on experimental pi-pi phase shifts and their treatment of complex zeros; lattice QCD remains limited to the Euclidean region.Source: §I · p. 1
• Traditional fits risk model bias, while neural-network interpolation does not ensure global S-matrix consistency and can suffer high-q^2 spectral starvation.The paper connects this reconstruction problem to uncertainties in the two-pion HVP contribution and conflicting lattice-QCD and CMD-3 results.Source: §I · p. 1

A. The low-energy expansion

• The low-energy expansion fixes F_pi(0) = 1 and relates its slope at the origin to the pion charge radius.Physically admissible parameterisations must reproduce both the unit charge and the observed slope.Source: A. The low-energy expansion · p. 2

B. Analyticity and the S matrix

• Analyticity confines form-factor singularities to physical hadronic cuts, with the two-pion threshold initiating the cut in the isovector vector channel.The absorptive part determines the discontinuity across the physical branch cut.Source: B. Analyticity and the S matrix · p. 2

C. Isospin breaking, the inelastic threshold, and Watson’s theorem

• Below the inelastic threshold, elastic pi-pi unitarity fixes the form-factor phase to the isospin-1 P-wave scattering phase shift.Electromagnetic isospin breaking mixes rho and omega contributions, producing interference patterns in e+e− data.Source: C. Isospin breaking, the inelastic… · pp. 2–3

D. Asymptotic pQCD scaling

• At large spacelike momentum transfer, perturbative QCD requires F_pi(s) to scale as 1/s, up to logarithmic corrections.The paper argues that enforcing normalisation, phase tracking, analytic continuity, and this asymptotic scaling together in a phenomenological model has been difficult.Source: D. Asymptotic pQCD scaling · p. 3
• The framework imposes S-matrix and QCD constraints through the loss functional rather than selecting a form-factor model.It uses a conformal coordinate as a geometric preconditioner; the stated stability argument applies to any bijection compactifying the domain into a finite region.Source: D. Asymptotic pQCD scaling · p. 3, A. Using conformal mapping to bypass… · p. 5

A. Using conformal mapping to bypass geometric instability

• Direct training in unbounded s can make gradients explode with periodic activations or vanish with tanh, starving the network of high-q^2 learning.The conformal map sends the cut physical plane to a bounded disk; the rho pole lies outside the disk, leaving the interior form factor holomorphic.Source: A. Using conformal mapping to bypass… · p. 4, Figure 1 · p. 4
• Mapping the physical domain into a bounded conformal coordinate bounds the loss Hessian independently of the unbounded physical variable and supports robust gradient-descent convergence.The loss decreases at every step under the stated learning-rate condition, unless a stationary point is reached.Source: A. Using conformal mapping to bypass… · pp. 4–5

B. Preserving the Neural Tangent Kernel

• The empirical NTK describes training dynamics and expressive bandwidth, while conformal mapping leaves the network able to approximate the physical form factor arbitrarily closely.The stated approximation argument uses a smooth target on a finite disk and non-polynomial activations such as the adaptive sine function.Source: B. Preserving the Neural Tangent Kernel · pp. 5–6
• In unbounded s-space, a periodic activation must shrink its internal weights to match the vanishing deep-Euclidean form factor, causing parameter gradients and the minimum NTK eigenvalue to vanish.The paper identifies this collapse as spectral starvation that stalls optimization.Source: B. Preserving the Neural Tangent Kernel · p. 5
• Mapping the infinite spacelike tail to the finite boundary z → 1 removes this weight-collapse mechanism and, for generic non-polynomial activations, preserves full-row-rank parameter Jacobians.The paper states that the empirical NTK remains positive definite during training, preventing kinematic regions from becoming learning dead zones.Source: B. Preserving the Neural Tangent Kernel · p. 5

