Physlib.Particles.BeyondTheStandardModel.TwoHDM.Potential

The zero instance for the potential parameters of the Two Higgs Doublet Model (2HDM) is defined by setting the mass squared parameters $m_{11}^2$, $m_{22}^2$, $m_{12}^2$ and the quartic coupling parameters $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6, \lambda_7$ all equal to $0$.

For the zero instance of the Two Higgs Doublet Model (2HDM) potential parameters, the mass squared parameter $m_{11}^2$ is equal to $0$.

For the zero instance of the Two Higgs Doublet Model (2HDM) potential parameters, the mass squared parameter $m_{22}^2$ is equal to $0$.

In the Two Higgs Doublet Model (2HDM), the mass squared parameter $m_{12}^2$ is equal to $0$ for the zero instance of the potential parameters.

In the Two Higgs Doublet Model (2HDM), the quartic coupling parameter $\lambda_1$ is equal to $0$ for the zero instance of the potential parameters.

In the Two Higgs Doublet Model (2HDM), the quartic coupling parameter $\lambda_2$ is equal to 0 for the zero instance of the potential parameters.

In the Two Higgs Doublet Model (2HDM), the quartic coupling parameter $\lambda_3$ is equal to $0$ for the instance where all potential parameters are set to zero.

In the Two Higgs Doublet Model (2HDM), the quartic coupling parameter $\lambda_4$ is equal to $0$ for the instance where all potential parameters are set to zero.

In the Two Higgs Doublet Model (2HDM), for the configuration where all potential parameters are set to zero, the quartic coupling parameter $\lambda_5$ is equal to $0$.

In the Two Higgs Doublet Model (2HDM), for the configuration where all potential parameters are set to zero, the quartic coupling parameter $\lambda_6$ is equal to $0$.

In the Two Higgs Doublet Model (2HDM), for the configuration where all potential parameters are set to zero, the quartic coupling parameter $\lambda_7$ is equal to $0$.

Given the parameters $P$ of a Two Higgs Doublet Model (2HDM) potential, the function $\xi$ defines a real-valued 4-vector $\xi_\mu$ indexed by $\mu \in \{0, 1, 2, 3\}$ that reparameterizes the quadratic mass terms. The components are defined in terms of the mass-squared parameters $m_{11}^2$, $m_{22}^2$, and $m_{12}^2$ as: - $\xi_0 = \frac{m_{11}^2 + m_{22}^2}{2}$ - $\xi_1 = -\text{Re}(m_{12}^2)$ - $\xi_2 = \text{Im}(m_{12}^2)$ - $\xi_3 = \frac{m_{11}^2 - m_{22}^2}{2}$

In the Two Higgs Doublet Model (2HDM), if all the potential parameters are set to zero, then the associated mass-term parameter vector $\xi$ is the zero vector, such that $\xi_\mu = 0$ for all $\mu \in \{0, 1, 2, 3\}$.

The function $\eta$ computes a symmetric $4 \times 4$ real matrix from the parameters $P$ of the Two-Higgs-Doublet Model (2HDM). This matrix represents the quartic couplings in the Gram vector formalism, where indices $\mu, \nu \in \{0, 1, 2, 3\}$ correspond to the basis elements of $\text{Fin } 1 \oplus \text{Fin } 3$. Given the quartic parameters $\lambda_1, \dots, \lambda_4 \in \mathbb{R}$ and $\lambda_5, \lambda_6, \lambda_7 \in \mathbb{C}$, the entries of $\eta$ are defined as: - $\eta_{00} = \frac{\lambda_1 + \lambda_2 + 2\lambda_3}{8}$ - $\eta_{01} = \eta_{10} = \frac{\text{Re}(\lambda_6 + \lambda_7)}{4}$ - $\eta_{02} = \eta_{20} = -\frac{\text{Im}(\lambda_6 + \lambda_7)}{4}$ - $\eta_{03} = \eta_{30} = \frac{\lambda_1 - \lambda_2}{8}$ - $\eta_{11} = \frac{\text{Re}(\lambda_5) + \lambda_4}{4}$ - $\eta_{12} = \eta_{21} = -\frac{\text{Im}(\lambda_5)}{4}$ - $\eta_{13} = \eta_{31} = \frac{\text{Re}(\lambda_6 - \lambda_7)}{4}$ - $\eta_{22} = \frac{\lambda_4 - \text{Re}(\lambda_5)}{4}$ - $\eta_{23} = \eta_{32} = \frac{\text{Im}(\lambda_7 - \lambda_6)}{4}$ - $\eta_{33} = \frac{\lambda_1 + \lambda_2 - 2\lambda_3}{8}$

