Physlib.Particles.BeyondTheStandardModel.PatiSalam.Basic

The gauge group of the Pati-Salam model, prior to any quotienting by $\mathbb{Z}_2$, is defined as the product of special unitary groups $SU(4) \times SU(2) \times SU(2)$.

The group homomorphism $\text{incl}_{SM}$ from the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ to the Pati-Salam gauge group $SU(4) \times SU(2) \times SU(2)$ maps the triple $(h, g, \alpha)$ to $(\text{blockdiag}(\alpha h, \alpha^{-3}), g, \text{diag}(\alpha^3, \alpha^{-3}))$, where $h \in SU(3)$, $g \in SU(2)$, and $\alpha \in U(1)$.

The kernel of the inclusion homomorphism $\text{incl}_{SM}$ from the Standard Model gauge group into the Pati-Salam gauge group is equal to the subgroup $\text{gaugeGroup}_{\mathbb{Z}_3}$ of the Standard Model gauge group.

The group embedding $$\text{embedSM}_{\mathbb{Z}_3} : (SU(3) \times SU(2) \times U(1)) / \mathbb{Z}_3 \to SU(4) \times SU(2) \times SU(2)$$ is the injective group homomorphism from the quotiented Standard Model gauge group to the Pati-Salam gauge group. This map is induced by the inclusion homomorphism $\text{incl}_{SM}$ by quotienting out its kernel, $\ker(\text{incl}_{SM}) = \mathbb{Z}_3$.

This definition establishes the group isomorphism between the un-quotiented Pati-Salam gauge group $\text{GaugeGroupI} = SU(4) \times SU(2) \times SU(2)$ and the product of spin groups $Spin(6) \times Spin(4)$. This equivalence utilizes the exceptional isomorphisms where $SU(4) \cong Spin(6)$ and $SU(2) \times SU(2) \cong Spin(4)$.

The definition identifies the $\mathbb{Z}_2$ subgroup of the un-quotiented Pati-Salam gauge group $SU(4) \times SU(2) \times SU(2)$ (often denoted as `GaugeGroupI`). This subgroup is generated by the non-trivial element $(-1, -1, -1)$, where each $-1$ represents the non-identity central element of the respective $SU(n)$ factors. Physically, this subgroup is characterized by the fact that it acts trivially on all particle representations in the Standard Model.

The gauge group of the Pati-Salam model, denoted as $G_{PS}/\mathbb{Z}_2$, is defined as the quotient of the group $SU(4) \times SU(2)_L \times SU(2)_R$ (referred to as `GaugeGroupI`) by its $\mathbb{Z}_2$ subgroup (referred to as `gaugeGroupℤ₂SubGroup`). This $\mathbb{Z}_2$ subgroup is generated by the element $(-1, -1, -1)$ in the product of the centers of the three $SU(n)$ factors. This specific quotient is physically relevant as it is the group that acts faithfully on the particle representations of the Standard Model.

The embedding homomorphism of the Standard Model gauge group into the Pati-Salam gauge group, $\iota: G_{SM} \to SU(4) \times SU(2)_L \times SU(2)_R$, maps the $\mathbb{Z}_6$ subgroup of the Standard Model into the $\mathbb{Z}_2$ subgroup of the Pati-Salam group. Specifically, the image of the subgroup `StandardModel.gaugeGroupℤ₆SubGroup` under the map `inclSM` is contained within the subgroup `PatiSalam.gaugeGroupℤ₂SubGroup`, allowing the embedding to descend to a homomorphism between the quotient groups $G_{SM}/\mathbb{Z}_6$ and $G_{PS}/\mathbb{Z}_2$.

This definition characterizes the group homomorphism from the Standard Model gauge group quotiented by $\mathbb{Z}_6$, denoted as $G_{SM}/\mathbb{Z}_6$, to the Pati-Salam gauge group quotiented by $\mathbb{Z}_2$, denoted as $G_{PS}/\mathbb{Z}_2$. This homomorphism is induced by the natural embedding of the Standard Model gauge group into the Pati-Salam gauge group.