$\lambda$ and $\rho$ trajectories for the doubly heavy baryons in the diquark picture

We present the explicit form of the Regge trajectory relations for the doubly heavy baryons ${\mathrm{\Xi }}_{Q{Q}^{\mathrm{\prime }}}$ and ${\mathrm{\Omega }}_{Q{Q}^{\mathrm{\prime }}}$ $(Q,Q^{\mathrm{\prime }}=b,c)$ in the diquark picture. Using the derived Regge trajectory relations, we estimate the masses of the $\lambda$-excited states and the $\rho$-excited states, which are consistent with other theoretical predictions. Both the $\lambda$-trajectories and $\rho$-trajectories are discussed. We show that the $\rho$-trajectories behave differently from the $\lambda$-trajectories. Specifically, the $\rho$-trajectories behave as $M\sim x_{\rho }^{2\mathrm{/}3}$ $(x_{\rho }=n_r,l)$, whereas the $\lambda$-trajectories follow $M\sim x_{\lambda }^{1\mathrm{/}2}$ $(x_{\lambda }=N_r,L)$. By using the obtained relations, the baryon Regge trajectory provides a straightforward and easy method for estimating the spectra of both the $\lambda$-excited states and $\rho$-excited states.