PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.Module

The addition of two elements $H_1$ and $H_2$ in the Two-Higgs-Doublet Model (2HDM) is defined component-wise. If $H_1 = (\Phi_1^{(1)}, \Phi_2^{(1)})$ and $H_2 = (\Phi_1^{(2)}, \Phi_2^{(2)})$, then their sum is given by: $$H_1 + H_2 = (\Phi_1^{(1)} + \Phi_1^{(2)}, \Phi_2^{(1)} + \Phi_2^{(2)})$$ where $\Phi_1$ and $\Phi_2$ are the two Higgs doublets, and the addition is performed in the Higgs vector space $\mathbb{C}^2$.

For any two elements $H_1$ and $H_2$ in the Two-Higgs-Doublet Model (2HDM), the first Higgs doublet $\Phi_1$ of their sum is equal to the sum of their respective first Higgs doublets: $$(H_1 + H_2).\Phi_1 = H_1.\Phi_1 + H_2.\Phi_1$$ where the addition on the right-hand side is performed in the Higgs vector space $\mathbb{C}^2$.

For any two elements $H_1$ and $H_2$ in the Two-Higgs-Doublet Model (2HDM), the second Higgs doublet of their sum is the sum of their individual second Higgs doublets: $$(H_1 + H_2).\Phi_2 = H_1.\Phi_2 + H_2.\Phi_2$$ where $\Phi_2$ denotes the second scalar field in the doublet pair, and the addition on the right-hand side is performed in the Higgs vector space $\mathbb{C}^2$.

The zero element of the Two-Higgs-Doublet Model (2HDM) is the configuration where both scalar fields $\Phi_1$ and $\Phi_2$ are equal to the zero vector in the Higgs vector space $\mathbb{C}^2$.

For the zero element $0$ of the Two-Higgs-Doublet Model, the first scalar field component $\Phi_1$ is equal to the zero vector of the Higgs vector space.

In the Two-Higgs-Doublet Model, the second scalar field $\Phi_2$ of the zero configuration is equal to the zero vector in the Higgs vector space $\mathbb{C}^2$.

For a complex number $c \in \mathbb{C}$ and an element $H$ of the Two Higgs Doublet Model space consisting of two fields $(\Phi_1, \Phi_2)$, the scalar multiplication $c \cdot H$ is defined component-wise as $(c \Phi_1, c \Phi_2)$.

For any complex number $c \in \mathbb{C}$ and any element $H$ of the Two Higgs Doublet Model space (consisting of two fields $\Phi_1, \Phi_2 \in \mathbb{C}^2$), the first component of the scalar multiplication $c \cdot H$ is equal to the scalar multiplication of $c$ and the first component $\Phi_1$. That is, $(c \cdot H).\Phi_1 = c \cdot \Phi_1$.

For any complex number $c \in \mathbb{C}$ and any element $H = (\Phi_1, \Phi_2)$ in the Two Higgs Doublet Model space, where $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the second component of the scalar product $c \cdot H$ is equal to $c \cdot \Phi_2$. That is, $(c \cdot H).\Phi_2 = c \cdot H.\Phi_2$.

For an element $H$ of the Two Higgs Doublet Model space, consisting of two Higgs fields $\Phi_1$ and $\Phi_2$ in $\mathbb{C}^2$, the negation $-H$ is defined by the component-wise negation of its fields: $-H = (-\Phi_1, -\Phi_2)$.

For any element $H$ in the Two Higgs Doublet Model space, consisting of two Higgs fields $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the first component of the negated doublet $-H$ is equal to the negation of its first component $\Phi_1$. That is, $(-H).\Phi_1 = -\Phi_1$.

For any element $H$ in the Two Higgs Doublet Model space, the second field component of the negated doublet $-H$ is equal to the negation of its second field component $H.\Phi_2$, which can be expressed as $(-H).\Phi_2 = -H.\Phi_2$.

The space of the Two-Higgs-Doublet Model (2HDM), where an element $H = (\Phi_1, \Phi_2)$ consists of two complex scalar fields $\Phi_1, \Phi_2 \in \mathbb{C}^2$, forms an additive abelian group. The group operations, including addition $H_1 + H_2$, negation $-H$, and the zero element $0$, are defined component-wise based on the additive structure of the Higgs vector space $\mathbb{C}^2$.

The space of the Two-Higgs-Doublet Model (2HDM), where an element $H = (\Phi_1, \Phi_2)$ consists of two complex scalar fields $\Phi_1, \Phi_2 \in \mathbb{C}^2$, forms a module over the field of complex numbers $\mathbb{C}$. This structure is defined using component-wise addition and scalar multiplication, where for any $c \in \mathbb{C}$ and $H = (\Phi_1, \Phi_2)$, the operation is given by $c \cdot H = (c\Phi_1, c\Phi_2)$.