Group Theoretical Analysis of the Wave Function of the $[{\bf 70},1^-]$ Nonstrange Baryons in the $1/N_c$ Expansion

TL;DR

This paper constructs an exact wave function for a specific baryon multiplet using group theory, compares it with the traditional asymmetric wave function, and evaluates the implications for the $1/N_c$ expansion in baryon spectroscopy.

Contribution

It provides a precise, symmetric wave function for the $[70,1^-]$ multiplet and compares two methods of analysis, highlighting the advantages of treating the system as a whole.

Findings

The symmetric wave function differs from the traditional one in matrix element calculations.

The whole-system approach simplifies the analysis and clarifies important operators.

Derived isoscalar factors of the permutation group as a function of $N_c$.

Abstract

Using standard group theoretical techniques we construct the exact wave function of the $[{\bf 70},1^-]$ multiplet in the orbital, spin and flavor space. This symmetric wave function is compared to that customarily used in the $1/N_c$ expansion, which is asymmetric. The comparison is made by analyzing the matrix elements of various operators entering the mass formula. These matrix elements are calculated by the help of isoscalar factors of the permutation group, specially derived for this purpose as a function of $N_c$. We also compare two distinct methods used in the study of the $[{\bf 70},1^-]$ multiplet. In the first method the generators are divided into two parts, one part acting on a subsystem of $N_c-1$ quarks called core and another on the separated quark. In the second method the system is treated as a whole. We show that the latter is simpler and allows to clearly reveal the physically important operators in the mass formula.