300px|right|thumb|The pattern of [[weak isospins, weak hypercharges, and strong charges for particles in the Georgi–Glashow model, rotated by the predicted weak mixing angle, showing electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes twelve colored X bosons, responsible for proton decay.]]
In particle physics, the Georgi–Glashow model is a particular Grand Unified Theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model, the Standard Model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5). The unified group SU(5) is then thought to be spontaneously broken into the Standard Model subgroup below a very high energy scale called the grand unification scale.
Since the Georgi–Glashow model combines leptons and quarks into single irreducible representations, there exist interactions which do not conserve baryon number, although they still conserve the quantum number B-L| associated with the symmetry of the common representation. This yields a mechanism for proton decay, and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. Nevertheless, the elegance of the model has led particle physicists to use it as the foundation for more complex models which yield longer proton lifetimes, particularly SO(10) in basic and SUSY variants.
(For a more elementary introduction to how the representation theory of Lie algebras are related to particle physics, see the article Particle physics and representation theory.)
Also, this model suffers from the doublet–triplet splitting problem.
Construction
thumb|Schematic representation of fermions and bosons in GUT showing split in the multiplets. Row for (the [[sterile neutrino singlet) is omitted, but would likewise be isolated. Neutral bosons (photon, Z-boson, and neutral gluons) are not shown but occupy the diagonal entries of the matrix in complex superpositions.]]
SU(5) acts on <math>\mathbb{C}^5</math> and hence on its exterior algebra <math>\wedge\mathbb{C}^5</math>. Choosing a <math>\mathbb{C}^2\oplus\mathbb{C}^3</math> splitting restricts SU(5) to , yielding matrices of the form
:<math>\begin{matrix}
\phi: & U(1)\times SU(2)\times SU(3) & \longrightarrow & S(U(2)\times U(3)) \subset SU(5) \\
& (\alpha, g, h) & \longmapsto &
\begin{pmatrix}
\alpha^3 g & 0\\
0 & \alpha^{-2}h
\end{pmatrix}\\
\end{matrix}</math>
with kernel <math>\{(\alpha, \alpha^{-3} \mathrm{Id}_2, \alpha^2 \mathrm{Id}_3) | \alpha \in \mathbb C , \alpha ^6 = 1 \}\cong \mathbb Z_6</math>, hence isomorphic to the Standard Model's true gauge group <math>SU(3)\times SU(2)\times U(1)/\mathbb{Z}_6</math>. For the zeroth power <math>{\textstyle\bigwedge}^0\mathbb{C}^5</math>, this acts trivially to match a left-handed neutrino, <math>\mathbb{C}_0\otimes\mathbb{C}\otimes\mathbb{C}</math>. For the first exterior power <math>{\textstyle\bigwedge}^1\mathbb{C}^5 \cong \mathbb{C}^5</math>, the Standard Model's group action preserves the splitting <math>\mathbb{C}^5 \cong \mathbb{C}^2\oplus\mathbb{C}^3</math>. The <math>\mathbb{C}^2</math> transforms trivially in , as a doublet in , and under the representation of (as weak hypercharge is conventionally normalized as ); this matches a right-handed anti-lepton, <math>\mathbb{C}_{\frac 1 2}\otimes\mathbb{C}^{2*}\otimes\mathbb{C}</math> (as <math>\mathbb{C}^{2}\cong\mathbb{C}^{2*}</math> in SU(2)). The <math>\mathbb{C}^3</math> transforms as a triplet in SU(3), a singlet in SU(2), and under the Y = − representation of U(1) (as ); this matches a right-handed down quark, <math>\mathbb{C}_{-\frac 1 3}\otimes\mathbb{C}\otimes\mathbb{C}^3</math>.
The second power <math>{\textstyle\bigwedge}^2\mathbb{C}^5</math> is obtained via the formula <math>{\textstyle\bigwedge}^2(V\oplus W)={\textstyle\bigwedge}^2 V^2 \oplus (V\otimes W) \oplus {\textstyle\bigwedge}^2 W^2</math>. As SU(5) preserves the canonical volume form of <math>\mathbb{C}^5</math>, Hodge duals give the upper three powers by <math>{\textstyle\bigwedge}^p\mathbb{C}^5\cong({\textstyle\bigwedge}^{5-p}\mathbb{C}^5)^*</math>. Thus the Standard Model's representation of one generation of fermions and antifermions lies within <math>\wedge\mathbb{C}^5</math>.
Similar motivations apply to the Pati–Salam model, and to SO(10), E6, and other supergroups of SU(5).
Explicit embedding of the Standard Model
Owing to its relatively simple gauge group <math> SU(5)</math> , GUTs can be written in terms of vectors and matrices which allows for an intuitive understanding of the Georgi–Glashow model. The fermion sector is then composed of an anti fundamental <math>\overline{\mathbf{5</math> and an antisymmetric <math>\mathbf{10}</math>. In terms of SM degrees of freedoms, this can be written as
:<math>
\overline{\mathbf{5_F=\begin{pmatrix}d_{1}^c\\d_{2}^c\\d_{3}^c\\e\\-\nu\end{pmatrix}</math>
and
:<math>
\mathbf {10}_F=\begin{pmatrix}
0&u_{3}^c&-u_{2}^c&u_1&d_1\\
-u_{3}^c&0&u_{1}^c&u_2&d_2\\
u_{2}^c&-u_{1}^c&0&u_3&d_3\\
-u_1&-u_2&-u_3&0&e_R\\
-d_1&-d_2&-d_3&-e_R&0
\end{pmatrix}</math>
with <math>d_i</math> and <math>u_i</math> the left-handed up and down type quark, <math>d_i^c</math> and <math>u_i^c</math> their righthanded counterparts, <math>\nu</math> the neutrino, <math>e</math> and <math>e_R</math> the left and right-handed electron, respectively.
In addition to the fermions, we need to break <math> SU(3)\times SU_L(2)\times U_Y(1)\rightarrow SU(3)\times U_{EM}(1)</math>; this is achieved in the Georgi–Glashow model via a fundamental <math>\mathbf{5}</math> which contains the SM Higgs,
:<math>
\mathbf{5}_H=(T_1,T_2,T_3,H^+,H^0)^T</math>
with <math> H^+</math> and <math>H^0</math> the charged and neutral components of the SM Higgs, respectively. Note that the <math>T_i</math> are not SM particles and are thus a prediction of the Georgi–Glashow model.
The SM gauge fields can be embedded explicitly as well. For that we recall a gauge field transforms as an adjoint, and thus can be written as <math>A^a_\mu T^a</math> with <math>T^a</math> the <math>SU(5)</math> generators. Now, if we restrict ourselves to generators with non-zero entries only in the upper <math>3\times 3</math> block, in the lower <math>2\times 2</math> block, or on the diagonal, we can identify
:<math>\begin{pmatrix}G^a_\mu T^a_3&0\\0&0\end{pmatrix}</math>
with the <math>SU(3)</math> colour gauge fields,
:<math>
\begin{pmatrix}0&0\\0&\frac{\sigma^a}{2}W^a_\mu\end{pmatrix}</math>
with the weak <math>SU(2)</math> fields, and
:<math> N\,B^0_\mu\operatorname{diag}\left(-1/3, -1/3, -1/3, 1/2, 1/2\right)</math>
with the <math>U(1)</math> hypercharge (up to some normalization <math>N</math>.)
Using the embedding, we can explicitly check that the fermionic fields transform as they should.
This explicit embedding can be found in Ref. or in the original paper by Georgi and Glashow.) which can work quite well in SUSY models.
A review of the DT splitting problem can be found in.