Classifying the pole of an amplitude using a deep neural network

Most of the exotic resonances observed in the past decade appear as a peak structure near some threshold. These near-threshold phenomena can be interpreted as genuine resonant states or enhanced threshold cusps. Apparently, there is no straightforward way of distinguishing the two structures. In this work, we employ the strength of deep feed-forward neural network in classifying objects with almost similar features. We construct a neural network model with scattering amplitude as input and the nature of a pole causing the enhancement as output. The training data is generated by an S-matrix satisfying the unitarity and analyticity requirements. Using the separable potential model, we generate a validation data set to measure the network’s predictive power. We find that our trained neural network model gives high accuracy when the cutoff parameter of the validation data is within 400–800 MeV. As a final test, we use the Nijmegen partial wave and potential models for nucleon-nucleon scattering and show that the network gives the correct nature of the pole.

Figure 1

Schematic of deep neural network for S-matrix pole classification.

Figure 2

Behavior of $\xi =\xi (c)$.

Figure 3

Configuration of S-matrix pole as $c$ is varied from zero to some negative value starting with (a) a pair of virtual states, (b) virtual state with widths and (c) resonance. The red dots represents the pole positions when $c=0$, the red line shows the trajectory and the blue line shows the direction of pole motion as $c$ becomes negative.

Figure 4

Virtual ($\diamond$) and resonance ($\star$) poles near threshold (a) and the corresponding line-shape (b). Poles far from the threshold but close to the imaginary axis of unphysical sheet (c) and the corresponding line-shape (d).

Figure 5

Testing accuracy of different neural networks with architecture (a) $[200+1,100+1,3]$, (b) $[200+1,150+1,3]$, (c) $[200+1,100+1,50+1,3]$ and (d) $[200+1,50+1,50+1,50+1,3]$

Figure 6

Testing accuracy of neural network model trained using set 1 (a) and the cost-function profile (b) in each training epoch.

Figure 7

Testing accuracy of neural network model trained using set 2 (a) and the cost-function profile (b) in each training epoch.

Figure 8

Pole trajectory of separable potential with energy-independent coupling (a), energy-dependent coupling with $M_{sep}>0$ (b) and with $M_{sep}<0$ (c). The dashed line shows the pole’s trajectory and the dotted line separates resonances with virtual states.

Figure 9

Accuracy of neural network model trained using set 1 with separable potential amplitude as input for different cutoff parameters. (a) Performance with energy-independent coupling, (b) for energy-dependent coupling with $M_{sep}>0$ and (c) for energy-dependent coupling with $M_{sep}<0$. Each horizontal bar correspond to 100,000 input s-wave cross section.

Figure 10

Accuracy of neural network model trained using set 2 with separable potential amplitude as input for different cutoff parameters. (a) Performance with energy-independent coupling, (b) for energy-dependent coupling with $M_{sep}>0$ and (c) for energy-dependent coupling with $M_{sep}<0$. Each horizontal bar correspond to 100, 000 input s-wave cross section.