Skyrmion semiclassical quantization in the presence of an isospin chemical potential

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The semiclassical description of Skyrmions at small isospin chemical potential ${\ensuremath{\mu}}_{I}$ 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 (${\ensuremath{\mu}}_{I}=0$), together with the ``natural'' assumption ${\ensuremath{\mu}}_{I}=\mathcal{O}({N}_{c}^{0})$, the proton and neutron masses are nonlinear in ${\ensuremath{\mu}}_{I}$ in the regime $|{\ensuremath{\mu}}_{I}|<{m}_{\ensuremath{\pi}}$. 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 ${\ensuremath{\mu}}_{I}$ the energy excitations above the vacuum are linear in ${\ensuremath{\mu}}_{I}$. 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 ${\ensuremath{\mu}}_{I}$. 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 semiclassical 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, $I\ensuremath{\sim}{N}_{c}$.