Editing Georgi–Glashow model (section)

==Explicit Embedding of the Standard Model (SM)==
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.<ref name=ms>{{cite book | author=M. Srednicki | year=2015 | title=Quantum Field Theory | publisher=Cambridge University Press | isbn=978-0-521-86449-7 }}</ref> or in the original paper by Georgi and Glashow.<ref name="GG" />