PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.OrbitRepresentative

Given a vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, this definition constructs a representative configuration for the gauge orbits of the two Higgs doublet model. The configuration consists of two complex scalar doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$ defined as: $$\Phi_1 = \begin{pmatrix} X_0 \\ 0 \end{pmatrix}, \quad \Phi_2 = \begin{pmatrix} X_1 + i X_2 \\ X_3 \end{pmatrix}$$ This family is polynomial in the parameters $X$, meaning its Gram vector components do not involve square roots, which facilitates the study of gauge-invariant potentials.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the first complex scalar doublet $\Phi_1$ of the polynomial gauge orbit representative $\text{repHiggs}(X)$ is given by $$\Phi_1 = \begin{pmatrix} X_0 \\ 0 \end{pmatrix}$$ where $X_0$ is the first component of the vector $X$.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the second complex scalar doublet $\Phi_2$ of the polynomial gauge orbit representative $\text{repHiggs}(X)$ in the two Higgs doublet model is given by: $$\Phi_2 = \begin{pmatrix} X_1 + i X_2 \\ X_3 \end{pmatrix}$$ where $i$ is the imaginary unit.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, let $\Phi_1$ be the first Higgs doublet of the polynomial representative configuration $\text{repHiggs}(X)$. The squared norm of this doublet is given by $\lVert \Phi_1 \rVert^2 = X_0^2$.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the squared norm of the second Higgs doublet $\Phi_2$ in the polynomial representative configuration $\text{repHiggs}(X)$ is given by $\|\Phi_2\|^2 = X_1^2 + X_2^2 + X_3^2$.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the complex inner product of the two Higgs doublets $\Phi_1$ and $\Phi_2$ in the polynomial gauge orbit representative $\text{repHiggs}(X)$ is given by: $$\langle \Phi_1, \Phi_2 \rangle_{\mathbb{C}} = X_0 (X_1 + i X_2)$$ where $i$ is the imaginary unit.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the 0-th component (indexed by `Sum.inl 0`) of the Gram vector of the polynomial gauge orbit representative $\text{repHiggs}(X)$ is equal to the sum of the squares of the parameters: $$(\text{repHiggs}(X)).\text{gramVector}_0 = X_0^2 + X_1^2 + X_2^2 + X_3^2$$

For any vector $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the component of the Gram vector $r$ of the polynomial gauge orbit representative $\text{repHiggs}(X)$ corresponding to the Pauli matrix $\sigma_1$ (indexed by $\mu=1$) is given by: $$r_1 = 2 X_0 X_1$$

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, let $\text{repHiggs}(X)$ be the representative configuration of the two Higgs doublet model. The component $r_2$ of its Gram vector (formally indexed as `Sum.inr 1` in the Pauli basis expansion $r_0, r_1, r_2, r_3$) is given by: $$r_2 = 2 X_0 X_2$$

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, let $\text{repHiggs}(X)$ be the representative configuration of the two Higgs doublet model. The third spatial component $r_3$ (indexed by `Sum.inr 2`) of its Gram vector $r$ is given by the formula: $$r_3 = X_0^2 - (X_1^2 + X_2^2 + X_3^2)$$

For any field configuration $\phi$ in the two Higgs doublet model, there exist a vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$ and a gauge transformation $g$ in the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ such that $g \cdot \phi = \text{repHiggs}(X)$. The representative configuration $\text{repHiggs}(X)$ consists of the two complex doublets: $$\Phi_1 = \begin{pmatrix} X_0 \\ 0 \end{pmatrix}, \quad \Phi_2 = \begin{pmatrix} X_1 + i X_2 \\ X_3 \end{pmatrix}$$