Confinement in QCD is due to a condensate of thick vortices with fluxes in the center of the gauge group (center vortices), as proposed long ago by the author and others. It is well-known that such vortices lead to an area law for fundamental-representation Wilson loops, but what happens for screened (e.g., adjoint) Wilson loops has been less clear, and problems have arisen over the large-$N$ limit. We study the adjoint and fundamental Wilson loops for gauge group $\mathrm{SU}\left(N\right)\mathrm{}$ with general $N$, where there are $N-1$ distinct vortices, whose properties (including collective coordinates and actions) we discuss. In $d=2\mathrm{}$ we construct a center-vortex model by hand so that it has a smooth large-$N$ limit of fundamental-representation Wilson loops and find, as expected, confinement. Extending an earlier work by the author, we construct the adjoint Wilson-loop potential in a related model for all $N$, as an expansion in powers of $\rho {/M}^2$, where $\rho$ is the vortex density per unit area and $M$ is the gauge-boson mass (inverse vortex size) and find, as expected, screening. (This is, in fact, unexpected in $d=2\mathrm{}$ QCD.) The leading term of the adjoint potential shows a roughly linear regime followed by string breaking when the potential energy is about $2M$. This leading potential is a universal $(N$-independent at fixed $K_F)$ function of the type ${(K}_F/M\left)U\right(\mathrm{MR})\mathrm{}$, where $R$ is the spacelike dimension of a rectangular adjoint Wilson loop and $K_F$ is the fundamental string tension. The linear-regime slope is not necessarily related to $K_F$ by Casimir eigenvalue ratios. We show that in $d=2\mathrm{}$ the dilute vortex model is essentially equivalent to true $d=2\mathrm{}$ QCD in the fundamental representation, but that this is not so for the adjoint representations; arguments to the contrary are based on illegal cumulant expansions which fail to represent the necessary periodicity of the Wilson loop in the vortex flux. Most or all of these arguments are expected to hold for $d=3,4\mathrm{}$ as well, but we cannot calculate explicitly in these dimensions (a proposal is made for another sort of approximation in $d=3\mathrm{}$, using earlier work where $d=3\mathrm{}$ vortices are mapped onto a scalar field theory).

• Received 6 January 1998

DOI:https://doi.org/10.1103/PhysRevD.57.7589