Asymptotically Non-Singular Extended Non-Dyonic Solutions of 't Hooft-Polyakov Monopole Violates Equations of Motion

TL;DR

The paper demonstrates that non-singular, extended non-Dyonic solutions of the 't Hooft-Polyakov monopole cannot exist outside the core without magnetic charge shielding, implying the inevitability of the Dirac string in such solutions.

Contribution

It proves that asymptotically non-singular non-Dyonic solutions violate equations of motion unless magnetic charge is shielded, clarifying the nature of static and Dyonic solutions.

Findings

Non-singular non-Dyonic solutions are forbidden outside the monopole core.

Dirac string is unavoidable for all admissible non-Dyonic solutions.

All admissible static solutions may be gauge transformed into Dyonic solutions.

Abstract

We show that based on the general solution, given by Corrigan, Olive, Fairlie and Nuyts, in the region outside the monopole's core; the equations of motion in the Higgs vacuum (i.e. outside the monopole's core) will not allow asymptotically non-singular extended non-trivial non-Dyonic (including, also, all static) solutions of the 't Hooft-Polyakov monopole. In other words, unless the monopole's magnetic charge is shielded (by some mechanism), the Dirac string is inevitable asymptotically, in the region outside the monopole's core, for all non-Dyonic solutions that are admissible by the equations of motion. That we show that the non-dyonic solutions (based on Corrigan et al) will include all "admissible" static solutions and their gauge transform might be interpreted as that all admissible dyonic solutions (based on Corrigan et al) are composite solutions.