For any parameters $P$ of the Two Higgs Doublet Model (2HDM) potential, the associated Gram parameter matrix $\eta$ is symmetric. That is, for all indices $\mu, \nu \in \{0, 1, 2, 3\}$, the entries satisfy $\eta_{\mu\nu} = \eta_{\nu\mu}$.

If the potential parameters $P$ of the Two Higgs Doublet Model (2HDM) are all zero (where $m_{11}^2 = m_{22}^2 = m_{12}^2 = 0$ and $\lambda_1 = \dots = \lambda_7 = 0$), then the resulting Gram parameter matrix $\eta$ is the zero matrix, such that all entries $\eta_{\mu\nu} = 0$ for $\mu, \nu \in \{0, 1, 2, 3\}$.

This definition specifies a set of parameters for the Two Higgs Doublet Model (2HDM) potential that serves as a counterexample to the stability condition (boundedness from below) provided in arXiv:hep-ph/0605184. Starting from the zero configuration, the parameters are defined as follows: the complex mass term parameter $m_{12}^2 = i$, the quartic couplings $\lambda_1 = \lambda_2 = \lambda_3 = \lambda_4 = \lambda_5 = 2$, and $\lambda_6 = \lambda_7 = -2$. These parameters correspond to the potential $$V(\Phi_1, \Phi_2) = 2 \operatorname{Im}(\langle \Phi_1, \Phi_2 \rangle_{\mathbb{C}}) + \|\Phi_1 - \Phi_2\|^4$$ where the quartic term is non-negative and vanishes only when the mass term is zero, yet the potential is not stable.

For the stability counterexample parameters of the Two Higgs Doublet Model (2HDM) potential, the components of the mass-term parameter vector $\xi_\mu$ are $\xi_0 = 0$, $\xi_1 = 0$, $\xi_2 = 1$, and $\xi_3 = 0$.

The Gram parameter matrix $\eta$ for the Two Higgs Doublet Model (2HDM) potential stability counterexample is the $4 \times 4$ symmetric matrix defined by: \[ \eta = \begin{pmatrix} 1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \] where the indices $\mu, \nu \in \{0, 1, 2, 3\}$ correspond to the components of the Gram vector formalism used to describe the quartic couplings of the potential.

The mass term of the Two Higgs Doublet Model (2HDM) potential is a function that, given the potential parameters $P$ and two Higgs doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$, returns the real value: \[ V_{\text{mass}}(P, \Phi_1, \Phi_2) = m_{11}^2 \|\Phi_1\|^2 + m_{22}^2 \|\Phi_2\|^2 - \text{Re} \left( m_{12}^2 \langle \Phi_1, \Phi_2 \rangle + (m_{12}^2)^* \langle \Phi_2, \Phi_1 \rangle \right) \] where: - $m_{11}^2, m_{22}^2 \in \mathbb{R}$ and $m_{12}^2 \in \mathbb{C}$ are the mass parameters defined in $P$. - $(m_{12}^2)^*$ denotes the complex conjugate of $m_{12}^2$. - $\|\cdot\|$ denotes the standard Euclidean norm on $\mathbb{C}^2$. - $\langle \cdot, \cdot \rangle$ denotes the standard complex inner product $\Phi_i^\dagger \Phi_j$ on $\mathbb{C}^2$. - $\text{Re}(\cdot)$ denotes the real part of a complex number.

