 Solution

At critical coupling, an $N$-vortex solution of the Abelian Higgs model satisfies the Bogomolny vortex equations. The Abelian Higgs vortex moduli space ${\mathcal{M}}_N$ consists of these solutions modulo gauge transformations. The unordered zeros of the Higgs field specify the $N$ vortex positions, soas a complex manifold. A tangent vector is represented by a linearized solution orthogonal to infinitesimal gauge transformations. Substitution into the field-theory kinetic energy giveswhich defines the natural $L^2$ Riemannian metric on the moduli space. Its geodesics give the slow-motion approximation.

For $N=1$, translation invariance makes ${\mathcal{M}}_1$ a flat plane. For $N=2$, the center-of-mass plane factors from a rotationally symmetric relative moduli space. Far from coincidence, exchanging the vortices identifies the relative separation $z$ with $-z$, so the relative space is asymptotic to a cone of angle $\pi$. The exact metric smoothly rounds its apparent tip at coincident vortices; the regular local coordinate is proportional to $z^2$.

A head-on relative geodesic passes smoothly through this coincidence point. In terms of the asymptotic separation coordinate, it emerges on the perpendicular axis, producing Right-angle scattering of Abelian Higgs vortices. For signed impact parameter $b$, the deflection is odd in $b$, approaches zero as $|b|\to \infty$, and approaches the two signed values $\pm \pi /2$ as $b\to 0^{\pm }$. Within the geodesic approximation the trajectory, and hence this graph, is independent of incoming speed; speed only changes its parametrization in time.

For identical point particles in a bounded repulsive central potential, the relative coordinate moves in an ordinary plane rather than the rounded exchange cone. At energy below the finite central barrier, a head-on encounter is reflected and gives backscattering through $\pi$, so the available deflections range from zero to nearly $\pi$. At energy above the barrier, a head-on trajectory passes through the center and has zero deflection; the magnitude reaches a maximum at nonzero $|b|$ and then returns to zero. Increasing speed reduces this maximum. Thus point particles can exhibit backscattering at low speed and near-transparent head-on passage at high speed, whereas slow identical vortices have the speed-independent right-angle limit forced by the geometry of ${\mathcal{M}}_2$.