Learning S-Matrix Phases with Neural Operators

Abstract

We use Fourier Neural Operators (FNOs) to study the relation between the modulus and phase of amplitudes in $2\to 2$ elastic scattering at fixed energies. Unlike previous approaches, we do not employ the integral relation imposed by unitarity, but instead train FNOs to discover it from many samples of amplitudes with finite partial wave expansions. When trained only on true samples, the FNO correctly predicts (unique or ambiguous) phases of amplitudes with infinite partial wave expansions. When also trained on false samples, it can rate the quality of its prediction by producing a true/false classifying index. We observe that the value of this index is strongly correlated with the violation of the unitarity constraint for the predicted phase, and present examples where it delineates the boundary between allowed and disallowed profiles of the modulus. Our application of FNOs is unconventional: it involves a simultaneous regression-classification task and emphasizes the role of statistics in ensembles of NOs. We comment on the merits and limitations of the approach and its potential as a new methodology in Theoretical Physics.

Figures (8)

• Figure 1: Plots of the ground truth $\sin\phi(z)$ (blue color) and FNO-predicted $\sin\phi(z)$ (orange color) for 3 randomly chosen samples of amplitudes within the 6K test-dataset. From top to bottom we list plots for amplitudes with finite partial wave expansions and $L=1,2,3$ respectively.
• Figure 2: The top two plots display the prediction of the trained NO for $\sin\phi(z)$ against the exact result for input modulus $B(z) = \frac{1}{10}(z+4)$. On the left are the actual functions, while on the right the point-wise relative difference. The bottom two plots display the corresponding data for input modulus $B(z) = \frac{1}{2}(z^2+1)$. Both cases refer to amplitudes with an infinite partial wave expansion.
• Figure 3: Heatmaps for the log base 10 loss of the NO prediction with respect to the unitarity condition \ref{['unitarity']}. The top left plot refers to linear moduli $B(z)=az+b$, the top right plot to quadratic moduli $B(z) = cz^2+d$ and the bottom plot to cubic moduli $B(z)=cz^3+d$. Analogous results for the top two plots were obtained with the use of PINNs in Ref. Dersy:2023job (see Figs. 3 and 5 of that paper). The thin black curves are the $\sin\mu=1$ boundaries while the thick gray curves express the dual bounds.
• Figure 4: Predictions for quadratic moduli $B(z)=cz^2+d$ by one of the 56 NOs trained on both true and false S-matrix phases. The heatmap on the left presents the $\log_{10}$ unitarity loss of the predictions. The heatmap on the right presents the value of the fidelity index ${\cal F}$. The colorbar scale for the latter focuses on values between 0.95 and 1. Values below 0.95 are depicted in deep blue and values above 1 are depicted in deep red. As in previous plots, we have included the curve at $\sin\mu=1$ (light black) and the curve of the dual bound (thick gray).
• Figure 5: The left heatmap depicts the value of the mean fidelity index $\bar{{\cal F}}$ on the landscape of linear moduli $B(z)=az+b$. The right heatmap depicts the corresponding values of the mean fidelity index for quadratic moduli $B(z)=cz^2+d$. Notice that the colorbar scale focuses on values between 0.95 and 1. We have also included the curve at $\sin\mu=1$ (light black) and the curve of the dual bound (thick gray).