Stability analysis of superconducting electroweak vortices

We carry out a detailed stability analysis of the superconducting vortex solutions in the Weinberg-Salam theory described in [J. Garaud, M.S. Volkov, Nucl. Phys. B 826 (2010) 174]. These vortices are characterized by constant electric current I and electric charge density I0, for I→0 they reduce to Z strings. We consider the generic field fluctuations around the vortex and apply the functional Jacobi criterion to detect the negative modes in the fluctuation operator spectrum. We find such modes and determine their dispersion relation, they turn out to be of two different types, according to their spatial behavior. There are non-periodic in space negative modes, which can contribute to the instability of infinitely long vortices, but they can be eliminated by imposing the periodic boundary conditions along the vortex. There are also periodic negative modes, but their wavelength is always larger than a certain minimal value, so that they cannot be accommodated by the short vortex segments. However, even for the latter there remains one negative mode responsible for the homogeneous expansion instability. This mode may probably be eliminated when the vortex segment is bent into a loop. This suggests that small vortex loops balanced against contraction by the centrifugal force could perhaps be stable.