Finite $N_c$ Results for $F/D$ Ratios of the Baryon Vertices and $I=J$ Rule

Abstract

We calculate the $F/D$ ratios of spin-nonflip baryon vertex for an arbitrary number of color degrees of freedom $N_c$ both in the non-relativistic quark model with the $SU(6)$ spin-flavor symmetry and in the chiral soliton model with $SU(3)$ flavor symmetry. We find that the spin-nonflip $F/D$ ratio tends to $-1$ in the limit of $N_c \to \infty$. We show that this leading value $F/D= -1$ of spin-nonflip baryon vertex in the $1/N_c$ expansion corresponds to the isoscalar dominance while the well known leading value $F/D=1/3$ of the spin-flip vertex corresponds to the isovector dominance. We discuss origins of the dominance of isovector in spin-flip and isoscalar in spin-nonflip baryon vertices, referred to as the $I=J$ rule. \par In terms of the matrix elements of the operator which transform as the generator $\lambda^8$ of the $SU(3)$ symmetry we derive the model independent isoscalar formula for baryon vertices and apply this to the mass formula and the isoscalar part of the baryon magnetic moments. The same Okubo-Gell-Mann mass relation and its refined relation among the octet baryons as the one for the case $N_c=3$ is derived model independently for arbitrary color degrees of freedom $N_c$. Contrary to $\lambda^8$,