System Energies for Pentaquark Family Using Thomas-Fermi Quark Model

TL;DR

Using a Thomas-Fermi quark model with three-region density functions $f_1(x)$, $f_2(x)$, and $\overline{f}_2(x)$, the paper analyzes system energies for pentaquark families built from $uudc\bar{c}$ from 5 quarks to 70 quarks. It derives two cases (Case I: $\eta=1$ and Case II: $2\le\eta\le14$) of region-boundary consistency conditions that reduce the TF equations to a set of single-ODEs with constants $\overline{Q}_2$, $Q_2$, and $Q_1$, then computes the kinetic, potential, and volume energies by solving these equations and integrating over the regions. The authors identify a pronounced energy stability for integral multiples of four pockets (an icosaquark, $\eta=4$), which has the lowest energy per quark among the studied configurations, suggesting a potentially stable multi-pentaquark state within the TF framework, though spin-energy and other quantum numbers require further work. They acknowledge that spin-energy and other quantum numbers are not fully captured in the present semi-classical treatment and propose future work to incorporate these aspects and explore broader quark content. Overall, the work demonstrates systematic trends in multi-quark bound-state energies and outlines a computational approach to map stability across large families of pentaquark-like states.

Abstract

This study calculates the system energies of families of pentaquarks, $uudc\bar{c}$, starting from 5 quarks up to 70 using Thomas-Fermi statistical quark model. The model assumes spherical symmetry with the particles continuously distributed with varying densities and boundaries. The particles interact with Coulombic forces and a discontinuity in the light quark density function is energetically favored. Total energies of the system are calculated for each multiquark system and compared to determine family stability characteristics. A remarkable stability for multiquark combinations with multiples of four pentaquarks, which we term an icosaquark, is identified. The first icosaquark system is found to have lowest energy per quark, suggesting stability of a system consisting of four charm, four anticharm and twelve light quarks.

Figures (4)

• Figure 1: The schematic diagram of the distribution of quarks for the case of (a) $\eta = 1$, and (b) $\eta = 2-10$ in three different regions. $\overline{x}_{2}$, $x_{2}$ and $x_{1}$ differentiate the boundary between these regions.
• Figure 2: The density function plot for (a) $\eta=1$ and (b) $\eta=2$ as a function of distance from the center.
• Figure 3: The black line shows physical distance (in $fm$) versus quark content, indicating the increase in distance with the increase in quark content. The green and red lines refer to the boundaries of anti-charm and charm, respectively.
• Figure 4: The plot of (a) kinetic energy (b) potential energy (c) volume energy, (d) total energy per quark (in MeV) without rest mass energy for the case of $\eta = 1\text{-}14$.