Small winding-number expansion: vortex solutions at critical coupling

We study an axially symmetric solution of a vortex in the Abelian-Higgs model at critical coupling in detail. Here we propose a new idea for a perturbative expansion of a solution, where the winding number of a vortex is naturally extended to be a real number and the solution is expanded with respect to it around its origin. We test this idea on three typical constants contained in the solution and confirm that this expansion works well with the help of the Fade approximation. For instance, we analytically reproduce the value of the scalar charge of the vortex with an error of O(10(-6)). This expansion is also powerful even for large winding numbers.