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The semiclassical description of Skyrmions at small isospin chemicalpotential $\mu_I$ is carefully analyzed. We show that when the calculation ofthe energy of a nucleon is performed using the straightforward generalizationof the vacuum sector techniques ($\mu_I=0$), together with the "natural"assumption $\mu_I = {\cal O} (N_c^0)$, the proton and neutron masses arenonlinear in $\mu_I$ in the regime $|\mu_I| < m_\pi$. Although thesenonlinearities turn out to be numerically quite small, such a result fails tostrictly agree with the very robust prediction that for those values of $\mui$the energy excitations above the vacuum are linear in $\mu_I$. The resolutionof 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 asimplified context devoid of the technical complications present in the Skyrmemodel but which fully displays the general scaling behavior with $N_c$. Theanalysis shows that the paradoxical result appears as a symptom of using thesemi-classical approach beyond its regime of validity and that, at a formallevel, the standard methods for dealing with the Skyrme model are only strictlyjustified for states of high isospin $I \sim N_c$.

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