  Recently, it has been shown that (4+1)-dimensional Yang-Mills theory may be
written as a (3+1)-dimensional BPS Skyrme model, in which the Skyrme field is
coupled to an infinite tower of vector mesons. Truncating this tower to a
single vector meson yields an extension of the standard Skyrme model to a
theory of pions coupled to the rho meson, with the significant simplification
that no additional free parameters are introduced. The present paper is
concerned with this truncated theory and results are presented for Skyrmions
with baryon numbers one to four. The approach involves the use of an extended
version of the Atiyah-Manton construction, in which the Skyrme field is
approximated by the holonomy of a Yang-Mills instanton. It is found that the
coupling to the rho meson significantly reduces Skyrmion binding energies, to
produce an improved comparison with the experimental data on nuclei. A
truncation that includes both a vector and an axial vector meson is also
investigated, providing a model of pions, the rho meson and the a1 meson.
Binding energies are further reduced by the inclusion of this additional meson,
shifting the Skyrmion energies a little closer to those of nuclei. Fixing the
energy unit by equating the energy of the baryon number four Skyrmion to the
He-4 mass, yields masses for all lower baryon numbers that are within 20 MeV of
the experimental values, which is an error that is four times smaller than in
the standard Skyrme model.
