From BPS geodesics to mode-driven dynamics in the scattering of multiple BPS vortices

Abstract

We analyze how the geodesic motion in the 3- and 4-vortex sectors of the Abelian-Higgs model at critical coupling is deformed by the excitation of a massive bound mode. We find that the geodesics corresponding to BPS solutions with enhanced symmetry remain unchanged, although the direction of the actual motion depends on the mode-generated force, i.e., a force arising from the change of the mode frequency along the geodesic. In a generic case, for example in head-on collisions between the axially symmetric 1- and 2-vortex or between two 2-vortices, the vortex trajectories can differ strongly from the BPS geodesic. This enhances the chaotic behavior in the formation of the final state.

Figures (14)

• Figure 1: This figure illustrates the definition for the values $a$ and $b$ for the four different quadrants in the $a$-$b$-plane.
• Figure 2: This plot shows the moduli space with the geodesics corresponding to the equilateral triangle (blue), collinear (orange), and 2+1 collisions (green). The labeled dots correspond to the time frames in Figure \ref{['fig:energy_density_1+2']}. The 2+1 collision was obtained from a numerical simulation with initial vortex velocities $v_2 = 0.01$ for the 2-vortex and $v_1 = -0.02$ for the 1-vortex, respectively.
• Figure 3: This figure displays five time frames of the energy density during the scattering of a $2$-vortex with a $1$-vortex. The animated version of the time evolution can be found in the following video: https://youtu.be/o-0fQn-0wlE
• Figure 4: This figure shows the flow of the spectrum along the BPS geodesic path of the 2+1 collision. The r.h.s. of the frequency axis corresponds to the part of the geodesic with $a>0$. The l.h.s. of the frequency axis corresponds to the part of the geodesic with $a<0$. The light-gray lines show the measured spectral flow obtained from the numerical simulation.
• Figure 5: This figure shows the trajectories of the $2+1$ scattering with the $\eta_1$ mode excited. The black curve illustrates the BPS geodesic. The colored trajectories represent cases in which the 2-vortex was initially excited with its lowest mode. The blue line corresponds to the case used to measure the spectral flow in the simulation. All the trajectories start from the positive $a$-direction ($b=0$, $a=7$).