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$λ$ and $ρ$ Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$ in the diquark-triquark picture

He Song, Xin-Ru Liu, Jia-Qi Xie, Jiao-Kai Chen

TL;DR

The paper develops Regge trajectory relations for the fully heavy pentaquark $P_{cc\bar{c}bb}$ in a diquark-triquark framework, accounting for four excitation channels $\rho_1$, $\rho_2$, $\lambda_1$, and $\lambda_2$. By combining a spinless Salpeter treatment with a Cornell-like potential, it derives four Regge relations and computes masses for radially and orbitally excited states in two configurations, finding that all four trajectories exhibit $M \sim x^{2/3}$ and are concave downward in the relevant plots. The complete trajectory expressions are lengthy, but simple fitted formulas closely reproduce the results, underscoring the need to incorporate pentaquark substructure rather than naively mirroring meson trajectories. The study provides rough mass estimates and a unified, predictive Regge framework that can extend to other fully heavy pentaquarks and guide experimental searches.

Abstract

We propose the Regge trajectory relations for the fully heavy pentaquark $P_{cc\bar{c}bb}$ utilizing both diquark and triquark Regge trajectory relations. Using these new relations, we discuss four series of Regge trajectories: the $ρ_1$-, $ρ_2$-, $λ_1$-, and $λ_2$-trajectories. We provide rough estimates for the masses of the $ρ_1$-, $ρ_2$-, $λ_1$-, and $λ_2$-excited states. Except for the $λ_1$-trajectories, the complete forms of the other three series of Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$ are lengthy and cumbersome. We show that the $ρ_1$-, $ρ_2$-, and $λ_2$-trajectories can not be obtained by simply imitating the meson Regge trajectories because mesons have no substructures. To derive these trajectories, pentaquark's structure and substructure should be taken into consideration. Otherwise, the $ρ_1$-, $ρ_2$-, and $λ_2$-trajectories must rely solely on fitting existing theoretical or future experimental data. Consequently, the fundamental relationship between the slopes of the obtained trajectories and constituents' masses and string tension will become unobvious, and the predictive power of the Regge trajectories would be compromised. Moreover, we show that the lengthy complete forms of the $ρ_1$-, $ρ_2$-, and $λ_2$-trajectories can be well approximated by the simple fitted formulas. Four series of Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$ all exhibit a behavior of $M{\sim}x^{2/3}$, where $x=n_{r_1},n_{r_2},l_1,l_2,N_{r_1},N_{r_2},L_1,L_2$. All four series of trajectories exhibit concave downward behavior in the $(M^2,\,x)$ plane.

$λ$ and $ρ$ Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$ in the diquark-triquark picture

TL;DR

The paper develops Regge trajectory relations for the fully heavy pentaquark in a diquark-triquark framework, accounting for four excitation channels , , , and . By combining a spinless Salpeter treatment with a Cornell-like potential, it derives four Regge relations and computes masses for radially and orbitally excited states in two configurations, finding that all four trajectories exhibit and are concave downward in the relevant plots. The complete trajectory expressions are lengthy, but simple fitted formulas closely reproduce the results, underscoring the need to incorporate pentaquark substructure rather than naively mirroring meson trajectories. The study provides rough mass estimates and a unified, predictive Regge framework that can extend to other fully heavy pentaquarks and guide experimental searches.

Abstract

We propose the Regge trajectory relations for the fully heavy pentaquark utilizing both diquark and triquark Regge trajectory relations. Using these new relations, we discuss four series of Regge trajectories: the -, -, -, and -trajectories. We provide rough estimates for the masses of the -, -, -, and -excited states. Except for the -trajectories, the complete forms of the other three series of Regge trajectories for the pentaquark are lengthy and cumbersome. We show that the -, -, and -trajectories can not be obtained by simply imitating the meson Regge trajectories because mesons have no substructures. To derive these trajectories, pentaquark's structure and substructure should be taken into consideration. Otherwise, the -, -, and -trajectories must rely solely on fitting existing theoretical or future experimental data. Consequently, the fundamental relationship between the slopes of the obtained trajectories and constituents' masses and string tension will become unobvious, and the predictive power of the Regge trajectories would be compromised. Moreover, we show that the lengthy complete forms of the -, -, and -trajectories can be well approximated by the simple fitted formulas. Four series of Regge trajectories for the pentaquark all exhibit a behavior of , where . All four series of trajectories exhibit concave downward behavior in the plane.

Paper Structure

This paper contains 11 sections, 26 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Regge trajectory relations for the fully heavy pentaquarks
  3. Preliminary
  4. Regge trajectory relations for the fully heavy pentaquarks
  5. Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$
  6. Parameters
  7. $\rho$-trajectories for the pentaquark $P_{cc\bar{c}bb}$
  8. $\lambda$-trajectories for the pentaquark $P_{cc\bar{c}bb}$
  9. Discussions
  10. Conclusions
  11. List of the Regge trajectory relations

Figures (3)

  • Figure 1: Schematic diagram of a pentaquark in the diquark-triquark picture. The left grey part represents the diquark $1$, composed of two quarks ($q_1$ and $q_2$). The right grey part represents a triquark composed of one antiquark ($\bar{q}_3$) and diquark $2$. Diquark $2$ is composed of quarks $q_4$ and $q_5$. The circles denote the quarks and the antiquark.
  • Figure 2: The $\rho_1$- and $\rho_2$-trajectories for the pentaquark $P_{cc\bar{c}bb}$ in the configurations $(cc)(\bar{c}(bb))$ and $(bb)(\bar{c}(cc))$. $n_{r_1}$ and ${l_1}$ are the radial and orbital quantum numbers for the $\rho_1$-mode, respectively, while $n_{r_2}$ and ${l_2}$ are the corresponding numbers for the $\rho_2$-mode. Circles represent the predicted data listed in Table \ref{['tab:rho']}. The black dashed lines correspond to the $\rho$-trajectories for the complete forms, obtained from Eqs. (\ref{['ppt2q']}) and (\ref{['pppa2qQ']}) or from Eqs. (\ref{['rtpf']}) and (\ref{['pppa2qQ']}). The pink lines correspond to the fitted formulas, obtained by linearly fitting the predicted data in Table \ref{['tab:rho']}; these formulas are listed in Table \ref{['tab:formulas']}. The blue dotted lines correspond to the main parts of the complete form, which are also listed in Table \ref{['tab:formulas']}.
  • Figure 3: The $\lambda_1$- and $\lambda_2$-trajectories for the pentaquark $P_{cc\bar{c}bb}$ in configurations $(cc)(\bar{c}(bb))$ and $(bb)(\bar{c}(cc))$. $N_{r_1}$ and ${L_1}$ are the radial and orbital quantum numbers for the $\lambda_1$-mode, respectively. $N_{r_2}$ and ${L_2}$ are the radial and orbital quantum numbers for the $\lambda_2$-mode, respectively. Circles represent the predicted data listed in Table \ref{['tab:lambda']}. The black dashed lines correspond to the $\lambda$-trajectories for the complete forms, obtained from Eqs. (\ref{['ppt2q']}) and (\ref{['pppa2qQ']}) or from Eqs. (\ref{['rtpf']}) and (\ref{['pppa2qQ']}). The pink lines correspond to the fitted formulas, obtained by linearly fitting the calculated data in Table \ref{['tab:lambda']}; these formulas are listed in Table \ref{['tab:formulas']}. The blue dotted lines are for the main parts of the full forms, which are also listed in Table \ref{['tab:formulas']}.