Physlib.Particles.BeyondTheStandardModel.TwoHDM.GramMatrix

The Gram matrix of a two Higgs doublet configuration $H$, consisting of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, is the $2 \times 2$ complex matrix defined as: $$G = \begin{pmatrix} \langle \Phi_1, \Phi_1 \rangle & \langle \Phi_2, \Phi_1 \rangle \\ \langle \Phi_1, \Phi_2 \rangle & \langle \Phi_2, \Phi_2 \rangle \end{pmatrix}$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$. This matrix is used to characterize the gauge orbits within the configuration space of the two Higgs doublet model.

For any configuration $H$ of the two Higgs doublet model consisting of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the Gram matrix $G$ defined by: $$G = \begin{pmatrix} \langle \Phi_1, \Phi_1 \rangle & \langle \Phi_2, \Phi_1 \rangle \\ \langle \Phi_1, \Phi_2 \rangle & \langle \Phi_2, \Phi_2 \rangle \end{pmatrix}$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb{C}^2$, is a self-adjoint (Hermitian) matrix.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and 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 complex doublets. For any gauge group element $g \in \mathcal{G}$, the Gram matrix of the gauge-transformed configuration $g \cdot H$ is equal to the Gram matrix of the original configuration $H$. That is, $$(g \cdot H).\text{gramMatrix} = H.\text{gramMatrix}$$ where the Gram matrix is the $2 \times 2$ matrix of Hermitian inner products $\langle \Phi_j, \Phi_i \rangle$.

In the two Higgs doublet model, let a configuration $H$ be defined by two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$. Let $G$ be the $2 \times 2$ Gram matrix associated with these doublets, whose entries are given by the Hermitian inner products of the fields. The determinant of this Gram matrix is given by: $$\det(G) = \|\Phi_1\|^2 \|\Phi_2\|^2 - |\langle \Phi_1, \Phi_2 \rangle|^2$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb{C}^2$ and $\|\cdot\|$ denotes the induced Euclidean norm.

In the two Higgs doublet model, let a configuration $H$ be defined by two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$. Let $G$ be the $2 \times 2$ complex Gram matrix associated with these doublets, whose entries are given by the Hermitian inner products of the fields. The real part of the determinant of this Gram matrix is given by: $$\text{Re}(\det(G)) = \|\Phi_1\|^2 \|\Phi_2\|^2 - |\langle \Phi_1, \Phi_2 \rangle|^2$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb{C}^2$ and $\|\cdot\|$ denotes the induced Euclidean norm.

In the two Higgs doublet model, let a configuration $H$ be defined by two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$. Let $G$ be the $2 \times 2$ complex Gram matrix associated with these doublets, whose entries are given by the Hermitian inner products of the fields. The real part of the determinant of this Gram matrix is non-negative, i.e., $$\text{Re}(\det(G)) \ge 0$$

For any two Higgs doublet configuration $H$ consisting of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, let $G$ be its Gram matrix defined as $$G = \begin{pmatrix} \langle \Phi_1, \Phi_1 \rangle & \langle \Phi_2, \Phi_1 \rangle \\ \langle \Phi_1, \Phi_2 \rangle & \langle \Phi_2, \Phi_2 \rangle \end{pmatrix}$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on the Higgs vector space. The real part of the trace of this Gram matrix is non-negative, i.e., $$\text{Re}(\text{tr}(G)) \ge 0$$

In the two Higgs doublet model, let a configuration $H$ be defined by two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$ with $\Phi_1 \neq 0$. Let $G$ be the $2 \times 2$ complex Gram matrix associated with these doublets. There exists a gauge transformation $g \in SU(3) \times SU(2) \times U(1)$ such that: 1. The transformed first doublet is $g \cdot \Phi_1 = \begin{pmatrix} \|\Phi_1\| \\ 0 \end{pmatrix}$. 2. The first component of the transformed second doublet $g \cdot \Phi_2$ is $\frac{\langle \Phi_1, \Phi_2 \rangle}{\|\Phi_1\|}$. 3. The norm of the second component of the transformed second doublet $g \cdot \Phi_2$ is $\frac{\sqrt{\text{Re}(\det(G))}}{\|\Phi_1\|}$. Here, $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb{C}^2$ and $\|\cdot\|$ denotes the induced Euclidean norm.

