  Color symmetry is spontaneously broken in quark matter at high density as a
consequence of di-quark condensations with exhibiting color superconductivity.
Non-Abelian vortices or color magnetic flux tubes stably exist in the
color-flavor locked phase at asymptotically high-density. The effective
worldsheet theory of a single non-Abelian vortex was previously calculated in
the singular gauge to obtain the ${\mathbb C}P^2$ model [ arXiv:0908.4470 ,
arXiv:0912.1352 ]. Here, we reconstruct the effective theory in a regular gauge
without taking a singular gauge, confirming the previous results in the
singular gauge. As a byproduct of our analysis, we find that non-Abelian
vortices in high-density QCD do not suffer from any obstruction for the global
definition of a symmetry breaking.
