\frac{\hbar}{2m_nr}\hat \theta .\end{aligned}$$ The divergence of the superfluid velocity when $r\to 0$ is avoided by introducing cut-off on scales of the order of the coherence length. The cut-off scale, for Fermi-liquids, can be understood by noting that an increase of ${\bf v}_n$ when $r\to 0$ causes an increase of the kinetic energy of Cooper pairs which eventually becomes larger than the binding energy of a pair. The broken pairs will perform a rigid-body rotation with an angular velocity which scales as $Cr$ and is regular in the $r\to 0$ limit (here $C$ is a constant). This crossover can be seen from the well-known solutions of the Ginzburg-Landau equation for the amplitude $f(r)$: $$\begin{aligned}
\label{eq:AS:nondim}
\frac{d^2f}{d\zeta^2} + \frac{1}{\zeta}\frac{df}{d\zeta} - 
\frac{1}{\zeta^2}f + f -f^3 = 0,\end{aligned}$$ where $\zeta = r/\xi_n$ and $\xi_n$ is the size of the vortex core. The asymptotic solutions of Eq. (\[eq:AS:nondim\]) are $$\begin{aligned}
f(\zeta)= \left\{
\begin{array}{cc} C\zeta & \zeta \ll 1,\\
1 - (2\zeta^2)^{-1} & \zeta \to \infty ,\\
\end{array} \right.\end{aligned}$$ while numerical solutions for intermediate values of $\zeta$ show that the condensate wave function is at half of its value in a homogeneous condensate when $\zeta = 1$. Note the long-range nature of the superfluid vortex velocity, 