Phase Structure of a Quantized Chiral Soliton on S^3

Abstract

A quantization of a breathing motion of a rotating chiral soliton on $S^3$ is performed in terms of a family of trial functions for a profile function of the hegdehog ansatz. We determine eigenenergies of the quantized $S^3$ skyrmion by solving the Schr\"odinger equation of the breathing mode for several lower spin and isospin states varying the Skyrme term constants $e$. When $S^3$ radius is smaller than $2/ef_\pi$, where $f_\pi$ is the pion decay constant, we always obtain a conformal map solution as the lowest eigenenergy state. In the conformal map case, allowed states have only symmetric or anti-symmetric wave function under inversion of a dynamical variable describing the breathing mode. As the $S^3$ radius increases the energy splitting between the symmetric and anti-symmetric states rapidly decreases and two states become completely degenerate state. When the $S^3$ radius larger than $3/ef_\pi$, for the small Skyrme term constant $e$ the lowest eigenenergy states are obtained with the profile function given by an arccosine form which is almost the same to those of usual $R^3$ skyrmion. When the effects of the Skyrme term are weak, i.e. large $e$, the lowest energy states are obtained by the profile function of conformal map, which correspond to the \lc\lc frozen states" for the $R^3$ skyrmion as the limit of $S^3$ radius $ \to \infty$.