Vortex renormalization in three space dimensions

A three-dimensional heuristic renormalization-group theory for vortex filaments in three space dimensions is developed. It is based on a low-fugacity assumption and follows a methodology inspired by the Kosterlitz-Thouless analysis in two dimensions and its extension to three dimensions by Williams and Shenoy. The results agree with recent numerical and theoretical analyses of sparse vortex systems and, in a certain inconsistent simplification, reproduce the Williams smooth vortex theory and a form of its ‘‘fractal’’ extension by Shenoy. It is also shown that this theory is incomplete in its present form, pending an accounting for the dense vortex systems that arise at the transition. Nevertheless, the theory provides a check on other work and constitutes a potentially useful computational tool.