PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.GaugeSlice

Given three complex numbers $z, w_0, w_1 \in \mathbb{C}$, the upper-triangular slice configuration is a two-Higgs-doublet state $(\Phi_1, \Phi_2)$ defined by the doublets $\Phi_1 = \begin{pmatrix} z \\ 0 \end{pmatrix}$ and $\Phi_2 = \begin{pmatrix} w_0 \\ w_1 \end{pmatrix}$. This configuration represents the gauge-fixed state where the first doublet is aligned with the first axis of the $SU(2)$ space.

For any complex numbers $z, w_0, w_1 \in \mathbb{C}$, the first Higgs doublet $\Phi_1$ of the state defined by the upper-triangular slice $\text{sliceHiggs}(z, w_0, w_1)$ is equal to the vector $\begin{pmatrix} z \\ 0 \end{pmatrix}$.

For any complex numbers $z, w_0, w_1 \in \mathbb{C}$, let the upper-triangular slice configuration of two Higgs doublets be denoted by $\text{sliceHiggs}(z, w_0, w_1) = (\Phi_1, \Phi_2)$. This theorem states that the second Higgs doublet $\Phi_2$ in this configuration is the vector $\begin{pmatrix} w_0 \\ w_1 \end{pmatrix}$.

Let $H = (\Phi_1, \Phi_2)$ be a configuration in the Two-Higgs-Doublet Model, where $\Phi_1, \Phi_2 \in \mathbb{C}^2$ are the two Higgs doublets. For any real number $c \in \mathbb{R}$, the first doublet of the scalar product $c \cdot H$ is equal to $c \cdot \Phi_1$.

In the Two-Higgs-Doublet Model (2HDM), where a configuration $H = (\Phi_1, \Phi_2)$ consists of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the second doublet of a configuration scaled by a real number $c \in \mathbb{R}$ is equal to the scalar product of $c$ and the original second doublet. That is, $(c \cdot H).\Phi_2 = c \cdot \Phi_2$.

The function $\text{sliceR}$ is a real-linear map from the space of six real parameters $\mathbb{R}^6$ to the Two-Higgs-Doublet Model space. For a vector $a = (a_0, a_1, a_2, a_3, a_4, a_5) \in \mathbb{R}^6$, the map yields the doublet configuration $(\Phi_1, \Phi_2)$ defined by: $$\Phi_1 = \begin{pmatrix} a_0 + i a_1 \\ 0 \end{pmatrix}, \quad \Phi_2 = \begin{pmatrix} a_2 + i a_3 \\ a_4 + i a_5 \end{pmatrix}$$ where $i$ is the imaginary unit. This map parameterizes the upper-triangular gauge slice by treating the components of the input vector as the real and imaginary parts of the three surviving complex field parameters.

For any vector of six real parameters $a = (a_0, a_1, a_2, a_3, a_4, a_5) \in \mathbb{R}^6$, the real-linear map $\text{sliceR}$ is equal to the upper-triangular slice configuration: $$\text{sliceR}(a) = \text{sliceHiggs}(a_0 + i a_1, a_2 + i a_3, a_4 + i a_5)$$ where $i$ is the imaginary unit and $\text{sliceHiggs}(z, w_0, w_1)$ is the two-Higgs-doublet state defined by doublets $\Phi_1 = \begin{pmatrix} z \\ 0 \end{pmatrix}$ and $\Phi_2 = \begin{pmatrix} w_0 \\ w_1 \end{pmatrix}$.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the polynomial gauge orbit representative configuration $\text{repHiggs}(X)$ is equal to the upper-triangular slice configuration $\text{sliceHiggs}(z, w_0, w_1)$ with the complex parameters $z = X_0$, $w_0 = X_1 + i X_2$, and $w_1 = X_3$, where $i$ is the imaginary unit.

For any complex phase $a \in U(1)$ and complex parameters $z, w_0, w_1 \in \mathbb{C}$, the action of the Cartan element $g_a = (I_3, \operatorname{diag}(a, \bar{a}), 1)$ of the Standard Model gauge group on the upper-triangular Higgs doublet configuration $\text{sliceHiggs}(z, w_0, w_1) = \left( \begin{pmatrix} z \\ 0 \end{pmatrix}, \begin{pmatrix} w_0 \\ w_1 \end{pmatrix} \right)$ is given by: $$g_a \cdot \text{sliceHiggs}(z, w_0, w_1) = \text{sliceHiggs}(a z, a w_0, \bar{a} w_1)$$ where $\bar{a}$ denotes the complex conjugate of $a$. This transformation multiplies the first components of the doublets by $a$ and the perpendicular second component of the second doublet by $\bar{a}$.