C. Enforcement of physical constraints

• The structural ansatz F_pi(z) = 1 + s(z)N(z) guarantees F_pi(0) = 1 independently of trainable parameters.The network also hardwires Schwarz reflection by routing the imaginary component with the sign of Im(z).Source: C. Enforcement of physical constraints · p. 6
• The composite objective weights spacelike and timelike data losses together with physics losses for analyticity, dispersion, moments, positivity, Watson’s theorem, monotonicity, and spacelike and timelike asymptotics.The weights are empirically determined and fixed.Source: C. Enforcement of physical constraints · p. 6
• Analyticity is enforced by penalizing Cauchy-Riemann violations using exact automatic-differentiation gradients at dynamically resampled interior points.The collocation points are sampled within |z| ≤ 0.999 to avoid the branch-cut boundary.Source: C. Enforcement of physical constraints · p. 6
• A twice-subtracted dispersion loss reconstructs the form factor from its absorptive part while damping high-energy contributions that could destabilize an unsubtracted integral.The network obtains F′_pi(0) dynamically rather than fixing the slope with an empirical parameter.Source: C. Enforcement of physical constraints · pp. 6–7
• Higher-moment sum rules constrain dynamically generated derivatives to limit unphysical spectral artefacts where data are sparse.Their integral bounds restrict how the spectral density flattens toward the perturbative-QCD continuum.Source: C. Enforcement of physical constraints · pp. 7–8
• A soft positivity penalty discourages negative absorptive strength, while Watson’s theorem constrains the elastic-region phase through geometric projections of the form factor.The projection avoids inverse-tangent singularities and discontinuities and includes a directional condition to select the correct phase branch.Source: C. Enforcement of physical constraints · p. 8
• The phase-monotonicity loss requires a non-negative angular derivative along the lower conformal boundary.It is intended to prevent unphysical phase oscillations or backtracking where phase-shift data are sparse or affected by inelastic openings.Source: C. Enforcement of physical constraints · p. 8
• The spacelike pQCD loss imposes asymptotic scaling without fixing its normalization, while the timelike asymptotic loss constrains the real part to fall as 1/s.The spacelike condition is applied for z ≥ 0.82, corresponding to Q^2 ≳ 8 GeV^2.Source: C. Enforcement of physical constraints · p. 8

• Timelike inputs include direct energy scans from SND, CMD-2, and CMD-3 and ISR measurements from BaBar, KLOE, and BESIII.The e+e− cross sections are used in bare or final-state-radiation-corrected form, as applicable.Source: §IV · p. 9
• BaBar and CMD-3 disagree across the ρ resonance, while older CMD-2 results agree with BaBar and τ-decay data.The passage identifies possible unaccounted radiative-photon corrections or beam-luminosity calibrations as sources, not established causes.Source: §IV · p. 9
• The τ channel probes a weak form factor distinct from the electromagnetic form factor, with their relation involving isospin-breaking corrections.The listed corrections include phase-space and long-distance electromagnetic effects, hadronic form-factor differences, and ρ–ω mixing.Source: §IV · p. 10
• The raw-τ channel switch uses αρ−ω = 0 for τ data and αρ−ω ≠ 0 for e+e− data, avoiding explicit RIB corrections to τ inputs.The study also compares this approach with training on τ data corrected explicitly for isospin violation and with a combined dataset.Source: §IV · pp. 10–11

A. Global analytic structure and the zero-free landscape

• The first-sheet reconstruction follows the two-pion branch cut and ρ(770) profile without unphysical poles or numerical artefacts away from the real axis.The cut begins at the two-pion threshold, with the resonance peak near Re(s) ≈ 0.6 GeV².Source: A. Global analytic structure and the… · p. 11
• The phase obeys Schwarz reflection, vanishes below elastic threshold, and follows the two-pion scattering phase through the ρ(770) region.In the elastic region it rises through π/2 near the resonance and continues toward π.Source: A. Global analytic structure and the… · p. 11
• The predicted form factor has no interior zeros despite being trained without a zero-free constraint, supporting the zero-free hypothesis.Its deep-spacelike and complex-UV magnitude also falls consistently with 1/s scaling expectations.Source: A. Global analytic structure and the… · p. 11, Figure 4 · p. 11
• The absolute residual between the PINN and dispersion-relation real parts remains tightly bounded, supporting a holomorphic first-sheet amplitude.The comparison covers s ∈ [−2.0, 4mπ²].Source: A. Global analytic structure and the… · p. 11, Figure 5 · p. 11

B. Form factor across spacelike and timelike regions

• The Set A curve satisfies charge normalization, passes smoothly through low-energy spacelike data, and reproduces the ρ(770) resonance.The dispersion relation links the timelike shape to the spacelike behavior.Source: B. Form factor across spacelike and… · pp. 11–12, Figure 6 · p. 12
• The conditional αρ−ω switch incorporates isoscalar ρ–ω mixing without changing the network core, producing a localized dip-bump on the steep ρ peak slope.With the switch active, the network tracks the phase swing and sub-percent interference ripple; setting it to zero removes the interference.Source: B. Form factor across spacelike and… · p. 12
• The spacelike prediction shows stable monotonic damping against electroproduction data, without unphysical oscillations or sudden upturns.The reported behavior is consistent with short-distance limits of quantum field theory.Source: B. Form factor across spacelike and… · p. 12, Figure 8 · p. 12

