Parity doublet model for baryon octets: diquark classifications and mass hierarchy based on the quark-line diagram

We construct ${\rm SU(3)}_{\rm L} \otimes {\rm SU(3)}_{\rm R}$ invariant parity doublet models within the linear realization of the chiral symmetry. Describing baryons as the superposition of linear representations should be useful description for transitions toward the chiral restoration. The major problem in the construction is that there are much more chiral representations for baryons than in the two-flavor cases. To reduce the number of possible baryon fields, we introduce a hierarchy between representations with good or bad diquarks (called soft and hard baryon representations, respectively). We use $(3,\bar3)+(\bar3,3)$ and $(8,1)+(1,8)$ as soft to construct a chiral invariant Lagrangian, while the $(3,6)+(6,3)$ representations are assumed to be integrated out, leaving some effective interactions. The mass splitting associated with the strange quark mass is analyzed in the first and second order in the meson fields $M$ in $(3,\bar3)+(\bar3,3)$ representations. We found that the chiral ${\rm SU(3)}_L \otimes {\rm SU(3)}_R$ constraints are far more restrictive than the ${\rm SU(3)}_V$ constraints used in conventional models for baryons. After extensive analyses within $(3,\bar3)+(\bar3,3)$ and $(8,1)+(1,8)$ models, we found that models in the first order of $M$ do not reproduce the mass hierarchy correctly, although the {\GO} is satisfied. In the second order, the masses of the positive parity channels are reproduced well up to the first radial excitations, while some problem in the mass ordering remains in a negative parity channel. Apparently the baryon dynamics is not well-saturated by just $(3,\bar3)+(\bar3,3)$ and $(8,1)+(1,8)$ representations, as indicated by the necessity of terms higher order in $M$.