Casimir scaling versus Abelian dominance in QCD string formation

We show that the hypothesis of Abelian dominance in the maximal Abelian gauge, which is known to work for Wilson loops in the fundamental representation, fails for Wilson loops in higher group representations. Monte Carlo simulations are performed on lattice SU(2) gauge theory, in $D=3\mathrm{}$ dimensions, in the maximal Abelian gauge, in the confined phase. It is well known that Creutz ratios extracted from loops in various group representations are proportional to the quadratic Casimir invariant of each representation, in a distance interval from the confinement scale to the point where color screening sets in. In contrast, we find numerically, in the same interval, that string tensions extracted from loops built from Abelian projected configurations are the same for the fundamental and $j=\frac{3}{2}$ representations, and vanish for the adjoint representation. In addition, we perform a lattice Monte Carlo simulation of the Georgi-Glashow model in $D=3\mathrm{}$ dimensions. We find that the representation dependence of string tensions is that of pure Yang-Mills theory in the symmetric phase, but changes abruptly to equal tensions for the $j=\frac{1}{2}, \frac{3}{2}$ representations, and zero tension for $j=1\mathrm{}$, at the transition to the Higgs phase. Our results indicate that an effective Abelian theory at the confinement scale, invoking only degrees of freedom (monopoles and photons) associated with a particular Cartan subalgebra, is inadequate to describe the actual interquark potential in an unbroken non-Abelian gauge theory.