For any complex phase $c \in U(1)$ and any configuration on the upper-triangular gauge slice $\text{sliceHiggs}(z, w_0, w_1)$, the action of the $U(1)$ subgroup element $\text{ofU1Subgroup}(c)$ is given by: $$\text{ofU1Subgroup}(c) \cdot \text{sliceHiggs}(z, w_0, w_1) = \text{sliceHiggs}(z, w_0, c^6 w_1)$$ This transformation leaves the first two complex components $z$ and $w_0$ invariant and multiplies the third ("perpendicular") component $w_1$ by the phase $c^6$.

Given a phase $u \in U(1) \subset \mathbb{C}$ and a vector of six real parameters $a = (a_0, a_1, a_2, a_3, a_4, a_5) \in \mathbb{R}^6$ representing the field components on the gauge slice, this function returns a new vector in $\mathbb{R}^6$. The transformation corresponds to rotating the three complex pairs formed by the real parameters: - The first pair $(a_0, a_1)$ is treated as a complex number $z_0 = a_0 + i a_1$ and multiplied by $u$. - The second pair $(a_2, a_3)$ is treated as $z_1 = a_2 + i a_3$ and multiplied by $u$. - The third ("perpendicular") pair $(a_4, a_5)$ is treated as $z_2 = a_4 + i a_5$ and multiplied by the complex conjugate $\bar{u}$. The resulting vector is $(\text{Re}(u z_0), \text{Im}(u z_0), \text{Re}(u z_1), \text{Im}(u z_1), \text{Re}(\bar{u} z_2), \text{Im}(\bar{u} z_2))$.

For any complex phase $u \in U(1) \subset \mathbb{C}$ and any vector of six real parameters $a \in \mathbb{R}^6$, the action of the $SU(2)$ Cartan element $\text{gaugeCartan}(u) = (I_3, \text{diag}(u, \bar{u}), 1)$ on the Higgs configuration $\text{sliceR}(a)$ is equivalent to the Higgs configuration defined by the rotated parameters $\text{cartanRotParam}(u, a)$. That is, $$\text{gaugeCartan}(u) \cdot \text{sliceR}(a) = \text{sliceR}(\text{cartanRotParam}(u, a))$$ where $\text{sliceR}$ maps the parameters to the upper-triangular gauge slice of the Two-Higgs-Doublet Model.

Given a phase $c \in U(1)$ and a vector of six real parameters $a \in \mathbb{R}^6$, this function leaves the first four components $(a_0, a_1, a_2, a_3)$ invariant and rotates the complex number formed by the last two components $a_4 + i a_5$ by the phase $c^6$. Specifically, the output is the vector $(a_0, a_1, a_2, a_3, \text{Re}(c^6(a_4 + i a_5)), \text{Im}(c^6(a_4 + i a_5)))$, where $i$ is the imaginary unit.

Let $c \in U(1)$ be a unitary complex number and $a = (a_0, a_1, a_2, a_3, a_4, a_5) \in \mathbb{R}^6$ be a vector of parameters. Let $\text{sliceR}(a)$ be the configuration of two Higgs doublets $(\Phi_1, \Phi_2)$ defined by the upper-triangular gauge slice: $$\Phi_1 = \begin{pmatrix} a_0 + i a_1 \\ 0 \end{pmatrix}, \quad \Phi_2 = \begin{pmatrix} a_2 + i a_3 \\ a_4 + i a_5 \end{pmatrix}$$ The action of the residual $U(1)$ gauge group element $g(c)$ (defined by the inclusion $U(1) \to SU(3) \times SU(2) \times U(1)$) on this configuration is equivalent to rotating the parameter vector $a$ such that the first four components $(a_0, a_1, a_2, a_3)$ remain invariant and the complex number formed by the last two components $a_4 + i a_5$ is rotated by the phase $c^6$. That is, $$g(c) \cdot \text{sliceR}(a) = \text{sliceR}(a_0, a_1, a_2, a_3, \text{Re}(c^6(a_4 + i a_5)), \text{Im}(c^6(a_4 + i a_5))).$$