Scattering Amplitude Analysis
Using Neural Networks
Denny Sombillo
National Institute of Physics, University of the Philippines Diliman

Yoichi ikeda
Toru Sato
Atsushi Hosaka
DLBS, YI, TS, AH PRD 102 016024 (2020)
DLBS, YI, TS, AH Few-Body Syst. 62, 52 (2021)
DLBS, YI, TS, AH PRD 104 036001 (2021)

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Approaches in hadron spectroscopy

JPAC
10.1016/j.ppnp.2022.103981
10.48550/arXiv.2112.13436

Top-down approach

Bottom-up strategy

Theory/ Model

Start with experimental data

Generate hadron
spectrumUse a generic amplitude ansatz

Slide Title

Is the theory a good
description of experiment?

Compare withexperimental data

Extract physics by fitting
In the bottom-up strategy, we are notinterested in explaining everything.
What is the experiment telling us?

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Approaches in hadron spectroscopy
The bottom-up
strategy can beaugmented by thedeep learning
approach.

Slide Title

JPAC
10.1016/j.ppnp.2022.103981
10.48550/arXiv.2112.13436

Bottom-up strategy
Start with experimental data
Trained DNN

Use different generic
amplitudes in training.

Extract physics
Classifier-type DNN already in use:
• Bound-virtual for single channel• Pc(4312)• Pole configuration
DLBSombillo / 2022.10.31-11.03

DLBS, YI, TS, AH PRD 102 016024 (2020)
DLBS, YI, TS, AH Few-Body Syst. 62, 52 (2021)
JPAC Collaboration PRD 105 L091501 (2020)
DLBS, YI, TS, AH PRD 104 036001 (2021)

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Deep learning: alternative analysis tool
Benchmarked on the known nucleon-nucleon bound state
Given only the s-wave cross section, the origin of enhancement can beunambiguously identified.
DLBS, YI, TS, AH PRD 102 016024 (2020)

Slide Title
Spin-singlet(virtual)

DLBS, YI, TS, AH Few-Body Syst. 62, 52 (2021)

Spin-triplet(bound)For near-threshold pole:k cot δ ∼ − 1/a (constant)| f(k) |In addition to the near-threshold pole, the S-matrix
can have distant singularities on the unphysicalsheet.
Use different (unitary, analytic) backgrounds to help
DNN distinguish bound and virtual enhancements.
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2= | k cot δ − ik | ∼ 2 + ka2

There is no way to discriminate a boundstate pole enhancement with a virtual
2enhancement using only | f(k) | on thescattering region.

k + iγS(k) = exp [2iδbg(k)]k − iγ4

Deep learning: alternative analysis tool
Optimize parameters of DNN using mock amplitudes
k + iγS(k) = exp [2iδbg(k)]k − iγSlide Title

VirtualBound
Resonance

Spin-singlet

DLBSombillo / 2022.10.31-11.03

Spin-tripletTrained and validated DNN
deployed to probe the nucleon-nucleon

Clushiq 2022 emmi workshop

5Pole structure of coupled channel system
The nature of enhancement can only be deduced
in a model-dependent way.

Badalyan, et. al., Phys. Rep. 82, 2, 31-177 (1982)

We can at least extract the pole configuration
in a model-independent way.

Slide Title

How many nearby poles in eachRiemann sheet are needed toreproduce the experimental data?

Inspired by:
Two-pole structure of Λ(1405)

T. Hyodo and U. -G. Meißner PDG 2021 review

Pole-counting argument - nature of f0(S*) (aka f0(980))
D. Morgan and M. R. Pennington PRD 48 1185 (1993)
D. Morgan Nuc. Phys. A 543 632-644 (1992)

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6Pole structure of coupled channel system
DNN task:
Count the number of nearby poles in eachRiemann sheet using only the scattering data.

