Attractive Interaction between Vortex and Anti-vortex in Holographic Superfluid

Annihilation process of a pair of vortices in holographic superfluid is numerically simulated. The process is found to consist of two stages which are amazingly separated by vortex size $2r$. The separation distance $\delta (t)$ between vortex and anti-vortex as a function of time is well fitted by $\alpha (t_{0}-t)^{n}$,wherethescalingexponent$n=1/2$for$\delta (t)>2r$,and$n=2/5$for$\delta(t)<2r$. Then the approaching velocity and acceleration as functions of time and as functions of separation distance are obtained. Thus the attractive force between vortex and anti-vortex is derived as $f(\delta )\propto 1\mathrm{/}{\delta }^3$ for the first stage, and $f(\delta)\propto 1/\delta^{4}$ for the second stage. In the end, we explained why the annihilation rate of vortices in turbulent superfluid system obeys the two-body decay law when the vortex density is low.