In the Two Higgs Doublet Model (2HDM), for a given set of potential parameters $P$ and Higgs doublets $H$, the mass term of the potential $V_{\text{mass}}(P, H)$ is equal to the scalar product of the mass-parameter 4-vector $\xi_\mu$ and the Higgs gram vector $r_\mu$: \[ V_{\text{mass}}(P, H) = \sum_{\mu=0}^3 \xi_\mu r_\mu \] where the components of $\xi_\mu$ are defined by the mass parameters $m_{11}^2, m_{22}^2, m_{12}^2$ as: - $\xi_0 = \frac{m_{11}^2 + m_{22}^2}{2}$ - $\xi_1 = -\text{Re}(m_{12}^2)$ - $\xi_2 = \text{Im}(m_{12}^2)$ - $\xi_3 = \frac{m_{11}^2 - m_{22}^2}{2}$ and $r_\mu$ is the gram vector (a vector of gauge-invariant bilinears) associated with the Higgs doublets.

For any gauge transformation $g$ in the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$, any set of potential parameters $P$, and any configuration of Higgs doublets $H = (\Phi_1, \Phi_2)$, the mass term of the Two Higgs Doublet Model (2HDM) potential is invariant under the action of the gauge group: \[ V_{\text{mass}}(P, g \cdot H) = V_{\text{mass}}(P, H) \] where $V_{\text{mass}}$ is the quadratic part of the potential and $g \cdot H$ denotes the gauge transformation of the Higgs fields.

If the potential parameters $P$ of the Two Higgs Doublet Model (2HDM) are all zero (meaning $m_{11}^2 = m_{22}^2 = m_{12}^2 = 0$), then the mass term $V_{\text{mass}}(P, H)$ of the potential is zero for any configuration of the Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$.

For any pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$ in the Two Higgs Doublet Model (2HDM), the mass term of the potential evaluated with the stability counterexample parameters $P_{ce}$ (which are defined by $m_{11}^2 = m_{22}^2 = 0$ and $m_{12}^2 = i$) is given by twice the imaginary part of their complex inner product: $$V_{\text{mass}}(P_{ce}, \Phi_1, \Phi_2) = 2 \operatorname{Im}(\langle \Phi_1, \Phi_2 \rangle)$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$.

The quartic term of the two Higgs doublet model (2HDM) potential, given the potential parameters $P = (\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6, \lambda_7)$ and a pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$, is defined as: $$V_4(P, H) = \frac{\lambda_1}{2} \|\Phi_1\|^4 + \frac{\lambda_2}{2} \|\Phi_2\|^4 + \lambda_3 \|\Phi_1\|^2 \|\Phi_2\|^2 + \lambda_4 |\langle \Phi_1, \Phi_2 \rangle|^2 + \text{Re} \left[ \frac{\lambda_5}{2} \langle \Phi_1, \Phi_2 \rangle^2 + \lambda_6 \|\Phi_1\|^2 \langle \Phi_1, \Phi_2 \rangle + \lambda_7 \|\Phi_2\|^2 \langle \Phi_1, \Phi_2 \rangle \right]$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$, and $\|\cdot\|$ is the induced norm.

For any potential parameters $P = (\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6, \lambda_7)$ and a pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$ in the two Higgs doublet model (2HDM), the quartic term of the potential $V_4(P, H)$ can be expressed as: $$\begin{aligned} V_4(P, H) = &\frac{1}{2} \lambda_1 \|\Phi_1\|^4 + \frac{1}{2} \lambda_2 \|\Phi_2\|^4 + \lambda_3 \|\Phi_1\|^2 \|\Phi_2\|^2 + \lambda_4 \text{Re}(\langle \Phi_1, \Phi_2 \rangle \langle \Phi_2, \Phi_1 \rangle) \\ &+ \text{Re} \left( \frac{1}{2} \lambda_5 \langle \Phi_1, \Phi_2 \rangle^2 + \frac{1}{2} \bar{\lambda}_5 \langle \Phi_2, \Phi_1 \rangle^2 \right) \\ &+ \text{Re} \left( \lambda_6 \|\Phi_1\|^2 \langle \Phi_1, \Phi_2 \rangle + \bar{\lambda}_6 \|\Phi_1\|^2 \langle \Phi_2, \Phi_1 \rangle \right) \\ &+ \text{Re} \left( \lambda_7 \|\Phi_2\|^2 \langle \Phi_1, \Phi_2 \rangle + \bar{\lambda}_7 \|\Phi_2\|^2 \langle \Phi_2, \Phi_1 \rangle \right) \end{aligned}$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb{C}^2$, $\|\cdot\|$ is the induced norm, and $\bar{\lambda}$ denotes the complex conjugate.

