Editing Georgi–Glashow model (section)

====The {{math|Φ}} sector====
:<math>\ W = Tr \left [a\Phi^2+b\Phi^3 \right ]\ </math>

The {{math|F}} zeros corresponds to finding the stationary points of {{math|W}} subject to the traceless constraint <math>\ Tr[\Phi]=0 ~.</math> So, <math>\ 2a \Phi+3b\Phi^2 = \lambda \mathbf{1}\ ,</math> where {{math|λ}} is a Lagrange multiplier.

Up to an SU(5) (unitary) transformation,

:<math>\Phi=\begin{cases}
  \operatorname{diag}(0,0,0,0,0)\\
  \operatorname{diag}(\frac{2a}{9b},\frac{2a}{9b},\frac{2a}{9b},\frac{2a}{9b},-\frac{8a}{9b})\\
  \operatorname{diag}(\frac{4a}{3b},\frac{4a}{3b},\frac{4a}{3b},-\frac{2a}{b},-\frac{2a}{b})
\end{cases}</math>

The three cases are called case I, II, and III and they break the gauge symmetry into <math>\ SU(5),\ \left[SU(4) \times U(1) \right]/\Z_4\ </math> and <math>\ \left[SU(3)\times SU(2) \times U(1)\right]/\Z_6</math> respectively (the stabilizer of the VEV).

In other words, there are at least three different superselection sections, which is typical for supersymmetric theories.

Only case III makes any [[wikt:phenomenon|phenomenological]] sense and so, we will focus on this case from now onwards.

It can be verified that this solution together with zero VEVs for all the other chiral multiplets is a zero of the [[F-term]]s and [[D-term]]s. The matter parity remains unbroken (right up to the TeV scale).