  We study the stability properties of the twisted vortex solutions in the
semilocal Abelian Higgs model with a global $\mathbf{SU}(2)$ invariance. This
model can be viewed as the Weinberg-Salam theory in the limit where the
non-Abelian gauge field decouples, or as a two component Ginzburg-Landau
theory. The twisted vortices are characterized by a constant global current
${\cal I}$, and for ${\cal I}\to 0$ they reduce to the semilocal strings, that
is to the Abrikosov-Nielsen-Olesen vortices embedded into the semilocal model.
Solutions with ${\cal I}\neq 0$ are more complex and, in particular, they are
{\it less energetic} than the semilocal strings, which makes one hope that they
could have better stability properties. We consider the generic field
fluctuations around the twisted vortex within the linear perturbation theory
and apply the Jacobi criterion to test the existence of the negative modes in
the spectrum of the fluctuation operator. We find that twisted vortices do not
have the homogeneous instability known for the semilocal strings, neither do
they have inhomogeneous instabilities whose wavelength is less than a certain
critical value. This implies that short enough vortex pieces are perturbatively
stable and suggests that small vortex loops could perhaps be stable as well.
For longer wavelength perturbations there is exactly one negative mode in the
spectrum whose growth entails a segmentation of the uniform vortex into a
non-uniform, `sausage like' structure. This instability is qualitatively
similar to the hydrodynamical Plateau-Rayleigh instability of a water jet or to
the Gregory-Laflamme instability of black strings in the theory of gravity in
higher dimensions.

High Energy Physics - Theory;Superconductivity;High Energy Physics - Phenomenology