# Casimir Scaling vs. Abelian Dominance in QCD String Formation

## TL;DR

This study challenges the abelian dominance hypothesis in QCD string formation by showing it fails for higher representations and that abelian projected configurations do not reproduce the expected Casimir scaling of string tensions.

## Contribution

The paper provides numerical evidence that abelian dominance does not hold for higher group representations in lattice QCD and demonstrates the limitations of abelian effective theories in describing confinement.

## Key findings

- String tensions from abelian projected loops do not follow Casimir scaling.
- String tensions are the same for fundamental and j=3/2 representations in abelian projections.
- In the Higgs phase, string tensions become equal or vanish depending on the representation.

## Abstract

We show that the hypothesis of abelian dominance in 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 $D=3$ 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 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 $j=3/2$ representations, and vanish for the adjoint representation. In addition, we perform a lattice Monte Carlo simulation of the Georgi-Glashow model in $D=3$ dimensions. We find that the representation-dependence of string tensions is that of pure Yang-Mills in the symmetric phase, but changes abruptly to equal tensions for the $j=1/2,~3/2$ representations, and zero tension for $j=1$, at the transition to the Higgs phase. Our results indicate that an effective abelian theory at the confinement scale, invoking {\it 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.