$\lambda$-$\rho$ mode mixing and radial Regge trajectories for triply heavy baryons ${\mathrm{\Omega }}_{ccc}$ and ${\mathrm{\Omega }}_{bbb}$ in the diquark picture

The mixed-mode trajectories arising from permutation-enforced $\lambda$-$\rho$ mode mixing are first investigated in the diquark picture. By accounting for the internal structure and substructure of baryons, we propose an approach to construct the mixed-mode Regge trajectories for baryons composed of three identical quarks. For triply heavy baryons ${\mathrm{\Omega }}_{ccc}$ and ${\mathrm{\Omega }}_{bbb}$, naive diquark-quark basis states without full permutation symmetrization violate the exchange symmetry required by the Pauli exclusion principle for three identical fermions; consequently, physical eigenstates arise as permutation-enforced $\lambda$-$\rho$ mode admixtures. Under the minimal-mixing assumption, we explicitly perform permutation symmetrization for the $J^P=(3\mathrm{/}2{)}^{+}$ radially-excited S-wave states and construct mixed-mode Regge trajectory formulas for these symmetrized states. Unlike triply heavy baryons ${\mathrm{\Omega }}_{ccb}\mathrm{/}{\mathrm{\Omega }}_{cbb}$ in the fixed diquark $(cc)b\mathrm{/}(bb)c$ picture where well-defined pure $\rho$- and $\lambda$-trajectories exist, genuine pure-mode trajectories no longer survive for ${\mathrm{\Omega }}_{ccc}$ and ${\mathrm{\Omega }}_{bbb}$ due to permutation-driven cross-mode mixing. The resulting effective radial trajectories exhibit a novel sprouting-branch-like behavior. Both the original unmixed trajectories and the mixed-mode effective trajectories exhibit behavior of $M\sim x^{2\mathrm{/}3}$ (where $x=n_r,N_r,n_r+N_r$). Radially excited masses are crudely estimated via the mixed-mode Regge trajectories, which show reasonable agreement with existing theoretical predictions.