Analytic solution of the approach of quantum vortices towards reconnection

Experimental and simulational studies of the dynamics of vortex reconnections in quantum fluids showed that the distance d between the reconnecting vortices is close to a universal time dependence d=D[κ|t(0)-t|](α) with α fluctuating around 1/2 and κ=h/m is the quantum of circulation. Dimensional analysis, based on the assumption that the quantum of circulation κ=h/m is the only relevant parameter in the problem, predicts α=1/2. The theoretical calculation of the dimensionless coefficient D in this formula remained an open problem. In this Letter we present an analytic calculation of D in terms of the given geometry of the reconnecting vortices. We start from the numerically observed generic geometry on the way to vortex reconnection and demonstrate that the dynamics is well described by a self-similar analytic solution which provides the wanted information.