Skyrmion semiclassical quantization in the presence of an isospin chemical potential

The semiclassical description of Skyrmions at small isospin chemical potential μI 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 (μI=0), together with the 'natural' assumption μI=O(Nc0), the proton and neutron masses are nonlinear in μI in the regime |μI|<mπ. 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 μI the energy excitations above the vacuum are linear in μI. The resolution of this paradox is achieved by studying the realization of the large Nc limit of QCD in the Skyrme model at finite μI. 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 Nc. 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∼Nc.