C. The spacelike asymptotics

• Through Q² = 20 GeV², the prediction bridges QCD regimes, tracks the continuum DSE benchmark and Curve C at Q² ≳ 7 GeV², and stays above the textbook asymptotic Curve D.At Q² ≲ 4 GeV² it interpolates spacelike measurements below the empirical monopole bound, Curve B.Source: C. The spacelike asymptotics · pp. 12–13

D. The complex phase and Watson’s theorem

• The full phase prediction tracks the Roy-equation solution, while disabling Watson’s loss produces a deviation that grows with energy.Analyticity and dispersion constrain global structure but do not alone fix the low-energy phase.Source: D. The complex phase and Watson’s… · pp. 13–14, Figure 10 · p. 13
• The inverse-amplitude zero is used to locate the second-sheet pole, turning a numerical divergence into a root-finding problem.The pole is extracted from the network’s second-sheet access rather than solely from the real-axis phase crossing.Source: D. The complex phase and Watson’s… · p. 14

E. Uncertainty quantification

• Statistical uncertainty combines data resampling and ordinary training variation across 500 independently trained bootstrap networks.Published covariance matrices are used for joint sampling of KLOE, BaBar, Belle, and CLEO measurements; other measurements are sampled independently.Source: E. Uncertainty quantification · p. 14
• Calibration uncertainty is the standard deviation across 64 converged networks sampled over a factor-of-two range for each of ten loss weights.The statistical and calibration ensembles probe orthogonal sources of uncertainty, are reported separately, and are combined in quadrature for total uncertainty.Source: E. Uncertainty quantification · pp. 14–15

F. Charge radius of the pion

• ⟨r2π⟩ = 0.435 ± 0.008stat ± 0.007cali fm2 agrees with the PDG world average and lattice QCD at 0.67σ.The statistical uncertainty is more than a factor of two smaller than the lattice extraction’s; the estimate uses the full spacelike and timelike spectrum through dispersive integrals.Source: F. Charge radius of the pion · p. 15, Table II · p. 15
• The fourth-order moment agrees with two-loop ChPT at 0.39σ, while its uncertainty is about two-thirds that from two-loop ChPT.It sits just above the cited dispersive-bound range.Source: F. Charge radius of the pion · p. 15
• The sixth-order moment is 4.01 ± 0.95stat ± 0.90cali fm6 and is consistent at 0.8σ with the cited dispersive bounds and estimate.The paper relates higher-order derivatives at s = 0 to non-local contributions from the timelike resonance spectrum and high-s asymptotic tail, coupled through S-matrix constraints.Source: F. Charge radius of the pion · p. 15

G. Implications for the muon anomalous magnetic moment

• The full-spectrum aππ_μ estimate, (506.48 ± 2.02_stat ± 1.70_cali) × 10^-10, is compatible with the Muon g−2 Theory Initiative benchmark, Ref. [56], within 0.34σ.Its total uncertainty is ±2.64 in quadrature, versus ±3.29 to ±3.8 for traditional evaluations. Below 0.63 GeV and in the [2mπ,1.0 GeV] ρ-resonance region, it differs from Colangelo et al. [44] by 0.14σ and 0.37σ, respectively.Source: G. Implications for the muon anomalous… · p. 16

H. Hyperparameter loss weighting and structural ablation

• Removing Watson’s theorem destabilizes the fit, inflates low-energy moments, and produces phase deviations of 7.70° at s0 and 11.45° at s1.The theorem ties the form-factor phase to elastic ππ scattering below the inelastic threshold.Source: H. Hyperparameter loss weighting and… · p. 16
• Removing analyticity or dispersion constraints raises higher-energy phase errors to about 2.5°–2.9°, while disabling monotonicity yields about 2.20° at s1.Without these constraints, local gradient noise can unbind the real and imaginary parts above the ρ peak; monotonicity prevents numerical backtracking and unphysical phase oscillations.Source: H. Hyperparameter loss weighting and… · p. 16
• Omitting the moment constraint sends ⟨r6_π⟩ to about 12.7 fm6, more than triple baseline, while the moment loss anchors low-energy curvature.The loss penalises unphysical flattening near s = 0.Source: H. Hyperparameter loss weighting and… · p. 16
• With all physics constraints disabled, the network fits discrete timelike points but develops high-frequency oscillations that corrupt derivative-based moments and phase alignment.The physics losses act as non-local regularisers for holomorphy, unitarity, and reliable physical derivatives.Source: H. Hyperparameter loss weighting and… · pp. 17–18
• Removing timelike data collapses aππ_μ and shifts phase by about 116°, while CMD-3-only Set B raises the integral to 532.27 × 10^-10 and shifts phase by 5.08° at s1.Removing spacelike data instead causes minor degradation in low-energy moments; the Set B shift correlates with CMD-3’s higher cross-section normalisation near the ρ peak.Source: H. Hyperparameter loss weighting and… · p. 18
• With raw τ data, the native I = 1 switch yields 502.83 × 10^-10; Set Craw and Set CRIB phase mismatches are 6.65° and 6.28°, respectively, versus Set A’s 1.72°.Source: H. Hyperparameter loss weighting and… · p. 18
• Combining e+e− and raw τ data in Set D gives 515.30 × 10^-10, with a 3.46° high-energy phase deviation, better than standalone τ sets (>6°).The authors attribute the upward shift relative to Set A to raw τ normalisation and CMD-3 data jointly pulling the fit toward higher spectral density.Source: H. Hyperparameter loss weighting and… · p. 18

