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

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.

💡 Research Summary

The paper introduces “DeepQuark,” the first implementation of a deep‑neural‑network (DNN) based variational Monte‑Carlo (VMC) method for solving the quantum many‑body problem of multiquark bound states. Multiquark systems (tetra‑ and penta‑quarks) are far more challenging than electronic or nuclear systems because they carry an additional SU(3) color degree of freedom, experience very strong short‑range correlations, and often involve non‑pairwise confinement mechanisms such as flux‑tube three‑body forces. Traditional approaches—basis‑expansion methods like the Gaussian Expansion Method (GEM) and diffusion Monte‑Carlo (DMC)—suffer from exponential growth of basis size, sign‑problem limitations, or prohibitive cost when handling complex potentials.

DeepQuark overcomes these obstacles by encoding the full color‑spin‑isospin symmetry directly into the neural‑network architecture. The authors first construct coupled bases for color, spin, and isospin (e.g., χ̄₃⊗3, χ₆⊗ ̄6 for doubly‑heavy tetraquarks) and map each basis to a normalized vector α in ℝⁿ. These vectors, together with the spatial coordinates of the quarks (r_i) and all inter‑quark distances |r_i−r_j|, form the input feature set x = (r_i, |r_i−r_j|, α_c, α_s, α_t). A four‑layer fully‑connected DNN with tanh activations outputs a scalar f_NN(x). The final wave function is built as Ψ = (1+π P̂) A