Comments on Charge Distribution in Multipion Production

Abstract

Pais and collaborators determined the expectation value of the number of positive, neutral, and negative pions for $N$-pion states corresponding to a given irreducible representation (IR) of the permutation group $\mathrm{S}\mathrm{}\left(N\right)\mathrm{}$ when the isospin is 0 or 1. In this comment we indicate how the problem can be reduced to the expectation value of the quadrupole moment for states characterized by IR's of the unitary group U(3) and its orthogonal subgroup O(3). The latter expectation value was determined by Elliott for any IR of O(3) that appears only once in the given IR of U(3). As the partitions characterizing the IR of U(3) and $\mathrm{S}\mathrm{}\left(N\right)\mathrm{}$ are the same and the IR of O(3) corresponds to the isospin, a generalization of Pais's results follows from Elliott's analysis.