For any potential parameters $P$ and any pair of Higgs doublets $H$ in the two Higgs doublet model (2HDM), the quartic term of the potential $V_4(P, H)$ is equal to the quadratic form: $$V_4(P, H) = \sum_{a, b} r_a r_b \eta_{ab}$$ where $r$ is the Gram vector associated with the Higgs doublets $H$, $r_a$ and $r_b$ are its components, and $\eta_{ab}$ are the entries of the $4 \times 4$ symmetric Gram parameter matrix $\eta$ derived from the potential parameters $P$.

For any gauge transformation $g$ in the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$, any set of potential parameters $P$ for the Two Higgs Doublet Model (2HDM), and any pair of Higgs doublets $H$, the quartic term of the potential $V_4$ is invariant under the gauge action, such that: $$V_4(P, g \cdot H) = V_4(P, H)$$ where $g \cdot H$ denotes the action of the gauge group on the Higgs doublets.

In the Two Higgs Doublet Model (2HDM), if the potential parameters $P$ are all zero (i.e., $\lambda_1 = \lambda_2 = \lambda_3 = \lambda_4 = \lambda_5 = \lambda_6 = \lambda_7 = 0$), then the quartic term of the potential $V_4(P, H)$ is zero for any pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$.

For the Two Higgs Doublet Model (2HDM) potential with the parameters defined in the stability counterexample, the quartic term $V_4$ for a pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$ is given by: $$V_4 = \left( \|\Phi_1\|^2 + \|\Phi_2\|^2 - 2 \operatorname{Re} \langle \Phi_1, \Phi_2 \rangle \right)^2$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$ and $\|\cdot\|$ is the induced norm.

For any pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$, the quartic term of the 2HDM potential $V_4$ evaluated with the stability counterexample parameters is equal to the fourth power of the norm of the difference between the two doublets: $$V_4 = \|\Phi_1 - \Phi_2\|^4$$ where $\|\cdot\|$ denotes the norm induced by the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$.

For any pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$, the quartic term of the 2HDM potential $V_4$ evaluated with the stability counterexample parameters is non-negative: $$V_4 \ge 0$$ where $V_4$ for these specific parameters is defined as $\|\Phi_1 - \Phi_2\|^4$, and $\|\cdot\|$ is the norm induced by the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$.

For any pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$ in the Two Higgs Doublet Model (2HDM), if the quartic term $V_4$ of the potential evaluated with the stability counterexample parameters is zero ($V_4(P_{ce}, H) = 0$), then the mass term $V_{\text{mass}}$ evaluated with the same parameters is also zero ($V_{\text{mass}}(P_{ce}, H) = 0$).

The potential of the Two Higgs Doublet Model (2HDM) is a real-valued function that depends on a set of potential parameters $P$ and a pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$. It is defined as the sum of the mass term $V_{\text{mass}}(P, H)$ and the quartic term $V_4(P, H)$: \[ V(P, H) = V_{\text{mass}}(P, H) + V_4(P, H) \] where $V_{\text{mass}}$ represents the quadratic part of the potential (the mass terms) and $V_4$ represents the quartic interactions of the Higgs fields.

For any gauge transformation $g$ in the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$, any set of potential parameters $P$, and any configuration of Higgs doublets $H = (\Phi_1, \Phi_2)$, the full potential of the Two Higgs Doublet Model (2HDM) is invariant under the action of the gauge group: \[ V(P, g \cdot H) = V(P, H) \] where $V(P, H)$ is the sum of the mass term and the quartic term, and $g \cdot H$ denotes the gauge transformation of the Higgs fields.

If the potential parameters $P$ of the Two Higgs Doublet Model (2HDM) are all zero, then the 2HDM potential $V(P, H)$ is zero for any pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$.

