DeepQuark: A Deep-Neural-Network Approach to Multiquark Bound States

Abstract

For the first time, we implement the deep-neural-network-based variational Monte Carlo approach for the multiquark bound states, whose complexity surpasses that of electron or nucleon systems due to strong SU(3) color interactions. We design a novel and high-efficiency architecture, DeepQuark, to address the unique challenges in multiquark systems such as stronger correlations, extra discrete quantum numbers, and intractable confinement interaction. Our method demonstrates competitive performance with state-of-the-art approaches, including diffusion Monte Carlo and Gaussian expansion method, in the nucleon, doubly heavy tetraquark, and fully heavy tetraquark systems. Notably, it outperforms existing calculations for pentaquarks, exemplified by the triply heavy pentaquark. For the nucleon, we successfully incorporate three-body flux-tube confinement interactions without additional computational costs. In tetraquark systems, we consistently describe hadronic molecule $T_{cc}$ and compact tetraquark $T_{bb}$ with an unbiased form of wave function ansatz. In the pentaquark sector, we obtain weakly bound $\bar D^*Ξ_{cc}^*$ molecule $P_{cc\bar c}(5715)$ with $S=\frac{5}{2}$ and its bottom partner $P_{bb\bar b}(15569)$. They can be viewed as the analogs of the molecular $T_{cc}$. We recommend experimental search of $P_{cc\bar c}(5715)$ in the D-wave $J/ψΛ_c$ channel. DeepQuark holds great promise for extension to larger multiquark systems, overcoming the computational barriers in conventional methods. It also serves as a powerful framework for exploring confining mechanism beyond two-body interactions in multiquark states, which may offer valuable insights into nonperturbative QCD and general many-body physics.

Figures (7)

• Figure 1: Architecture of the DeepQuark wave function, taking isoscalar doubly heavy tetraquark as an example. The physical inputs in spatial, color, spin and isospin degrees of freedom are transformed as the encoded inputs before fed into the DNN. Boundary condition, fermionic antisymmetrization ($\mathcal{A}$) and parity projection $(1+\pi\hat{P})$ are imposed on the scalar output of the DNN $f_{NN}(\boldsymbol{x})$ to obtain the multiquark wave function $\Psi_{\text{A}}^\pi(\boldsymbol{x})$.
• Figure 2: The energy estimate as a function of iteration steps for the (a) nucleon in the AL1 potential and flux-tube confinement interaction (FT), (b) isoscalar vector doubly charmed tetraquark, (c) scalar fully charmed tetraquark, and (d) isoscalar triply bottomed tetraquark systems in the optimization progress of DeepQuark (DQ). The Monte Carlo standard errors of the energies are shown by the shaded area, which are very tiny. The lowest two-body dissociation thresholds are represented by the black dashed lines. The ground-state energies given by the GEM Ma:2022vqfMeng:2023jqk and DMC Ma:2022vqf are respectively displayed by the blue and purple dashed lines for comparison.
• Figure S1: Two confinement scenarios for the baryons. The left and right panels represent the pairwise confinement mechanism and the flux-tube confinement mechanism, respectively.
• Figure S2: Distributions of the AL1 model parameter values in the 50 selected parameter sets. The red and green dashed lines represent the mean values over the 50 parameter sets and the original values in Ref. SilvestreBrac1996, respectively.
• Figure S3: Distributions of meson mass spectra (in MeV) calculated using the 50 selected sets of AL1 parameters. The red dashed lines represent the mean values. The red shaded area corresponds to $\pm 1$ standard deviation.