On Skyrmion semiclassical quantization in the presence of an isospin chemical potential

Abstract

The semiclassical description of Skyrmions at small isospin chemical potential $μ_I$ is carefully analyzed. We show that when the calculation of the energy of a nucleon is performed using the straightforward generalization of the vacuum sector techniques ($μ_I=0$), together with the "natural" assumption $μ_I = {\cal O} (N_c^0)$, the proton and neutron masses are nonlinear in $μ_I$ in the regime $|μ_I| < m_π$. Although these nonlinearities turn out to be numerically quite small, such a result fails to strictly agree with the very robust prediction that for those values of $\mui$ the energy excitations above the vacuum are linear in $μ_I$. The resolution of this paradox is achieved by studying the realization of the large $N_c$ limit of $QCD$ in the Skyrme model at finite $\mui$. This is done in a simplified context devoid of the technical complications present in the Skyrme model but which fully displays the general scaling behavior with $N_c$. The analysis shows that the paradoxical result appears as a symptom of using the semi-classical approach beyond its regime of validity and that, at a formal level, the standard methods for dealing with the Skyrme model are only strictly justified for states of high isospin $I \sim N_c$.