We discuss the statistical properties of a single vortex line in a perfect fluid. The partition function is calculated up to the end in the thin vortex approximation. It turns out that corresponding theory is renormalizable, and the renormalization law for the core size of the vortex is found. The direction of renormalization group flow makes the thin vortex approximation to be valid for the interest cases and this result does not depend on the choice of infrared regularization. The expressions for some gauge-invariant correlators are obtained to demonstrate the developed formalism at work.