Unraveling the Symmetry of the Gaussian Constants

We show that the $SU(2)$ Gaussian scalar field theory with the U(1) gauge group has a group symmetry at the level of the Gaussian potential and, in particular, an algebraic symmetric group. This group symmetry has many implications in the interpretation of the scalar fields. We discuss the possible meaning of this symmetry in terms of its effect on the evolution of the Gaussian potential. We argue that, regardless of the gauge group being used to describe the Gaussian scalar fields, this symmetry can be understood as the result of the stabilities of the scalar potential.