Slide Title

Model space restriction:
Maximum of 4 poles, distributed in any of theunphysical sheets.
Two-channel case: 35 possible pole configurations

It is reasonable to expect that more than
one pole-based model can describe thedata due to the error bars.
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7Pole structure of coupled channel system
DL approach:
Generate the training dataset•Use only the general properties of S-matrix•Include the energy uncertainty
Optimize the parameters of the deep neural network

Slide Title

Input layer:•Energy points•Real part of amplitude•Imaginary part of amplitude

Output layer:•Pole config. 0•Pole config. 1•Pole config. 2•…
•…
•Pole config. 34

Deploy the trained DNN to extract model from the experimental data.
Sample result:• X% 2[bt]-0[bb]-0[tb]• Y% 0[bt]-2[bb]-0[tb]• Z% 0[bt]-1[bb]-3[tb]• ...
DSombillo / 2022.11.18
DLBSombillo/ 2022.10.31-11.03

Clushiq NSTAR
2022 emmi
2022
workshop8Training dataset (generation of model space)
General form of S-matrix:•Hermiticity below the lowest threshold•Unitarity
Dm(−p1, p2)
S11(p1, p2) =•Analyticity∏ D (p , p )Slide Titlemm

1

S11 = 1 + 2iT112KJ Le Couteur, Proc. Roy. Soc (London) A256 (1960)
RG Newton J. Math. Phys. 2, 188 (1961)

Dm(p1, − p2)
S22(p1, p2) =∏ Dm(p1, p2)
m2
S11S22 − S12 =∏m

Dm(−p1, − p2)
Dm(p1, p2)

The available experimental data will determine the relevant matrix element.
Ensure that only one nearby pole Em is generated by each Dm(p1, p2).
Dm(p1, p2) = [(p1 − iβ1m)2222− α1m] + λm [(p2 − iβ2m) − α2m]

α1m, α2m, β1m, and β2m are related to Em.

Do not let β1m and β2m be both positive. (Analyticity)
The RS is determined by the signs of β1m and β2m.
Extra parameter λm to push the other pole far fromthe scattering region without violating analyticity.
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9Training dataset (generation of model space)
Incorporate uncertainty in the energy:

Slide Title

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10Optimization of DNN modelChosen DNN architecture

We adopted the curriculum method totrain the DNN using the noisy dataset.

Slide Title

After ∼ 31,000 epochs the final training and testing
accuracies are 76.5 % and 80.4 % , respectively.

Performance in curriculum training

Noticeable saturation
Can this be improved?

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Intrinsic ambiguity in the lineshape
Amplitude with onepole in [bt] sheet.

Slide Title

Amplitude with onepole in each unphysicalsheet. The two extra
poles have the sameenergy values.
Identical lower
channel amplitude.

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Higher channel
amplitude can be
distinguished.

The only way to improve the DNN
performance is to include the higher(or off-diagonal) channel amplitude.

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12Inference stage: application
Slide Title•Draw points from each error bar using a Gaussian Interference on 10^6 amplitudes• 44.6% 1[bt]-1[bb]-2[tb]distribution.
• 34.1% 1[bt]-1[bb]-1[tb]•Construct inference amplitudes from theexperimental data using the drawn points.
• 16.4% 0[bt]-1[bb]-3[tb]•Feed the inference amplitudes to the trained DNN. • 04.9% 0[bt]-1[bb]-2[tb]DLBSombillo / 2022.10.31-11.03

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Inference stage: application
Slide Title

Interference on 10^6 amplitudes
Using uniform distribution• 60.3% 1[bt]-1[bb]-2[tb]• 30.9% 1[bt]-1[bb]-1[tb]• 07.5% 0[bt]-1[bb]-3[tb]• 01.3% 0[bt]-1[bb]-2[tb]DLBSombillo / 2022.10.31-11.03

Clushiq 2022 emmi workshop

14Summary•We can teach DNN:
•to distinguish threshold enhancement in single-channel scattering•to determine the pole configuration of a coupled-channel amplitude

Slide Title•Deep learning approach can be used as a model-selection framework.
•The results of deep learning method can be used to design an amplitude
ansatz appropriate for a given experimental data.

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Acknowledgement
We would like to acknowledge The Mathematical Analysis Unit of theCenter for Infectious Disease Education and Research (CiDER), Osaka
University for the conference and travel support.

Slide Title

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