Short range intervortex forces

An explicit formula for the interaction energy of $n$ vortices in the abelian Higgs (or Ginzburg-Landau) model is derived, valid in the regime where all vortices are close to one another. An immediate consequence of this formula is that the interaction energy of a vortex pair with separation $d$ varies as $d^4$, not $d^2$. The formula contains $n-1$ real coefficients which are fixed by certain spectral data of the Jacobi operator of the cocentred $n$-vortex. The coefficients are computed numerically for $n=2$ and $n=3$ for couplings $0.1\le \lambda \le 2.5$. The resulting short range interaction potentials are compared with the results of full field theory simulations for $\lambda =0.5$ and $\lambda =2$, with excellent agreement at small to moderate vortex separation.