Editing Georgi–Glashow model (section)

==Minimal supersymmetric SU(5)==
The minimal supersymmetric SU(5) model assigns a <math>\Z_2</math> [[matter parity]] to the chiral superfields with the matter fields having odd parity and the Higgs having even parity to protect the electroweak Higgs from quadratic radiative mass corrections (the [[hierarchy problem]]). In the non-supersymmetric version the action is invariant under a similar <math>\Z_2</math> symmetry because the matter fields are all [[fermion]]ic and thus must appear in the action in pairs, while the Higgs fields are [[boson]]ic.

===Chiral superfields===
As complex representations:

{| class="wikitable" style="text-align:center;"
|-
! label !! description !! multiplicity !! SU(5) rep !! <math>\Z_2</math> rep
|-
| {{math|Φ}} || {{left|GUT Higgs field}} || 1 || '''24''' || +
|-
| {{math|H}}{{sub|u}} || {{left|electroweak Higgs field}} || 1 || '''5''' || +
|-
| {{math|H}}{{sub|d}} || {{left|electroweak Higgs field}} || 1 || <math>\overline{\mathbf{5}}</math>||+
|-
| <math>\overline{\mathbf{5}}</math> || {{left|matter fields}} || 3 || <math>\overline{\mathbf{5}}</math> || &minus;
|-
| '''10''' || {{left|matter fields}} || 3 || '''10''' || &minus;
|-
| {{math|N}}{{sup|c}} || {{left|sterile neutrinos}} || (fractal) {{sfrac| 1 |2}} || '''1''' || &minus;
|}

===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)]].

===Vacua===
The vacua correspond to the mutual zeros of the {{math|F}} and {{math|D}} terms. Let's first look at the case where the VEVs of all the chiral fields are zero except for {{math|Φ}}.

====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).

====Decomposition====
The gauge algebra '''24''' decomposes as

: <math>\begin{pmatrix}(8,1)_0\\(1,3)_0\\(1,1)_0\\(3,2)_{-\frac{5}{6}}\\(\bar{3},2)_{\frac{5}{6}}\end{pmatrix} ~.</math>

This '''24''' is a real representation, so the last two terms need explanation. Both <math>(3,2)_{-\frac{5}{6}}</math> and <math>\ (\bar{3},2)_{\frac{5}{6}}\ </math> are complex representations. However, the direct sum of both representation decomposes into two irreducible real representations and we only take half of the direct sum, i.e. one of the two real irreducible copies.  The first three components are left unbroken. The adjoint Higgs also has a similar decomposition, except that it is complex. The [[Higgs mechanism]] causes one real HALF of the <math>\ (3,2)_{-\frac{5}{6}}\ </math> and <math>\ (\bar{3},2)_{\frac{5}{6}}\ </math> of the adjoint Higgs to be absorbed. The other real half acquires a mass coming from the [[D-term]]s. And the other three components of the adjoint Higgs, <math>\ (8,1)_0, (1,3)_0\ </math> and <math>\ (1,1)_0\ </math> acquire GUT scale masses coming from self pairings of the superpotential, <math>\ a\Phi^2 +b <\Phi>\Phi^2 ~.</math>

The sterile neutrinos, if any exist, would also acquire a GUT scale Majorana mass coming from the superpotential coupling {{mvar|ν}}{{sup|c }}{{sup| 2 }} .

Because of matter parity, the matter representations <math>\ \overline{\mathbf{5}}\ </math> and '''10''' remain chiral.

It is the Higgs fields 5{{sub|H}} and <math>\ \overline{\mathbf{5}}_\mathrm{H}\ </math> which are interesting.

{| style="padding-left:3em"
|-
| <math> 5_\mathrm{H} </math>
|
| <math> \bar{5}_\mathrm{H} </math>
|-
| <math>
\begin{pmatrix}
(3,1)_{-\tfrac{1}{3}}\\
(1,2)_{\tfrac{1}{2}}
\end{pmatrix}
</math>
| <math>
\begin{matrix}
\_\_\_\\
???
\end{matrix}
</math>
| <math>
\begin{pmatrix}
(\bar{3},1)_{\tfrac{1}{3}}\\
(1,2)_{-\tfrac{1}{2}}
\end{pmatrix}
</math>
|}

The two relevant superpotential terms here are <math>\ 5_\mathrm{H}\ \bar{5}_\mathrm{H}\ </math> and <math>\ \langle24 \rangle5_\mathrm{H}\ \bar{5}_\mathrm{H} ~.</math> Unless there happens to be some [[Fine-tuning (physics)|fine tuning]], we would expect both the triplet terms and the doublet terms to pair up, leaving us with no light electroweak doublets. This is in complete disagreement with phenomenology. See [[doublet-triplet splitting problem]] for more details.

====Fermion masses====
{{Further|Georgi–Jarlskog mass relation}}