In the two Higgs doublet model, let a configuration $H$ be defined by two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$ with $\Phi_1 \neq 0$. Let $G$ be the $2 \times 2$ complex Gram matrix associated with these doublets. There exists a gauge transformation $g \in SU(3) \times SU(2) \times U(1)$ such that the transformed doublets take the following canonical form: 1. The first transformed doublet is $g \cdot \Phi_1 = \begin{pmatrix} \|\Phi_1\| \\ 0 \end{pmatrix}$. 2. The second transformed doublet is $g \cdot \Phi_2 = \begin{pmatrix} \frac{\langle \Phi_1, \Phi_2 \rangle}{\|\Phi_1\|} \\ \frac{\sqrt{\text{Re}(\det G)}}{\|\Phi_1\|} \end{pmatrix}$. Here, $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb{C}^2$, $\|\cdot\|$ denotes the induced Euclidean norm, and $\text{Re}(\det G)$ is the real part of the determinant of the Gram matrix.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and let $H_1, H_2$ be two configurations in the two Higgs doublet model, where each configuration consists of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$. Then $H_1$ belongs to the gauge orbit of $H_2$ under the action of $\mathcal{G}$ if and only if their Gram matrices are identical: $$H_1 \in \text{orb}_{\mathcal{G}}(H_2) \iff H_1.\text{gramMatrix} = H_2.\text{gramMatrix}$$ where the Gram matrix of a configuration $H = (\Phi_1, \Phi_2)$ is the $2 \times 2$ matrix $G_{ij} = \langle \Phi_j, \Phi_i \rangle_{\mathbb{C}}$ defined by the Hermitian inner products of the doublets.

Let $K$ be a $2 \times 2$ complex matrix. If $K$ is self-adjoint, the real part of its determinant is non-negative ($\text{Re}(\det K) \ge 0$), and the real part of its trace is non-negative ($\text{Re}(\text{tr } K) \ge 0$), then there exists a two Higgs doublet configuration $H$ (consisting of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$) such that the Gram matrix of $H$ is equal to $K$.

For a configuration $H$ of the two Higgs doublet model, the Gram vector is the vector in $\mathbb{R}^4$ whose components $r_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) are twice the coordinates of the Gram matrix $G$ when expanded in the Pauli basis $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$. Specifically, the Gram matrix $G$ is related to the Gram vector $r$ by the relation: $$G = \frac{1}{2} \sum_{\mu=0}^3 r_\mu \sigma_\mu$$ where $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices.

For any configuration $H$ of the two Higgs doublet model, the components $r_\mu$ (where $\mu \in \{0, 1, 2, 3\}$) of the Gram vector are equal to twice the coordinates of the Gram matrix $G$ when expanded in the Pauli basis $\mathcal{B} = \{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$. That is, $$r_\mu = 2 \cdot [\text{repr}_{\mathcal{B}}(G)]_\mu$$ where $G$ is the $2 \times 2$ self-adjoint complex matrix representing the inner products of the Higgs doublets, and $\text{repr}_{\mathcal{B}}(G)$ denotes the coordinate vector of $G$ with respect to the Pauli basis.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and let $H$ be a configuration in the two Higgs doublet model. For any gauge transformation $g \in \mathcal{G}$ and any index $\mu \in \{0, 1, 2, 3\}$ (represented by the set $\text{Fin } 1 \oplus \text{Fin } 3$), the $\mu$-th component of the Gram vector $r_\mu$ is invariant under the gauge transformation. That is, $$(g \cdot H).\text{gramVector}_\mu = H.\text{gramVector}_\mu$$ where the Gram vector components $r_\mu$ are the coordinates of the Gram matrix when expanded in the Pauli basis $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$.

For a configuration $H$ of the two Higgs doublet model, let $G$ be its Gram matrix and $r_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) be the components of its Gram vector. The Gram matrix is equal to half the sum of the Gram vector components weighted by the Pauli matrices $\sigma_\mu$: $$G = \frac{1}{2} \sum_{\mu=0}^3 r_\mu \sigma_\mu$$ where $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the standard spatial Pauli matrices.

