Editing Georgi–Glashow model (section)

===Superpotential===
A generic invariant [[renormalizable]] [[superpotential]] is a (complex) <math>SU(5)\times\Z_2</math> invariant cubic polynomial in the superfields. It is a linear combination of the following terms:

:<math>
\begin{matrix}
\Phi^2     & & \Phi^A_B \Phi^B_A \\[4pt]
\Phi^3     & & \Phi^A_B \Phi^B_C \Phi^C_A \\[4pt]
\mathrm{H}_\mathsf{d}\ \mathrm{H}_\mathsf{u}    & & {\mathrm{H}_\mathsf{d}}_A\ {\mathrm{H}_\mathsf{u}}^A \\[4pt]
\mathrm{H}_\mathsf{d}\ \Phi\ \mathrm{H}_\mathsf{u}                                 & & {\mathrm{H}_\mathsf{d}}_A\ \Phi^A_B\ {\mathrm{H}_\mathsf{u}}^B \\[4pt]
\mathrm{H}_\mathsf{u}\ \mathbf{10}_i \mathbf{10}_j              & & \epsilon_{ABCDE}\ {\mathrm{H}_\mathsf{u}}^A\ \mathbf{10}^{BC}_i\ \mathbf{10}^{DE}_j \\[4pt]
\mathrm{H}_\mathsf{d}\ \overline{\mathbf{5}}_i \mathbf{10}_j    & & {\mathrm{H}_\mathsf{d}}_A\ \overline{\mathbf{5}}_{Bi}\ \mathbf{10}^{AB}_j \\[4pt]
\mathrm{H}_\mathsf{u}\ \overline{\mathbf{5}}_i\ {\mathrm{N}^\mathsf{c}}_j            & & {\mathrm{H}_\mathsf{u}}^A\ \overline{\mathbf{5}}_{Ai}\ {\mathrm{N}^\mathsf{c}}_j \\[4pt]
{\mathrm{N}^\mathsf{c}}_i\ {\mathrm{N}^\mathsf{c}}_j                                  & & {\mathrm{N}^\mathsf{c}}_i\ {\mathrm{N}^\mathsf{c}}_j \\
\end{matrix}
</math>

The first column is an Abbreviation of the second column (neglecting proper normalization factors), where capital indices are SU(5) indices, and {{mvar|i}} and {{mvar|j}} are the generation indices.

The last two rows presupposes the multiplicity of <math>\ \mathrm{N}^\mathsf{c}\ </math> is not zero (i.e. that a [[sterile neutrino]] exists). The coupling <math>\ \mathrm{H}_\mathsf{u}\ \mathbf{10}_i\ \mathbf{10}_j\ </math> has coefficients which are symmetric in {{mvar|i}} and {{mvar|j}}. The coupling <math>\ \mathrm{N}^\mathsf{c}_i\ \mathrm{N}^\mathsf{c}_j\ </math> has coefficients which are symmetric in {{mvar|i}} and {{mvar|j}}. The number of sterile neutrino [[Generation (particle physics)|generation]]s need not be three, unless the SU(5) is embedded in a higher unification scheme such as [[SO(10) (physics)|SO(10)]].