Nature of the $P_c$ states from compositeness criteria

TL;DR

The paper investigates the nature of four hidden-charm P_c states using three compositeness criteria within a dynamical coupled-channel framework fitted to LHCb $J/ψ p$ data. It refines pole positions and residues with improved fits and applies the pole-counting rule, spectral density functions, and Gamow-state formalism to quantify elementariness and compositeness without model bias. All three methods consistently indicate that the P_c states are predominantly molecular, with the main components being $ar{D}Σ_c$ for $P_c(4312)$, $ar{D}Σ_c^*$ for $P_c(4380)$, and $ar{D}^*Σ_c$ for $P_c(4440)$ and $P_c(4457)$, while elementariness remains small. The results provide robust support for the molecular interpretation and highlight the importance of coupled-channel dynamics, suggesting future photoproduction measurements to further test the structure of these states.

Abstract

Based on a coupled-channel approach, we investigate the structures of four $P_c$ states through compositeness criteria. Toward a more precise description of the states, we have obtained refined fit results of the LHCb data on the $J/ψp$ invariant mass distribution of the $Λ_b^0\to J/ψp K^-$ decay. Allowing for the fact that each of the four $P_c$ states couples strongly to a nearby $S$-wave channel, three criteria on the compositeness/elementariness are adopted in this study: the pole-counting rule, the spectral density function, and the Gamow wave function. Compositeness information is extracted from the scattering amplitudes and the pole parameters (pole positions and residues), without any preconceived assumptions on the nature of the $P_c$ states, and without any dependence on the model parametrization. Consistently within the framework of all the three methods, it has been found that the $P_c(4312)\,1/2^-$ is mainly composed by $\bar{D}Σ_c$, $P_c(4380)\,3/2^-$ by $\bar{D}Σ_c^*$, while the $P_c(4440)\,1/2^-$ and $P_c(4457)\,3/2^-$ states both turn out as composite states of $\bar{D}^*Σ_c$. The upper limits of the values of their elementariness are estimated to be rather small. This paper provides an additional confirmation of the molecular interpretation for the $P_c$ states in the literature.

Figures (5)

• Figure 1: The relevant scattering channels and the thresholds. The four pentagrams label the four $P_c$ poles from the fit A of Ref. Shen:2024nck: from left to right, they are $P_c(4312)\,1/2^-$, $P_c(4380)\,3/2^-$, $P_c(4440)\,1/2^-$, and $P_c(4457)\,3/2^-$, respectively.
• Figure 2: The three refined fit solutions in this study. The data are from Ref. LHCb:2019kea.
• Figure 3: The modulus square of the amplitude $T(\bar{D}\Sigma_c\to \bar{D}\Sigma_c)$ with complex energy $z$ (in units of MeV). The $|T|^2$ is in arbitrary units.
• Figure 4: The SDFs constructed for the four $P_c$ states in the three fit solutions. In each subfigure, the three lines on the top are the Breit-Wigner (BW) SDFs, while the three on the bottom are the locally constructed (L.C.) SDFs for each pole. Note that the $y$-axis features a logarithmic scale.
• Figure 5: The subtraction of the $\bar{D}^*\Sigma_c$ threshold cusp in the SDF of $P_c(4457)$ (fit A).