For a configuration $H$ of the two Higgs doublet model, let $G$ be the Gram matrix and $r_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) be the components of its Gram vector. The Gram matrix can be expressed explicitly in terms of these components as: $$G = \frac{1}{2} \begin{pmatrix} r_0 + r_3 & r_1 - i r_2 \\ r_1 + i r_2 & r_0 - r_3 \end{pmatrix}$$ where $i$ is the imaginary unit, $r_0$ corresponds to the index `Sum.inl 0`, and $r_1, r_2, r_3$ correspond to the indices `Sum.inr 0`, `Sum.inr 1`, and `Sum.inr 2` respectively.

For a configuration $H$ of the two Higgs doublet model, let $G$ be its Gram matrix and $r_0$ be the 0th component of its Gram vector (corresponding to the index `Sum.inl 0`). The 0th component $r_0$ is equal to the real part of the trace of the Gram matrix: $$r_0 = \operatorname{Re}(\operatorname{Tr}(G))$$

For a configuration $H$ of the two Higgs doublet model, let $r_0$ be the 0th component of its Gram vector $r \in \mathbb{R}^4$ (corresponding to the index `Sum.inl 0`). This component is non-negative: $$r_0 \ge 0$$

For a configuration $H$ in the two Higgs doublet model consisting of doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the squared norm of the first Higgs doublet $\Phi_1$ is expressed in terms of the components of the Gram vector $r$ as: $$\|\Phi_1\|^2 = \frac{1}{2}(r_0 + r_3)$$ where $r_0$ is the component of the Gram vector indexed by `Sum.inl 0` and $r_3$ is the component indexed by `Sum.inr 2`.

For a configuration $H$ of the two Higgs doublet model with Higgs doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, let $r_0$ and $r_3$ be the components of the Gram vector $r \in \mathbb{R}^4$ corresponding to the indices `Sum.inl 0` and `Sum.inr 2` respectively. The squared norm of the second Higgs doublet $\Phi_2$ is given by: $$\|\Phi_2\|^2 = \frac{1}{2}(r_0 - r_3)$$

For a configuration $H$ of the two Higgs doublet model, let $\Phi_1, \Phi_2 \in \mathbb{C}^2$ be the two Higgs doublets and $r_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) be the components of its Gram vector. The complex Hermitian inner product of $\Phi_1$ and $\Phi_2$ is given by: $$\langle \Phi_1, \Phi_2 \rangle = \frac{1}{2} (r_1 + i r_2)$$ where $i$ is the imaginary unit, and $r_1, r_2$ correspond to the Gram vector indices `Sum.inr 0` and `Sum.inr 1` respectively.

For a configuration $H$ of the two Higgs doublet model consisting of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the Hermitian inner product of $\Phi_2$ and $\Phi_1$ is expressed in terms of the components of the Gram vector $r_\mu$ as: $$\langle \Phi_2, \Phi_1 \rangle = \frac{1}{2} (r_1 - i r_2)$$ where $\langle \cdot, \cdot \rangle$ is the standard Hermitian inner product, $i$ is the imaginary unit, and $r_1, r_2$ are the components of the Gram vector corresponding to the indices `Sum.inr 0` and `Sum.inr 1` respectively.

For a configuration $H$ of the two Higgs doublet model consisting of two complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the squared magnitude of their Hermitian inner product is expressed in terms of the components of the Gram vector $r$ as: $$|\langle \Phi_1, \Phi_2 \rangle|^2 = \frac{1}{4} (r_1^2 + r_2^2)$$ where $r_1$ and $r_2$ are the components of the Gram vector corresponding to the indices `Sum.inr 0` and `Sum.inr 1` respectively.

In the two Higgs doublet model, for a configuration $H$ consisting of the complex doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, the zeroth component $r_0$ of the Gram vector (indexed by `Sum.inl 0`) is equal to the sum of the squared norms of the two Higgs doublets: $$r_0 = \|\Phi_1\|^2 + \|\Phi_2\|^2$$

For a configuration $H$ of the two Higgs doublet model, let $G$ be the $2 \times 2$ Gram matrix and $r$ be the Gram vector. The zeroth component of the Gram vector $r_0$ (indexed by `Sum.inl 0`) is equal to the sum of the real parts of the diagonal entries of the Gram matrix: $$r_0 = \operatorname{Re}(G_{00}) + \operatorname{Re}(G_{11})$$

