Regge trajectories for the doubly heavy triquarks $((Qq){\bar{Q}}^{\mathrm{\prime }})$

We attempt to apply the Regge trajectory approach to the doubly heavy triquarks $((Qq){\bar{Q}}^{\mathrm{\prime }})$ $(Q,Q^{\mathrm{\prime }}=b,c;q=u,d,s)$. We propose the Regge trajectory relations for the doubly heavy triquarks, and then employ them to crudely estimate the spectra of the triquarks $((cu)\bar{c})$, $((cu)\bar{b})$, $((cs)\bar{c})$, $((cs)\bar{b})$, $((bu)\bar{c})$, $((bu)\bar{b})$, $((bs)\bar{c})$, and $((bs)\bar{b})$. The $\lambda$-trajectories and the $\rho$-trajectories are investigated. The triquark Regge trajectory becomes a new and very simple approach for estimating the spectra of triquarks. It also provides a simple method to investigate the $\rho$-mode and $\sigma$-mode excitations of pentaquarks and hexaquarks in the triquark picutre. Moreover, the spin-averaged masses of the ground states of pentaquarks $(\bar{c}(cu))(cu)$, $(\bar{b}(bu))(bu)$ and $(\bar{c}(cu))(bu)$ are estimated, which are consistent with other theoretical predictions.