Georgi–Glashow model

The N = 1 superspace extension of 3+1 Minkowski spacetime.

N = 1 SUSY over 3+1 Minkowski spacetime without R-symmetry.

SU(5).

${\displaystyle \mathbb {Z} _{2}}$ (matter parity)

To prevent unwanted couplings in the supersymmetric version of the model, we assign a ${\displaystyle \mathbb {Z} _{2}}$ matter parity to the chiral superfields with the matter fields having odd parity and the Higgs having even parity. This is unnecessary in the non-supersymmetric version, but then, we can't protect the electroweak Higgs from quadratic radiative mass corrections. See hierarchy problem. In the non-supersymmetric version the action is invariant under a similar ${\displaystyle \mathbb {Z} _{2}}$ symmetry because the matter fields are all fermionic and thus must appear in the action in pairs, while the Higgs fields are bosonic.

Those associated with the SU(5) gauge symmetry

As complex representations:

label	description	multiplicity	SU(5) rep	${\displaystyle \mathbb {Z} _{2}}$ rep
Φ	GUT Higgs field	1	24	+
Hu	electroweak Higgs field	1	5	+
Hd	electroweak Higgs field	1	${\displaystyle {\overline {\mathbf {5} }}}$	+
${\displaystyle {\overline {\mathbf {5} }}}$	matter fields	3	${\displaystyle {\overline {\mathbf {5} }}}$	-
10	matter fields	3	10	-
Nc	sterile neutrinos	?	1	-

A generic invariant renormalizable superpotential is a (complex) ${\displaystyle SU(5)\times \mathbb {Z} _{2}}$ invariant cubic polynomial in the superfields. It is a linear combination of the following terms:

${\displaystyle {\begin{matrix}\Phi ^{2}&&\Phi _{B}^{A}\Phi _{A}^{B}\\[4pt]\Phi ^{3}&&\Phi _{B}^{A}\Phi _{C}^{B}\Phi _{A}^{C}\\[4pt]H_{d}H_{u}&&{H_{d}}_{A}H_{u}^{A}\\[4pt]H_{d}\Phi H_{u}&&{H_{d}}_{A}\Phi _{B}^{A}H_{u}^{B}\\[4pt]H_{u}\mathbf {10} _{i}\mathbf {10} _{j}&&\epsilon _{ABCDE}H_{u}^{A}\mathbf {10} _{i}^{BC}\mathbf {10} _{j}^{DE}\\[4pt]H_{d}{\overline {\mathbf {5} }}_{i}\mathbf {10} _{j}&&{H_{d}}_{A}{\overline {\mathbf {5} }}_{Bi}\mathbf {10} _{j}^{AB}\\[4pt]H_{u}{\overline {\mathbf {5} }}_{i}N_{j}^{c}&&H_{u}^{A}{\overline {\mathbf {5} }}_{Ai}N_{j}^{c}\\[4pt]N_{i}^{c}N_{j}^{c}&&N_{i}^{c}N_{j}^{c}\\\end{matrix}}}$

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

The last two rows presupposes the multiplicity of ${\displaystyle N^{c}}$ is not zero (i.e. that a sterile neutrino exists). The coupling ${\displaystyle H_{u}\mathbf {10} _{i}\mathbf {10} _{j}}$ has coefficients which are symmetric in i and j. The coupling ${\displaystyle N_{i}^{c}N_{j}^{c}}$ has coefficients which are symmetric in i and j. The number of sterile neutrino generations need not be three, unless the SU(5) is embedded in a higher unification scheme such as SO(10).

