Matter-coupled Yang-Mills system in Minkowski space. I. Invariant solutions in the presence of scalar fields

We obtain classical solutions to the Minkowskian field equations of a Yang-Mills system in the presence of an isotriplet of massless scalar fields with a quartic self-interaction. The gauge group is SU(2). The conformal properties of the equations allow their mapping on compactified Minkowski space [diffeomorphic to SU(2)×U(1)] upon which SU(2,2) acts globally. The solutions are obtained by using as Ansätze fields invariant under certain subgroups of SU(2,2). The characterization of invariant fields in the context of gauge theories is reviewed and the theory is applied to explicitly construct the invariant Ansätze. The following (or combinations of the following) group actions on SU(2)×U(1) are used: (i) Left $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L$ translations, (ii) right $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R$ translations, (iii) left action of the product $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R$, (iv) U(1) translations. A class of $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$-invariant solutions is found. In addition, all $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_R\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$-invariant solutions (both real complex) are determined. These new solutions represent interaction modes for some of the previously found solutions to the pure Yang-Mills equations. They lead to everywhere regular configurations with finite energy on ordinary Minkowski space.