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

Abstract

We consider configuration mixing for the nonstrange positive parity excited baryons in the $[\mathbf{56'},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 $K=0,1,1',2,2',3$. We find that the mixing of the quark model states is much simpler than what is naively expected. 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 $\mu_{K=2}$.