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

TL;DR

This paper investigates the semiclassical quantization of Skyrmions with small isospin chemical potential, revealing nonlinear mass corrections and clarifying the limits of the semiclassical approach in large N_c QCD.

Contribution

It demonstrates that standard semiclassical methods are only valid for states with large isospin, resolving a paradox related to the linearity of energy in chemical potential.

Findings

Nonlinear mass corrections are small but present at small isospin chemical potential.

The paradox of nonlinear versus linear energy dependence is resolved.

Semiclassical methods are justified only for high isospin states.

Abstract

The semiclassical description of Skyrmions at small isospin chemical potential $\mu_I$ 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 ($\mu_I=0$), together with the "natural" assumption $\mu_I = {\cal O} (N_c^0)$, the proton and neutron masses are nonlinear in $\mu_I$ in the regime $|\mu_I| < m_\pi$. 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 $\mui$ the energy excitations above the vacuum are linear in $\mu_I$. 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 $\mui$. 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 $I \sim N_c$.