$λ$, $ρ$, and $σ$ 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 ${bb\bar{u}cc}$: $M=2m_{b}+2m_{c}+m_{u}+5C/2 +β_{x_λ}(x_λ+c_{0x_λ})^{2/3} +β_{x_{ρ_1}}(x_{ρ_1}+c_{0x_{ρ_1}})^{2/3}+β_{x_{ρ_2}}\sqrt{x_{ρ_2}+c_{0x_{ρ_2}}} +β_{x_σ}(x_σ+c_{0x_σ})^{2/3}$. Four series of Regge trajectories, namely the $λ$-, $ρ_1$-, $ρ_2$-, and $σ$-trajectories, are investigated. We demonstrate that accounting for the internal structure and substructure of pentaquarks is indispensable for constructing the $ρ_1$-, $ρ_2$-, and $σ$-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 (embedded within the triquark) do not correspond one-to-one to the $ρ_1$-, $ρ_2$-, and $σ$-trajectories. Nevertheless, these trajectories govern the behaviors of the respective $ρ_1$-, $ρ_2$-, and $σ$-trajectories. For both configurations $(bb)(\bar{u}(cc))$ and $(cc)(\bar{u}(bb))$, the $λ$-, $ρ_1$-, and $σ$-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 $ρ_2$-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 $λ$-, $ρ_1$-, $ρ_2$-, and $σ$-excited states.