Fermion structure of non-Abelian vortices in high density QCD

Abstract

We study the internal structure of a non-Abelian vortex in color superconductivity with respect to quark degrees of freedom. Stable non-Abelian vortices appear in the color-flavor-locked phase whose symmetry $SU(3{)}_{\mathrm{c}+\mathrm{L}+\mathrm{R}}$ is further broken to $SU(2{)}_{\mathrm{c}+\mathrm{L}+\mathrm{R}}\ensuremath{\bigotimes}U(1{)}_{\mathrm{c}+\mathrm{L}+\mathrm{R}}$ at the vortex cores. Microscopic structure of vortices at scales shorter than the coherence length can be analyzed by the Bogoliubov-de Gennes equation (rather than the Ginzburg-Landau equation). We obtain quark spectra from the Bogoliubov-de Gennes equation by treating the diquark gap having the vortex configuration as a background field. We find that there are massless modes (zero modes) well-localized around a vortex, in the triplet and singlet states of the unbroken symmetry $SU(2{)}_{\mathrm{c}+\mathrm{L}+\mathrm{R}}\ensuremath{\bigotimes}U(1{)}_{\mathrm{c}+\mathrm{L}+\mathrm{R}}$. The velocities ${v}_{i}$ of the massless modes ($i=t$, $s$ for triplet and singlet) change at finite chemical potential $\ensuremath{\mu}\ensuremath{\ne}0$, and decrease as $\ensuremath{\mu}$ becomes large. Therefore, low energy excitations in the vicinity of the vortices are effectively described by $1+1$ dimensional massless fermions whose velocities are reduced ${v}_{i}<1$.