What becomes of semilocal non-Abelian strings in $\mathcal{N}=1$ supersymmetric QCD

We study non-Abelian vortex strings in $\mathcal{N}=2$ supersymmetric QCD with the gauge group $\mathrm{U}(N)$ deformed by the mass $\mu$ of the adjoint matter. This deformation breaks $\mathcal{N}=2$ supersymmetry down to $\mathcal{N}=1$ and in the limit of large $\mu$ the theory flows to $\mathcal{N}=1$ QCD. Non-Abelian strings in addition to translational zero modes have orientation moduli. In the $\mathcal{N}=2$ limit of small $\mu$ the dynamics of orientational moduli is described by the two-dimensional $CP(N-1)$ model for QCD with $N_f=N$ flavors of quark hypermultiplets. For the case of $N_f>N$ the non-Abelian string becomes semilocal developing additional size moduli that modify the effective two-dimensional $\sigma$-model on the string making its target space noncompact. In this paper we consider the $\mu$-deformed theory with $N_f>N$ eventually making $\mu$ large. We show that size moduli develop a potential that forces the string transverse size to shrink. Eventually in the large $\mu$ limit size moduli decouple and the effective theory on the string reduces to the $CP(N-1)$ model. We also comment on physics of confined monopoles.