Deep learning topological inference-guided $T_{cc}^{+}$ pole parameter extraction

Abstract

We perform a data-driven study of the doubly charmed tetraquark candidate $T_{cc}^+$. An ensemble of deep neural network classifiers, trained on synthetic amplitudes with controlled analytic structures, identifies a dominant pole topology characterized by an isolated pole on the $[bt]$ Riemann sheet which is robust against left-hand cut effects. A subsequent pole parameter extraction was performed via the uniformized $\mathcal{S}$-matrix and a complementary $\mathcal{K}$-matrix parameterization, which respectively provides a model-independent baseline and dynamical insight on the pole position and trajectory of the resonant state. Using this two-pronged approach, we submit that the $T_{cc}^{+}$ is a shallow $D^0D^{*+}$ bound state in the second Riemann sheet of the complex plane.

Figures (9)

• Figure 1: Mapping of the four-sheeted complex $s$-plane onto the single-sheeted complex $\omega$-plane. The red dot labeled as $\tau$ correspond to the branch point associated with left-hand cut that arises from meson exchange. The maroon point at $\omega = i$ and the black point at $\omega = 1$ correspond to the first and second channel thresholds, $\varepsilon_1$ and $\varepsilon_2$, respectively. The physical scattering region between the thresholds is mapped onto the upper arc of the unit circle, while the region above the second threshold is represented by the segment of the real axis extending beyond $\omega = 1$.
• Figure 2: Schematic workflow for inference using trained DNN classifiers on empirical data. (Left) The experimental invariant mass distribution for the $D^{0}D^{0}\pi^{+}$ final state as reported by the LHCb CollaborationLHCb:2021vvqLHCb:2021auc, including statistical and detector-induced uncertainties. (Right) A diagram of trained DNN model with an input layer of $114$ nodes corresponding to concatenated energy-amplitude pairs. The illustrated model contains three hidden layers, though architectures with up to five hidden layers were also explored. The output layer contains two nodes, the dimensionality was reduced from $35$ as a result of preliminary coarse-grained classification and finalist selection strategy. The LHCb data was extracted from hepdata.Tcc, and the TikZ code for the DNN figure was taken from neutelings:2021; both are reproduced under the terms of the Creative Commons Attribution $4.0$ International License (CC BY $4.0$).
• Figure 3: Schematic overview of the coarse-grained classification and finalist selection strategy. The original $35$ pole classes (top row) were grouped into seven structural families (second row) to address degeneracies in the line shapes. Dominant classes from each family (third row) formed an $8$-class candidate set for amplitudes without lhc effects and $4$-class candidate set for amplitudes with lhc effects. These classes were randomly grouped into two sets to select the dominant class from each set (last row) for an extensive binary classification analysis using an ensemble of $50$ DNNs.
• Figure 4: Mean validation accuracy of DNN models across $7$ structural families. Each bar represents the average over three independently-trained manually tuned and Optuna-optimized DNN models per family. The left panel shows the classification performance of DNNs trained on amplitudes without lhc effects and the right panel displays the classification performance of DNNs trained on amplitudes with lhc effects. The fuchsia horizontal dashed-dotted line marks the $70\%$ validation accuracy threshold; i.e., predictions below this threshold are insufficiently reliable for subsequent inference using LHCb data.
• Figure 5: Molecular vs. compact tetraquark picture. Ensemble-averaged predicted class distributions from $100$ DNN models trained on (a) amplitudes without lhc effects and (b) on amplitudes with lhc effects. Predictions are made on LHCb data using two amplitude sampling assumptions: Gaussian and uniform distribution. The mean classification counts are obtained from $50{,}000$ LHCb amplitude realizations, with predictions aggregated across an ensemble of $50$ DNNs. Error bars represent standard errors of the ensemble means.