Note that, when we make the spatial integration, the last term in (43) drops off as it is a total derivative term. For massive pions, the pion profile of the Skyrmion always decays exponentially asymptotically, so the surface integral derived from the integration of this total derivative term always vanish.

When we defined the isospin charge $I_3$ in (34), we have not included the gauged WZW term (35). However, it can be shown that the isospin charge coming from (35) vanishes, as follows. Let us act the isospin transformation in the pion field $U$ in the gauged WZW term (35): $U\to U^{\prime }\equiv GU{G}^{†}$ where $G=\exp iϵ{\tau }_3$. To obtain the current, following the standard Noether’s method, we regard the transformation parameter as a spacetime-dependent variable, $G=G(x)$. Then we find $L\to G{U}^{†}{G}^{†}(\partial G)U{G}^{†}+G{U}^{†}(\partial U)G^{†}+G\partial G^{†}$, and $R\to GU(\partial G^{†})G{U}^{†}{G}^{†}+G(\partial U)U^{†}{G}^{†}+(\partial G)G^{†}$. Using $[G,{\tau }_3]=0$, $(\partial G)G^{†}+G\partial G^{†}=\partial (G{G}^{†})=0$, and $G^{†}\partial G=-\partial G^{†}G=i{\tau }_3\partial ϵ(x)$, we find that $\mathrm{tr}[{\tau }_3(L+R)]$ is invariant under this local transformation. So, there is no additional contribution to the isospin charge from the gauged WZW term. The result is natural, since the gauged WZW term is basically ${\pi }_0\to 2\gamma$ and this ${\pi }_0$ does not carry the isospin charge. We would like to thank the referee for pointing out the effect of the WZW term.

As long as we think of the massless pion as the limit $m_{\pi }\to 0$, the surface term contribution can be always ignored.

It is possible that the last term in (43) gives additional multipoles, although it is negligible for the total charge. However, the term itself is not gauge-invariant and once combined with the baryon number term (the second term in (43)) it becomes gauge-invariant. In this paper we evaluate only the gauge-invariant $j_{\mathrm{anm}}^{\mu }$ for the quadrupole moment, and the other terms (which can be evaluated if a backreaction to the Skyrmion profile can be computed) are left for our future work.