  In the context of scalar-tensor theories, the inclusion of new degrees of
freedom coupled non-minimally to the gravitational sector might produce some
appealing effects on the cosmic expansion history. We investigate this premise
by including a canonical SU(2) Yang- Mills field to the total energy budget of
the universe coupled to the standard quintessential field by a disformal
transformation. From the dynamical system analysis, we study three cases of
cosmological interest that span most of the physical phase space of the model:
the uncoupled limit, the isotropic, and the Abelian cases. New scaling
solutions with a non-vanishing gauge field are found in all cases which can be
interesting for early cosmological scenarios. Some of these scaling solutions
even exhibit anisotropic features. Also, the background evolution of the
universe is studied by means of numerical analysis. As an interesting result,
the disformal coupling changes the equation of state of the gauge field from
radiation to matter at some stages of the evolution of the universe, thereby
the gauge field can contribute to some fraction of the total dark matter. We
have also quantified the redshift-dependent contribution of the gauge field in
the form of dark radiation during the radiation era to the effective number of
relativistic species. This depends essentially on the initial conditions and,
more importantly, on the disformal coupling function.
