S-matrix informed neural networks for amplitude analysis

Technical Summary: S-matrix Informed Neural Networks for Amplitude Analysis

Problem Statement
Reconstructing scattering amplitudes from finite, noisy, and mutually inconsistent experimental measurements is an ill-posed inverse problem central to particle physics, particularly for understanding hadron interactions and searching for New Physics. Traditional analyses rely on fixed functional forms (e.g., Breit-Wigner, K-matrix, dispersive fits) to parameterize amplitudes. These approaches suffer from inherent model dependencies: the extracted resonance parameters (masses, widths, couplings) and uncertainties are conditional on the chosen ansatz. Furthermore, global datasets often contain tensions where different experiments disagree beyond their quoted uncertainties. Standard parameterizations are often too rigid to fit conflicting data without ad hoc decisions, and they struggle to simultaneously satisfy the fundamental S-matrix principles of unitarity, analyticity, and crossing symmetry without introducing hidden biases.

Methodology: S-matrix Informed Neural Networks (SINNs)
The authors introduce a framework utilizing S-matrix Informed Neural Networks (SINNs) to learn scattering amplitudes directly from data while strictly adhering to first principles. The methodology consists of four integrated components:

1. Hybrid Neural Architecture:

   • Structure: A multi-task architecture featuring a shared "trunk" network connected to multiple specialized "head" networks. The trunk learns a common representation of the scattering amplitude (enforcing crossing symmetry), while the heads produce specific partial-wave outputs (phase shifts and inelasticities).
   • Hard Constraints: A post-processing layer algebraically enforces "hard" physical constraints. This includes threshold behavior (via barrier factors $q^{2\ell+1}$) and unitarity (via inelasticity maps ensuring $0 \le \eta \le 1$). This reduces the search space and improves training stability compared to enforcing constraints solely through loss functions.
   • Soft Constraints: Analyticity and crossing symmetry are imposed numerically via a multi-objective loss function based on the Roy equations.

2. Curriculum Training and Loss Design:

   • The training minimizes a composite loss function comprising data fidelity ($L_{data}$), Roy equation residuals ($L_{Roy}$), and scattering length consistency terms ($L_{con}$, $L_{SL}$).
   • A six-phase curriculum is employed to stabilize optimization. Training begins with data-driven learning (ignoring physics constraints initially) and gradually introduces the Roy equation penalties and scattering length consistency terms. This prevents the optimizer from converging to trivial solutions that satisfy physics constraints but fail to describe the data.
   • The loss utilizes a weighted sum of objectives, with specific weights ($w_\chi$) interpolating between unnormalized least squares and standard $\chi^2$ to handle varying experimental uncertainties.

3. Spectral Response Clustering for Data Selection:

   • To address inconsistent datasets, the authors propose a novel, reproducible data selection procedure.
   • Mechanism: An ensemble of constrained SINNs is trained. For each target experiment, the network is retrained with that experiment upweighted. The resulting change in the fit quality ($\chi^2$) of all other experiments is measured to construct a "response matrix."
   • Clustering: This matrix is embedded into a reduced Euclidean space via spectral decomposition. Ward linkage clustering groups experiments based on their mutual compatibility. Groups showing positive tension (incompatibility) are iteratively removed until a consistent subset remains.

4. Uncertainty Quantification:

   • Uncertainties are estimated using a Gaussian replica method combined with deep ensembles. Data points are resampled according to their experimental uncertainties (including split-normal distributions for asymmetric errors like DIRAC), and new networks are trained on these replicas.
   • The final uncertainty bands represent the empirical distribution of constrained amplitudes, capturing both statistical noise and model variations.

Key Results
The framework is applied to low-energy $\pi\pi$ scattering, a system characterized by historical data inconsistencies and complex dynamics (including the broad $\sigma/f_0(500)$ and narrow $f_0(980)$).

• Amplitude Reconstruction: The SINN ensemble produces smooth, model-independent partial-wave amplitudes ($S_0, P_1, S_2, D_0, D_2, F_1, G_2$) that describe the selected experimental data with high fidelity. The results reproduce known features, such as the $\rho(770)$ resonance and the behavior near the $K\bar{K}$ threshold.
• Dispersive Fulfillment: The reconstructed amplitudes satisfy the Roy equations (the quantitative realization of unitarity, analyticity, and crossing) within the domain of validity ($\sqrt{s} < 1.1$ GeV). The residuals between the direct SINN output and the Roy-reconstructed amplitude are consistent with zero.
• Scattering Lengths: The extracted scattering lengths ($a_0^0 = 0.222^{+0.014}_{-0.014}$ and $a_0^2 = -0.041^{+0.005}_{-0.006}$ in units of $m_\pi^{-1}$) agree with previous dispersive analyses (Colangelo et al., García-Martín et al.) and fall within the "universal band" and DIRAC constraints.
• Data Selection: The spectral response clustering successfully identified and removed incompatible datasets (e.g., Protopopescu VI/XIII, specific Grayer and Hyams subsets), reducing the total loss significantly and resolving tensions that plagued previous global fits.

Validation and Robustness
The authors perform rigorous validation to ensure the results are not artifacts of the specific neural architecture or training choices:

• Architectural Independence: A sweep over 21 hyperparameter dimensions and 100 distinct architectures yielded central values for observables statistically indistinguishable from the production model, indicating negligible model architecture bias.
• Closure Test: The pipeline was applied to synthetic data generated from a known Constrained Fit to Data (CFD) parametrization. The SINN successfully reconstructed the input amplitudes and scattering lengths with minimal bias ($|z| < 0.4$), confirming the pipeline's ability to recover known truths.
• Ablation Study: Removing the Roy equation constraints ($L_{Roy}=0$) resulted in a significant shift in scattering lengths and larger uncertainties, demonstrating that the dispersive constraints are essential for regularizing the threshold behavior and providing physically meaningful results, even though the data fit quality ($L_{data}$) remained unchanged.

Significance and Claims
The paper claims to present a unified workflow that integrates physics-constrained representation learning, automated data selection, and rigorous uncertainty quantification.

• Model Independence: By removing fixed functional forms, the method allows the data to dictate the amplitude shape while respecting S-matrix principles, thereby reducing epistemic uncertainties associated with model choice.
• Handling Inconsistency: The spectral response clustering provides an objective, reproducible method to select compatible data from inconsistent global datasets, a common issue in hadronic physics.
• Transferability: The authors assert that this workflow is transferable to other scattering processes (e.g., $\pi K$, $\pi N$) and other constrained physics problems limited by inconsistent data.
• Future Utility: The constrained amplitudes and correlated uncertainties produced are suitable for propagation into precision observables, such as the hadronic vacuum polarization contribution to the muon $g-2$, and can be analytically continued to the complex plane for resonance pole extraction.

The work concludes that SINNs offer a robust, systematic alternative to traditional parameterizations for amplitude analysis, successfully balancing the competing demands of fitting noisy data, satisfying first principles, and quantifying uncertainties without relying on rigid ansatzes.