Dynamics of Abelian Higgs vortices in the near Bogomolny regime
The paper treats a known model of the classical field theory: two- dimensional Maxwell's equations coupled to an equation for a complex scalar Higgs field with the cubic nonlinearity, which lends an effective mass to a ``photon'' in this model via the mechanism called spontaneous symmetry breaking. The model's Hamiltonian is \[ H = {1 \over 2} \int_{\mathbb{R}^ 2} [E_ 1^ 2 + E_ 2^ 2 + B^ 2 + | D_ A \Phi |^ 2 + {1 \over 4} \lambda (1- | \Phi |^ 2)^ 2], \tag{1} \] where the electric and magnetic fields \(E_{1,2}\) and \(B\) are expressed through the components \(A_{1,2}\) and \(A_ 0\) of the electromagnetic vector potential \(E_{1,2} = \partial_ t A_{1,2} - \partial_{1,2} A_ 0\), \(B = \partial_ 1 A_ 2 - \partial_ 2A_ 1\), \(\Phi\) is the Higgs field, \(D_ A = \nabla - iA\) is the gauge- invariant gradient operator, and \(\lambda\) is a real coupling constant. It is known that the model (1) admits a vortex solution with the local amplitude of the Higgs field vanishing at the central point of the vortex and taking an asymptotically constant value, 1, at infinity. It was also known that, at \(\lambda = 1\) (the Bogomolny point), the equations of motion ensuing from the Hamiltonian (1) had a class of remarkable exact static solutions in the form of a system of vortices with their centers constituting an arbitrary set of points on the plane. In this work, the case of small but finite values of \(\lambda-1\) is considered by means of the perturbation theory. It is demonstrated that, in this case, the model's dynamics in an \(N\)-vortex sector may be reduced, on a basis of a well-controlled approximation, to a finite-dimensional dynamical system in the so-called moduli space, i.e., the space of parameters of the \(N\)- vortex configuration modulo the gauge transformations. In the course of the derivation, estimates of the approximation errors are based on using the energy conservation. The main result obtained is that, while the interaction forces between the vortices are absent at \(\lambda = 1\), the vortices, respectively, repel and attract each other at \(\lambda - 1\) positive and negative. Another result is that, at \(\lambda\) close to 1, two colliding vortices undergo the right-angle scattering, which was previously observed in numerical simulations of the model.
scientific article; zbMATH DE number 509891