For a configuration $H$ of the two-Higgs-doublet model with Higgs doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, let $r$ be the corresponding Gram vector in $\mathbb{R}^4$. The component $r_1$ (indexed by `Sum.inr 0`) is equal to twice the real part of the complex Hermitian inner product of $\Phi_1$ and $\Phi_2$: $$r_1 = 2 \operatorname{Re} \langle \Phi_1, \Phi_2 \rangle$$

For a configuration $H$ of the two-Higgs-doublet model, let $G$ be the Gram matrix and $r$ be the Gram vector. The component $r_1$ of the Gram vector (indexed by `Sum.inr 0`) is equal to twice the real part of the entry $G_{10}$ in the Gram matrix: $$r_1 = 2 \operatorname{Re} G_{10}$$

For a configuration $H$ of the two-Higgs-doublet model with Higgs doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, let $r \in \mathbb{R}^4$ be the corresponding Gram vector. The component of the Gram vector $r_2$ (indexed by `Sum.inr 1`) is equal to twice the imaginary part of the complex Hermitian inner product of $\Phi_1$ and $\Phi_2$: $$r_2 = 2 \operatorname{Im} \langle \Phi_1, \Phi_2 \rangle$$

For a configuration $H$ of the two Higgs doublet model, let $G$ be the $2 \times 2$ Gram matrix and $r \in \mathbb{R}^4$ be the Gram vector. The component $r_2$ of the Gram vector (indexed by `Sum.inr 1`) is equal to twice the imaginary part of the entry $G_{10}$ of the Gram matrix: $$r_2 = 2 \operatorname{Im}(G_{10})$$

For a configuration $H$ of the two Higgs doublet model with Higgs doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, let $r_3$ be the component of the Gram vector $r \in \mathbb{R}^4$ corresponding to the index `Sum.inr 2`. This component is equal to the difference of the squared norms of the Higgs doublets: $$r_3 = \|\Phi_1\|^2 - \|\Phi_2\|^2$$

For a configuration $H$ of the two Higgs doublet model, let $G$ be the $2 \times 2$ Gram matrix and $r \in \mathbb{R}^4$ be the Gram vector. The component $r_3$ of the Gram vector (corresponding to the index `Sum.inr 2`) is equal to the difference between the real parts of the diagonal elements of the Gram matrix: $$r_3 = \text{Re}(G_{00}) - \text{Re}(G_{11})$$

For a configuration $H$ in the two Higgs doublet model, let $G$ be the $2 \times 2$ complex Gram matrix and $r \in \mathbb{R}^4$ be its associated Gram vector. The real part of the determinant of the Gram matrix is given by: $$\text{Re}(\det G) = \frac{1}{4} \left( r_0^2 - \sum_{i=1}^3 r_i^2 \right)$$ where $r_0$ is the component of the Gram vector indexed by `Sum.inl 0`, and $r_1, r_2, r_3$ are the components indexed by `Sum.inr 0`, `Sum.inr 1`, and `Sum.inr 2` respectively.

For a configuration $H$ in the two Higgs doublet model, let $r \in \mathbb{R}^4$ be the associated Gram vector. Let $r_0$ be the component indexed by `Sum.inl 0`, and $r_1, r_2, r_3$ be the components indexed by `Sum.inr 0`, `Sum.inr 1`, and `Sum.inr 2` respectively. The sum of the squares of the components $r_1, r_2, r_3$ is less than or equal to the square of the component $r_0$: $$\sum_{i=1}^3 r_i^2 \le r_0^2$$

Let $v: (\text{Fin } 1 \oplus \text{Fin } 3) \to \mathbb{R}$ be a vector in $\mathbb{R}^4$ with components $(r_0, r_1, r_2, r_3)$. If the first component $r_0 \ge 0$ and the remaining components satisfy the inequality $\sum_{i=1}^3 r_i^2 \le r_0^2$, then there exists a configuration $H$ of the two Higgs doublet model such that the Gram vector of $H$ is equal to $v$.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and let $H_1, H_2$ be two configurations in the two Higgs doublet model. $H_1$ belongs to the gauge orbit of $H_2$ under the action of $\mathcal{G}$ if and only if their Gram vectors are equal: $$H_1 \in \text{orb}_{\mathcal{G}}(H_2) \iff H_1.\text{gramVector} = H_2.\text{gramVector}$$ The Gram vector $r \in \mathbb{R}^4$ of a configuration is the vector whose components $r_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) are twice the coordinates of the Gram matrix $G$ when expanded in the Pauli basis $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$.