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Hidden charm pentaquarks and the nature of $P_{c}$ states observed at LHCb

This paper employs a semirelativistic potential quark model to demonstrate that the hidden-charm pentaquark states observed by LHCb are predominantly hadronic molecules formed via $\sigma$ - and $\rho$ -meson exchanges, successfully identifying the $P_c(4312)^+$ , $P_c(4440)^+$ , and $P_c(4457)^+$ as $\Sigma_c\bar{D}$ and $\Sigma_c\bar{D}^*$ bound states while offering a molecular interpretation for the $P_c(4380)^+$ despite a discrepancy in its predicted width.

Original authors: Zhi-Biao Liang, Jun-Jie Liu, Mu-Yang Chen, Xian-Hui Zhong

Published 2026-09-18

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CC BY 4.0

Original authors: Zhi-Biao Liang, Jun-Jie Liu, Mu-Yang Chen, Xian-Hui Zhong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, matter is built from a small family of fundamental particles called quarks. For decades, physicists understood that these particles usually group together in twos or threes to form the protons and neutrons inside an atom's nucleus. However, the laws of physics do not forbid quarks from gathering in larger, more exotic groups. When five quarks bind together, they form a particle known as a pentaquark. For years, these were only theoretical possibilities, but in recent years, experiments at the Large Hadron Collider have provided strong evidence that they actually exist. The big question that remains is what these particles are made of and how they hold together. Do they act like a single, tight knot of five quarks, or are they more like two smaller particles loosely orbiting each other, similar to a planet and its moon?

A team of researchers has taken a fresh look at this puzzle by simulating the behavior of these hidden-charm pentaquarks. They focused on a specific set of particles recently discovered by the LHCb collaboration, which contain a mix of up, down, and charm quarks. Using a sophisticated computer model that treats quarks as objects moving under the influence of specific forces, the team calculated the possible energy levels and masses of these particles. They tested two competing ideas: one where the five quarks are packed tightly into a single compact object, and another where they form a "hadronic molecule," a loose association of a baryon (a three-quark particle) and a meson (a two-quark particle). The researchers also calculated how these particles would likely break apart, a process that helps distinguish between the two structural possibilities.

The study began by calculating the properties of the compact pentaquark scenario. The results showed that if these particles were tight knots of five quarks, they would be significantly heavier than what has been observed in experiments. The model predicted that such compact states would have masses around 4,600 to 4,800 MeV, which is roughly 200 to 300 MeV heavier than the specific particles the LHCb team found. Furthermore, the researchers found that these compact states would be very narrow and stable, but their mass mismatch with the experimental data is too large to ignore. This effectively rules out the idea that the recently discovered particles are compact five-quark knots.

Instead, the researchers turned their attention to the hadronic molecule picture. In this scenario, the pentaquark is not a single tight unit but a pair of smaller particles held together by the exchange of other particles, much like how atoms are held together by the exchange of photons, though the forces here are different. The team found that this model aligns perfectly with the experimental observations. They identified that the particle known as Pc(4312)+ is likely a molecule made of a Sigma_c baryon and an anti-D meson. Similarly, the two closely spaced particles, Pc(4440)+ and Pc(4457)+, were identified as molecules formed by a Sigma_c baryon and an anti-D* meson, but with slightly different internal arrangements of their spins. The calculations showed that the forces holding these pairs together are primarily driven by the exchange of specific mesons, particularly the sigma and rho mesons, which act as the glue binding the two halves of the molecule.

The study also revisited an earlier, broader signal reported in 2015 called Pc(4380)+. While the original data suggested this was a wide, fuzzy structure, the new analysis suggests it could actually be a narrow molecule formed by a Sigma_c* baryon and an anti-D meson. The discrepancy in width might be due to how the data was analyzed in the past, and the researchers suggest that future, more precise measurements could confirm this narrower nature. The team also noted that while some very loosely bound states might exist in other configurations, their existence depends heavily on the strength of the attractive forces, which are not yet perfectly pinned down by experiment.

Ultimately, this work provides a unified explanation for the hidden-charm pentaquarks observed at the LHC. By demonstrating that the hadronic molecule model fits the observed masses and decay patterns while the compact model does not, the researchers have clarified the nature of these exotic particles. They are not tight knots of five quarks, but rather delicate, loosely bound pairs of smaller particles. This finding not only solves a specific mystery about the particles seen in 2015 and 2019 but also offers a roadmap for future experiments, suggesting where scientists should look to find more of these molecular states and how they might decay. The study confirms that the subatomic world is rich with complex structures that go beyond the simple combinations we see in everyday matter.

Technical Summary: Hidden Charm Pentaquarks and the Nature of Pc States Observed at LHCb

Problem Statement
The LHCb collaboration has reported the observation of several hidden-charm pentaquark candidates ( $P_c$ states) in the $pJ/\psi$ invariant mass spectrum, specifically $P_c(4312)^+$ , $P_c(4440)^+$ , $P_c(4457)^+$ , and the earlier broad structure $P_c(4380)^+$ . A central challenge in hadron physics is determining the internal structure of these states: whether they are compact multiquark configurations ( $qqqc\bar{c}$ ) or loosely bound hadronic molecules composed of a charmed baryon and a charmed meson. Existing interpretations vary, including compact states, hadronic molecules, and kinematic threshold effects. This work aims to provide a unified theoretical framework to distinguish between these scenarios by calculating the mass spectra and fall-apart decay properties of both compact and molecular configurations within the same model.

Methodology
The authors employ a unified semirelativistic potential quark model to investigate the low-lying $1S$ -wave states of the $qqqc\bar{c}$ system ( $q=u, d$ ).

