$\lambda$, $\rho$, and $\sigma$ Regge trajectories for the quadruply heavy pentaquark $bb\bar{u}cc$ in the diquark-triquark picture

Abstract

Systematic investigations of four series of Regge trajectories for quadruply heavy pentaquarks are still lacking. Using the diquark and triquark Regge trajectory relations, we propose the Regge trajectory relations for the quadruply heavy pentaquark ${bb\bar{u}cc}$: $M=2m_{b}+2m_{c}+m_{u}+5C/2 +\beta_{x_{\lambda}}(x_{\lambda}+c_{0x_{\lambda}})^{2/3}$$+\beta_{x_{\rho_1}}(x_{\rho_1}+c_{0x_{\rho_1}})^{2/3}+\beta_{x_{\rho_2}}\sqrt{x_{\rho_2}+c_{0x_{\rho_2}}} +\beta_{x_{\sigma}}(x_{\sigma}+c_{0x_{\sigma}})^{2/3}$. Four series of Regge trajectories, namely the $\lambda$-, $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories, are investigated. We demonstrate that accounting for the internal structure and substructure of pentaquarks is indispensable for constructing the $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories; without such structural considerations, functional form and trajectory parameters can only be obtained via pure fitting against theoretical or experimental data. We further prove that the Regge trajectories of diquark 1, triquark, and diquark 2 (emdedded within the triquark) do not correspond one-to-one to the $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories. Nevertheless, these trajectories govern the behaviors of the respective $\rho_1$-, $\rho_2$-, and $\sigma$-trajectories. For both configurations $(bb)(\bar{u}(cc))$ and $(cc)(\bar{u}(bb))$, the $\lambda$-, $\rho_1$-, and $\sigma$-trajectories exhibit behavior of $M{\sim}x^{2/3}$ ($x=n_{r_1},n_{r_3},l_1,l_3,N_{r},L$), whereas the $\rho_2$-trajectories exhibit behavior of $M{\sim}\sqrt{x}$ ($x=n_{r_2},\,l_2$). The functional behavior of Regge trajectories for diquarks and triquark offers guidance for fitting the pentaquark Regge trajectories. Additionally, we provide rough estimates for spin-averaged masses of the $\lambda$-, $\rho_1$-, $\rho_2$-, and $\sigma$-excited states.