Model independent analysis of coupled-channel scattering: A deep learning approach

We develop a robust method to extract the pole configuration of a given partial-wave amplitude. In our approach, a deep neural network is constructed where the statistical errors of the experimental data are taken into account. The teaching dataset is constructed using a generic S-matrix parametrization, ensuring that all the poles produced are independent of each other. The inclusion of statistical error results into a noisy classification dataset which we should solve using the curriculum method. As an application, we use the elastic $\pi N$ amplitude in the $I(J^P)=1/2(1/2^{-})$ sector where ${10}^6$ amplitudes are produced by combining points in each error bar of the experimental data. We fed the amplitudes to the trained deep neural network and find that the enhancements in the $\pi N$ amplitude are caused by one pole in each nearby unphysical sheet and at most two poles in the distant sheet. Finally, we show that the extracted pole configurations are independent of the way points in each error bar are drawn and combined, demonstrating the statistical robustness of our method.

Figure 1

The elastic $\pi N$ amplitude of the GW-SAID in [26]. The two-hadron thresholds are shown as vertical thin lines. Only the $\pi N$ and $\eta N$ channels are considered in the present study. The $K\mathrm{\Lambda }$ and $K\mathrm{\Sigma }$ thresholds are shown for reference purposes only.

Figure 2

Relevant regions of two-channel Riemann sheets. The red ray, with two dots, represents the scattering region lying on the upper $[tt]$. The curve surface in (a) shows the $[tt]-[bb]$ connection, while the curve in (b) shows the $[bt]-[tb]$ connection.

Figure 3

The partial cross section (a) and the corresponding pole configuration (b). The pole in $[bt]$ sheet is fixed and is the same for all cases with real part equal to $T_2$. The $[bb]$ pole is added without assuming any trajectory.

Figure 4

DNN architecture. The circles are for input nodes, crossed circles are for bias nodes, single cross circles are for hidden layer nodes, and shaded circles are for output nodes. The lines are the weights and biases.

Figure 5

Performance of the DNN models for the first three curricula. The blue lines are for the training performance while the orange lines are for the testing performance.

Figure 6

Training (blue) and testing (orange) performance of our DNN model. Curriculum learning is used until epoch 3200 (red vertical solid line). The scale of horizontal axis is changed to $\mathrm{epoch}/20$ after the curriculum learning at the vertical red line.

Figure 7

DNN inference on amplitudes generated from experimental data with points drawn from Gaussian distribution in each error bar. The legend shows the identified pole configurations by the DNN.