Abstract In this paper we present a rigorous mathematical analysis for Abrikosov–Nielsen–Olesen (ANO) string solutions of the Abelian–Higgs (AH) model with a Coleman–Weinberg (CW) potential. While previous studies have primarily focused on ANO vortices with the conventional quadratic–quartic potential, we investigate the case where spontaneous symmetry breaking is induced by the logarithmic CW potential, which arises naturally in quantum field theories with classical scale invariance. Using a direct minimization approach, we establish the existence of smooth, finite-energy vortex solutions satisfying appropriate boundary conditions. We derive detailed asymptotic properties of these solutions, including exponential decay at infinity and power-law behaviour near the origin. Furthermore, we prove several integral relations, including an exact partition between electromagnetic and potential energy components. Our results show that the CW-ANO strings share many qualitative features with their conventional counterparts while exhibiting distinct quantitative characteristics owing to the logarithmic nonlinearity. This analysis provides a rigorous foundation for recent numerical investigations of CW-ANO strings and opens new avenues for studying topological defects in scale-invariant field theories.
Han et al. (2026) studied this question.