Towers of positive parity excited baryons and their mixing in the large $N_c$ limit

We consider configuration mixing for the nonstrange positive parity excited baryons in the $[{\mathbf{56}}^{\prime },0^{+}]$, $[\mathbf{56},2^{+}]$, $[\mathbf{70},0^{+}]$, and $[\mathbf{70},2^{+}]$ quark model $SU(6)\times O(3)$ multiplets contained in the $N=2$ band. Starting from the effective mass operator for these states, we show by an explicit calculation that in the large $N_c$ limit they fall into six towers of degenerate states labeled by $K=0,1,1^{\prime },2,2^{\prime },3$. We find that the mixing of the spin-flavor states is much simpler than what is naively expected in the quark model. To leading order in $N_c$, only states carrying the same $K$ label can mix, which implies that, for the spin-flavor states we started with, configuration mixing can be parameterized by just two constants: ${\mu }_{K=1}$ and ${\mu }_{K=2}$. The obtained mass degeneracies and mixing pattern constitute a signature of the contracted spin-flavor symmetry for baryons in the large $N_c$ limit.