Curious properties of the spherically symmetric Yang-Mills Euclidean solutions

Abstract

The general O(4) spherically symmetric solution to the Yang-Mills equations in 4-dimensional Euclidean space is obtained. It is found that the only finite-action solution is necessarily self-dual. Any other symmetric solution must have an infinite action. Furthermore, the corresponding equation of the motion is found to be the same as the differential equation which gives the static kink solution of the two-dimensional ${\ensuremath{\varphi}}^{4}$ scalar field theory. By exploiting this analogy the patching of isolated pseudoparticle solutions to obtain multipseudoparticle configurations is discussed. Their existence then becomes evident. Finally, the inclusion of a scalar O(4)-symmetric field is considered. It is found that only the pseudoparticle solution has finite action; however, it is unstable. This instability property implies the absence of vacuum tunneling in the symmetric theory, at least in the semiclassical approximation.