Center symmetry and area laws

TL;DR

This paper explores how the invariance under center subgroup transformations in SU(N_c) gauge theories influences the area law behavior of Wilson loops, providing a formula for string tensions across representations.

Contribution

It derives a general formula for string tensions in SU(N_c) gauge theories with matter fields, linking center symmetry invariance to confinement criteria.

Findings

String tensions depend on the center subgroup p and representation n-ality.

Invariance under a nontrivial center subgroup is necessary for area law behavior.

Results hold regardless of the number of colors N_c.

Abstract

SU($N_c$) gauge theories containing matter fields may be invariant under transformations of some subgroup of the $\mathbb{Z}_{N_c}$ center; the maximum such subgroup is $\mathbb{Z}_{p}$, with $p$ depending on $N_c$ and the representations of the various matter fields in the theory. Confining SU($N_c$) gauge theories in either 3+1 or 2+1 space-time dimensions and with matter fields in any representation have string tensions for representation $R$ given by $\sigma_R =\sigma_f \, \, \frac{p_R (p-p_R) \, \, g\left (p_R (p-p_R) \right )}{(p-1) \, \, g(p -1 )} $ with $p_R={n_R \, \rm mod}(p)$, where $\sigma_f $ is the string tension for the fundamental representation, $g$ is a positive finite function and $n_R$ is the n-ality of $R$. This implies that a necessary condition for a theory in this class to have an area law is invariance of the theory under a nontrivial subgroup of the center. Significantly, these results depend on $p$ regardless of the value of $N_c$.