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

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 $\rho_1$-, $\rho_2$-, $\lambda_1$-, and $\lambda_2$-trajectories. We provide rough estimates for the masses of the $\rho_1$-, $\rho_2$-, $\lambda_1$-, and $\lambda_2$-excited states. Except for the $\lambda_1$-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 $\rho_1$-, $\rho_2$-, and $\lambda_2$-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 $\rho_1$-, $\rho_2$-, and $\lambda_2$-trajectories must rely solely on fitting existing theoretical or future experimental data. The fundamental relationship between the slopes of the obtained trajectories and constituents' masses and string tension will remain unclear, and the predictive power of the Regge trajectories would be compromised. Moreover, we show that the lengthy complete forms of the $\rho_1$-, $\rho_2$-, and $\lambda_2$-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.