VI. CLOSED-FORM APPROXIMATION OF THE FORM FACTOR

• The fitted denominator roots all lie outside the unit disk, making the approximant holomorphic within it; its structure enforces Schwarz reflection.The reflection property pairs a pole at zρ with its mirror at z∗ρ.Source: §VI · p. 18
• Near the real axis and ρ(770) pole, Padé discrepancies are small; on the physical axis the curves are visually indistinguishable through the peak.Further into the complex plane, magnitude discrepancies grow to several per cent and phase discrepancies to a few degrees.Source: §VI · pp. 18–19, Figure 13 · p. 20
• The Padé pole gives an independent extraction, agreeing with the direct network extraction to within approximately 0.1% in mass and 3% in width.The second sheet is reached using zII(s) = 1/zI(s), mapping the fitted pole to its position through the inverse conformal map.Source: §VI · p. 19
• 3.4% mean relative deviation at [4] marks the timelike-cut plateau; [4/4] is selected as the minimal order saturating achievable accuracy.The deviation is 3.4% at and 3.2% at, while extracted resonance parameters stabilize from [4] onward.Source: §VI · pp. 19–20
• Padé-derived aππ values agree closely with the PINN, while higher moments and the phase deviation at s1 show larger residual shifts.The approximant compresses the continuous representation into a rational form that gives access to resonance poles and low-energy derivative expansions.Source: §VI · p. 20
• Minimal-order truncation slightly shifts low-energy moments by redistributing far-UV asymptotic tails, so the Padé is an analytic bridge rather than a solver.The full PINN solves the inverse problem; the rational approximant compresses its continuous representation.Source: §VI · p. 20

VII. DISCUSSION AND FUTURE DIRECTIONS

• The zero-free structure and CMD-3/BaBar preference emerge without direct loss terms, but are evidence from this architecture and constraint set, not independent physical proof.The default result excludes CMD-3; global holomorphy and dispersion constraints fit it less comfortably than BaBar and τ-decay data, and the framework alone does not resolve the tension.Source: §VII · p. 20
• The extracted pole width is 4–8% below the PDG world average, a deficit the authors associate with the elastic-unitarity approximation used to reach the second sheet.They state that resolving it will require an explicit treatment of inelastic channels.Source: §VII · p. 20
• Multi-channel form factors could capture inelastic effects beyond 1 GeV2; unified momentum-dependent ρ–ω mixing and π±–π0 mass splitting could address τ-decay and e+e− differences without external corrections.The authors also propose adapting the conformal PINN to the pion transition form factor for hadronic light-by-light scattering.Source: §VII · p. 20

Appendix A: Proofs of theorems

• With distinct points in the unit disk and a real-analytic, non-polynomial activation, the empirical NTK is positive definite generically when the network has at least as many hidden neurons as points.The proof establishes full row rank of the parameter Jacobian, which makes the empirical NTK positive definite.Source: Appendix A · pp. 20–21
• A single sine neuron can supply rank 3 at three distinct points for non-special point choices; the authors’ 512-neuron, four-layer network exceeds the stated conservative bound.The conservative bound counts at least one useful direction per neuron.Source: Appendix A · p. 21
• Under bounded activation derivatives, weights, and labels, the MSE-loss Hessian is uniformly bounded in z-space, independently of the original physical coordinate s.The bound depends on network size and weight bounds.Source: Appendix A · p. 21

Appendix B: The NLO pQCD constraint for the asymptotic loss functional

• For the deep-spacelike pQCD constraint, the authors choose the asymptotic pion distribution amplitude φas(x) = 6x(1−x), making factorization-scale dependence disappear and the NLO evolutional correction negligible.The correction is approximately 1% and is omitted when establishing a deep-spacelike boundary.Source: Appendix B · p. 22