$k$-strings with exact Casimir law and Abelian-like profiles

We explore vortex solutions for a class of dual $SU(N)$ Yang-Mills models with $N^2-1$ Higgs fields in the adjoint representation. Initially, we show that there is a collective behavior that can be expressed in terms of a small $N$-independent number of field profiles. Then, we find a region in parameter space where the nontrivial profiles coincide with those of the Nielsen-Olesen vortex, and the energy scales exactly with the quadratic Casimir. Out of this region, we solve the ansatz equations numerically and find very small deviations from the Casimir law. The coexistence of Abelian-like string profiles and non-Abelian scaling features is welcome, as these properties have been approximately observed in pure Yang-Mills lattice simulations.

Figure 1

Profiles $a(\rho )$ and $h(\rho )$ for various ${\overline{\mu }}^2$. The profile $a$ is that with a linear behavior around $\rho =0$.

Figure 2

Profile $h_1(\rho )$ for various ${\overline{\mu }}^2$.

Figure 3

Plot of ${\mathrm{\Delta }}_C(k)$ with $N=8$. Notice the region depicted is that where the SSB takes place, including positive ${\overline{\mu }}^2$.

Figure 4

Plot of ${\mathrm{\Delta }}_S(k)$ with $N=8$. The deviations are much larger in the whole region explored.