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

${\displaystyle W=Tr\left[a\Phi ^{2}+b\Phi ^{3}\right]}$

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

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

${\displaystyle \Phi ={\begin{cases}\operatorname {diag} (0,0,0,0,0)\\\operatorname {diag} ({\frac {2a}{9b}},{\frac {2a}{9b}},{\frac {2a}{9b}},{\frac {2a}{9b}},-{\frac {8a}{9b}})\\\operatorname {diag} ({\frac {4a}{3b}},{\frac {4a}{3b}},{\frac {4a}{3b}},-{\frac {2a}{b}},-{\frac {2a}{b}})\end{cases}}}$

The three cases are called case I, II and III and they break the gauge symmetry into ${\displaystyle SU(5),[SU(4)\times U(1)]/\mathbb {Z} _{4}}$ and ${\displaystyle [SU(3)\times SU(2)\times U(1)]/\mathbb {Z} _{6}}$ 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 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-terms and D-terms. The matter parity remains unbroken (right up to the TeV scale).

The gauge algebra 24 decomposes as

${\displaystyle {\begin{pmatrix}(8,1)_{0}\\(1,3)_{0}\\(1,1)_{0}\\(3,2)_{-{\frac {5}{6}}}\\({\bar {3}},2)_{\frac {5}{6}}\end{pmatrix}}.}$

This 24 is a real representation, so the last two terms need explanation. Both ${\displaystyle (3,2)_{-{\frac {5}{6}}}}$ and ${\displaystyle ({\bar {3}},2)_{\frac {5}{6}}}$ 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 ${\displaystyle (3,2)_{-{\frac {5}{6}}}}$ and ${\displaystyle ({\bar {3}},2)_{\frac {5}{6}}}$ of the adjoint Higgs to be absorbed. The other real half acquires a mass coming from the D-terms. And the other three components of the adjoint Higgs, ${\displaystyle (8,1)_{0},(1,3)_{0}}$ and ${\displaystyle (1,1)_{0}}$ acquire GUT scale masses coming from self pairings of the superpotential, ${\displaystyle a\Phi ^{2}+b<\Phi >\Phi ^{2}.}$

The sterile neutrinos, if any exists, would also acquire a GUT scale Majorana mass coming from the superpotential coupling ν^c2.

Because of matter parity, the matter representations ${\displaystyle {\overline {\mathbf {5} }}}$ and 10 remain chiral.

It is the Higgs fields 5_H and ${\displaystyle {\overline {\mathbf {5} }}_{H}}$ which are interesting.

${\displaystyle 5_{H}}$		${\displaystyle {\bar {5}}_{H}}$
${\displaystyle {\begin{pmatrix}(3,1)_{-{\frac {1}{3}}}\\(1,2)_{\frac {1}{2}}\end{pmatrix}}}$	${\displaystyle {\begin{matrix}\_\_\_\\???\end{matrix}}}$	${\displaystyle {\begin{pmatrix}({\bar {3}},1)_{\frac {1}{3}}\\(1,2)_{-{\frac {1}{2}}}\end{pmatrix}}}$

The two relevant superpotential terms here are ${\displaystyle 5_{H}{\bar {5}}_{H}}$ and ${\displaystyle <24>5_{H}{\bar {5}}_{H}}$. Unless there happens to be some 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.

In the lowest-order Feynman diagram corresponding to the simplest source of proton decay in SU(5), a left-handed and a right-handed up quark annihilate, yielding an X⁺ boson, which decays to a right-handed (or left-handed) positron and a left-handed (or right-handed) anti-down quark:

${\displaystyle u_{L}+u_{R}\to X^{+}\to e_{R}^{+}+{\bar {d}}_{L}}$,

${\displaystyle u_{L}+u_{R}\to X^{+}\to e_{L}^{+}+{\bar {d}}_{R}}$.

This process conserves weak isospin, weak hypercharge, and color. GUTs equate anti-color with having 2 colors, ${\displaystyle {\bar {g}}\equiv rb}$, and SU(5) defines left-handed normal leptons as "white" and right-handed antileptons as "black." The first vertex only involves fermions of the 10 representation, while the second only involves fermions in the 5̅ (or 10), demonstrating the preservation of SU(5) symmetry.