Monopole-charge instability

TL;DR

This paper analyzes the stability of monopoles with nonvanishing Higgs potential, showing that stability reduces to a gauge theory problem on the two-sphere and identifying unique stable charges and negative modes.

Contribution

It provides a reduction of the stability problem to a gauge theory on the two-sphere and explicitly constructs stable monopole charges and negative modes.

Findings

Each topological sector has a unique stable monopole charge.

Unstable monopoles have a calculable number of negative modes.

Explicit spectrum and negative modes are constructed on the 2-sphere.

Abstract

For monopoles with nonvanishing Higgs potential it is shown that with respect to "Brandt-Neri-Coleman type" variations (a) the stability problem reduces to that of a pure gauge theory on the two-sphere (b) each topological sector admits one, and only one, stable monopole charge, and (c) each unstable monopole admits $2\sum_{q<0} (2|q|-1)$ negative modes, where the sum goes over all negative eigenvalues $q$ of the non-Abelian charge $Q$. An explicit construction for (i) the unique stable charge (ii) the negative modes and (iii) the spectrum of the Hessian, on the 2-sphere, is then given. The relation to loops in the residual group is explained. The negative modes are tangent to suitable energy-reducing two-spheres. The general theory is illustrated for the little groups U(2), U(3), SU(3)/Z_3 and O(5).