Physlib.Particles.BeyondTheStandardModel.GeorgiGlashow.Basic

The gauge group of the Georgi-Glashow model is defined as the special unitary group $SU(5)$.

The group homomorphism from the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ into the Georgi-Glashow gauge group $SU(5)$ maps the triple $(h, g, \alpha)$, where $h \in SU(3)$, $g \in SU(2)$, and $\alpha \in U(1)$, to a $5 \times 5$ block-diagonal matrix: \[ \Phi(h, g, \alpha) = \begin{pmatrix} \alpha^3 g & 0 \\ 0 & \alpha^{-2} h \end{pmatrix} \] where $\alpha^3 g$ is a $2 \times 2$ matrix and $\alpha^{-2} h$ is a $3 \times 3$ matrix.

The kernel of the inclusion homomorphism $\Phi : SU(3) \times SU(2) \times U(1) \to SU(5)$, which embeds the factors of the Standard Model gauge group into the Georgi-Glashow $SU(5)$ group, is equal to the subgroup $\mathbb{Z}_6$ (represented by the term `StandardModel.gaugeGroupℤ₆SubGroup`).

The group embedding $\bar{\Phi} : (SU(3) \times SU(2) \times U(1)) / \mathbb{Z}_6 \hookrightarrow SU(5)$ is the injective homomorphism from the physical Standard Model gauge group into the Georgi-Glashow $SU(5)$ gauge group. It is induced by the homomorphism $\Phi : SU(3) \times SU(2) \times U(1) \to SU(5)$, which maps $(h, g, \alpha)$ to the block-diagonal matrix $\text{diag}(\alpha^3 g, \alpha^{-2} h)$ for $h \in SU(3)$, $g \in SU(2)$, and $\alpha \in U(1)$, by quotienting out the kernel $\ker(\Phi) = \mathbb{Z}_6$.