$ F_4$ , $E_6$ and $G_2$ Exceptional Gauge Groups in Vacuum Domain Structure Model

Abstract

Using vacuum domain structure model, trivial static potentials in various representations of $F_4$, $E_6$ and $G_2$ exceptional groups are calculated by means of the unit center element. Due to the absence of the non-trivial center elements, the potential of every representation is screened at far distances. However, the linear part is observed at intermediate quark separations which is investigated by the decomposition of the exceptional group to its maximal subgroups. Comparing the group factor of the super-group with the corresponding one obtained from the non-trivial center elements of $SU(3)$ subgroup, shows that $SU(3)$ is not the direct cause of temporary confinement in any of the exceptional groups. However, the trivial potential obtained from the group decomposition to the $SU(3)$ subgroup is the same as the potential of the super-group itself. In addition, any regular or singular decomposition to the $SU(2)$ subgroup which produces the Cartan generator with the same elements as $h_1$, in any exceptional group, leads to the linear intermediate potential of the exceptional gauge groups. The other $SU(2)$ decompositions with the Cartan generator different from $h_1$, are still able to describe the linear potential if the number of $SU(2)$ non-trivial center element which emerge in the decompositions is the same. As a result, it is the center vortices quantized in terms of non-trivial center element of the $SU(2)$ subgroup which give rise to the intermediate confinement in the static potentials.