Hamiltonian and Potentials: The mass spectrum is calculated using a Hamiltonian that includes:
- One-Gluon Exchange (OGE): Provides short-range confinement (linear potential), color-Coulomb interaction, and chromomagnetic (spin-spin) interactions.
- One-Boson Exchange (OBE): Includes long-range interactions mediated by pseudoscalar ( $\pi, K, \eta, \eta'$ ), scalar ( $\sigma$ ), and vector ( $\rho, \omega, K^*, \phi$ ) mesons. This component is crucial for describing hadronic molecules.
- The model parameters (constituent quark masses, coupling constants, cutoff parameters) are fixed by fitting light meson and baryon spectra, as well as lattice QCD predictions for doubly-charmed baryons.
Wave Function Construction:
- Compact States: The wave functions are constructed to satisfy the Pauli exclusion principle for identical light quarks, utilizing $S_3$ permutation group representations for flavor, color, and spin. Three independent color-singlet configurations ( $C_1, C_2, C_3$ ) are considered, involving both color-singlet triquark- $q\bar{q}$ and color-octet triquark-octet $q\bar{q}$ couplings.
- Molecular States: Treated as loosely bound systems of two color-singlet clusters (e.g., $\Sigma_c \bar{D}$ ). Permutation symmetry is not imposed between quarks belonging to different clusters. Only the color-singlet configuration contributes, and binding is driven primarily by OBE potentials.
Numerical Solution: The five-body Schrödinger equation is solved using the Explicitly Correlated Gaussian (ECG) method. The spatial wave functions are expanded in a basis of correlated Gaussians with variational parameters optimized to minimize the energy.
Decay Analysis: For states lying above the relevant two-body thresholds, fall-apart decay widths are evaluated using a quark-exchange model. In this framework, decays into baryon-meson pairs are induced by short-range OGE interactions via quark rearrangement.

Key Contributions and Results

Compact Pentaquark Spectrum:
- The model predicts a set of compact $qqqc\bar{c}$ states with masses significantly higher than the observed $P_c$ peaks.
- Seven states with isospin $I=1/2$ are found in the mass range of $\sim 4610 - 4700$ MeV.
- Three states with isospin $I=3/2$ are found in the range of $\sim 4760 - 4800$ MeV.
- These compact states are predicted to have narrow fall-apart decay widths (a few MeV).
- Conclusion on Compactness: The observed $P_c(4312)^+$ , $P_c(4440)^+$ , and $P_c(4457)^+$ cannot be assigned to these compact states, as the predicted masses are $\sim 200-300$ MeV too high.
Hadronic Molecule Spectrum:
- The study calculates the binding energies and masses for $\Lambda_c \bar{D}^{(*)}$ and $\Sigma^{(*)}_c \bar{D}^{(*)}$ molecular systems.
- $P_c(4312)^+$ : Identified as the $\Sigma_c \bar{D}$ molecule with $I^J = 1/2^-$ , denoted as $[\Sigma_c \bar{D}]^{1/2}_{1/2^-}(4318)$ . It has a binding energy of $\sim 6$ MeV and a large root-mean-square radius ( $\sim 3.8$ fm).
- $P_c(4440)^+$ and $P_c(4457)^+$ : Identified as the two $\Sigma_c \bar{D}^*$ molecular states with $I^J = 1/2^-$ . Specifically, $P_c(4440)^+$ corresponds to the deeper bound state $[\Sigma_c \bar{D}^*]^{1/2}_{1/2^-}(4437)$ (B.E. $\sim 24$ MeV), and $P_c(4457)^+$ corresponds to the shallower state $[\Sigma_c \bar{D}^*]^{1/2}_{3/2^-}(4458)$ (B.E. $\sim 2.7$ MeV).
- $P_c(4380)^+$ : The broad structure reported in 2015 is re-evaluated. The model predicts a narrow molecular state $[\Sigma^*_c \bar{D}]^{1/2}_{3/2^-}(4382)$ with a binding energy of $\sim 5.6$ MeV. The authors suggest the experimental observation of a broad structure may need re-examination or that the state is narrower than previously reported, potentially to be confirmed by Run 3 data.
Role of Interactions:
- The formation of $\Sigma^{(*)}_c \bar{D}^{(*)}$ bound states with $I=1/2$ is driven primarily by attractive $\sigma$ - and $\rho$ -meson exchanges.
- For the $I=3/2$ sector and $\Lambda_c \bar{D}^{(*)}$ systems, the existence of bound states depends critically on the strength of the $\sigma$ -exchange potential. With the central parameters used, very shallow bound states (B.E. $\sim 200-600$ keV) may exist, but they are highly sensitive to model variations.
Decay Properties:
- The calculated partial decay widths for the molecular assignments into $J/\psi p$ channels are consistent with experimental branching fraction constraints.
- For instance, the predicted branching fraction $Br[P_c(4312)^+ \to J/\psi p]$ is $\sim 4.1 \times 10^{-3}$ , consistent with LHCb and GlueX measurements.
- The study suggests that observing decays into $\Lambda_c \bar{D}^*$ and $\eta_c p$ channels would provide further confirmation of the molecular nature of these states.

Significance
The paper claims to provide a unified theoretical description that successfully distinguishes between compact and molecular interpretations of hidden-charm pentaquarks. By demonstrating that the observed $P_c$ masses align with hadronic molecule predictions while being inconsistent with compact quark model predictions, the study supports the molecular picture for $P_c(4312)^+$ , $P_c(4440)^+$ , and $P_c(4457)^+$ . Furthermore, it offers specific predictions for the nature of the $P_c(4380)^+$ state and identifies potential decay channels ( $\Lambda_c \bar{D}^*$ , $\eta_c p$ ) that future experiments should target to further explore the nature of these exotic hadrons. The work emphasizes that the $\sigma$ -meson exchange is a critical factor in the formation of these bound states.

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