Physlib.Particles.BeyondTheStandardModel.TwoHDM.Basic

Let $H_1$ and $H_2$ be two configurations in the two Higgs doublet model. If their first Higgs doublets are equal, $H_1.\Phi_1 = H_2.\Phi_1$, and their second Higgs doublets are equal, $H_1.\Phi_2 = H_2.\Phi_2$, then the configurations are identical, $H_1 = H_2$.

The group action (scalar multiplication) of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the two Higgs doublet configuration space $H = (\Phi_1, \Phi_2)$ is defined by acting on each doublet independently. For an element $g = (g_3, g_2, g_1) \in \mathcal{G}$ and a configuration $H = (\Phi_1, \Phi_2) \in \mathbb{C}^2 \times \mathbb{C}^2$, the action is given by: \[ g \cdot (\Phi_1, \Phi_2) = (g \cdot \Phi_1, g \cdot \Phi_2) \] where the action on each individual doublet $\Phi_i$ is $g \cdot \Phi_i = g_1^3 (g_2 \Phi_i)$. Here, $g_2 \in SU(2)$ acts via standard matrix-vector multiplication, $g_1 \in U(1)$ acts as a complex phase multiplication raised to the third power, and the $SU(3)$ component $g_3$ acts trivially.

For any element $g$ of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ and any configuration $H = (\Phi_1, \Phi_2)$ in the Two Higgs Doublet Model, the first Higgs doublet component of the gauge-transformed configuration $g \cdot H$ is equal to the gauge action of $g$ on the individual first doublet $\Phi_1$. That is, $(g \cdot H).\Phi_1 = g \cdot \Phi_1$.

For any element $g$ of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ and any configuration $H = (\Phi_1, \Phi_2)$ in the two Higgs doublet model (where $\Phi_1, \Phi_2 \in \mathbb{C}^2$), the second doublet of the configuration resulting from the gauge group action $g \cdot H$ is equal to the action of $g$ on the second doublet $\Phi_2$. That is, $(g \cdot H)_2 = g \cdot \Phi_2$.

The definition establishes that the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ (without discrete quotients) forms a group action on the configuration space of the two Higgs doublet model $H = (\Phi_1, \Phi_2) \in \mathbb{C}^2 \times \mathbb{C}^2$. Specifically, it proves that the scalar multiplication defined for this system satisfies the required axioms: the identity element of the gauge group acts as the identity map on the doublets ($1 \cdot H = H$), and the action is associative with respect to the group multiplication ($(g_1 g_2) \cdot H = g_1 \cdot (g_2 \cdot H)$).