For any pair of Higgs doublets $H = (\Phi_1, \Phi_2) \in (\mathbb{C}^2)^2$ in the Two Higgs Doublet Model (2HDM), the potential $V$ evaluated with the stability counterexample parameters $P_{ce}$ is given by the sum of twice the imaginary part of their complex inner product and the fourth power of the norm of their difference: $$V(P_{ce}, \Phi_1, \Phi_2) = 2 \operatorname{Im}(\langle \Phi_1, \Phi_2 \rangle) + \|\Phi_1 - \Phi_2\|^4$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$, $\operatorname{Im}(\cdot)$ denotes the imaginary part, and $\|\cdot\|$ denotes the induced norm.

For any potential parameters $P$ and any pair of Higgs doublets $H$ in the Two Higgs Doublet Model (2HDM), the potential $V(P, H)$ can be expressed in terms of the Higgs Gram vector $r$ as: \[ V(P, H) = \sum_{\mu} \xi_\mu r_\mu + \sum_{a, b} r_a r_b \eta_{ab} \] where $r$ is the Gram vector associated with the Higgs doublets, $\xi$ is the mass-parameter vector, and $\eta$ is the $4 \times 4$ symmetric Gram parameter matrix representing the quartic couplings.

A set of potential parameters $P$ in the Two Higgs Doublet Model (2HDM) is said to be stable if the associated potential function $V(P, H)$ is bounded from below. Formally, this condition is satisfied if there exists a real constant $c \in \mathbb{R}$ such that for all possible configurations of the Higgs doublets $H$, the potential satisfies: \[ V(P, H) \geq c \]

The potential of the Two Higgs Doublet Model (2HDM) with the parameters $P_{ce}$ defined by the stability counterexample (where $m_{11}^2 = m_{22}^2 = 0$, $m_{12}^2 = i$, $\lambda_1 = \lambda_2 = \lambda_3 = \lambda_4 = \lambda_5 = 2$, and $\lambda_6 = \lambda_7 = -2$) is not stable. That is, the potential function \[ V(\Phi_1, \Phi_2) = 2 \operatorname{Im}(\langle \Phi_1, \Phi_2 \rangle) + \|\Phi_1 - \Phi_2\|^4 \] is not bounded from below for Higgs doublets $\Phi_1, \Phi_2 \in \mathbb{C}^2$.

For a given set of potential parameters $P$ in the Two Higgs Doublet Model (2HDM), the reduced mass term is a real-valued function of a vector $\mathbf{k} \in \mathbb{R}^3$ (represented as an element of `EuclideanSpace ℝ (Fin 3)`). This function is equivalent to the term $J_2$ used in the stability analysis of the potential and is defined as: $$J_2(P, \mathbf{k}) = \xi_0 + \sum_{i=1}^3 \xi_i k_i$$ where $\xi_0$ is the component of the mass-term parameter vector $\xi_\mu$ corresponding to the index `Sum.inl 0`, and $\xi_1, \xi_2, \xi_3$ are the components corresponding to `Sum.inr` indices.

For any set of potential parameters $P$ in the Two Higgs Doublet Model (2HDM) and any vector $\mathbf{k} \in \mathbb{R}^3$ satisfying $\|\mathbf{k}\|^2 \leq 1$, the reduced mass term $J_2(P, \mathbf{k})$ is bounded below such that: $$\xi_0 - \sqrt{\sum_{i=1}^3 \xi_i^2} \leq J_2(P, \mathbf{k})$$ where $\xi_0$ is the scalar component of the mass-term parameter vector $\xi_\mu$ (indexed by `Sum.inl 0`) and $\xi_1, \xi_2, \xi_3$ are the spatial components (indexed by `Sum.inr`).

In the Two Higgs Doublet Model (2HDM), if the potential parameters $P$ are all zero, then the reduced mass term $J_2(P, \mathbf{k})$ is zero for all $\mathbf{k} \in \mathbb{R}^3$.

For the potential parameters $P$ of the stability counterexample in the Two Higgs Doublet Model (2HDM), the reduced mass term $J_2(P, \mathbf{k})$ evaluated for any vector $\mathbf{k} \in \mathbb{R}^3$ is equal to the second component of the vector, $k_1$.

The function `quarticTermReduced` (often denoted as $J_4$) evaluates the quartic part of the Two-Higgs-Doublet Model (2HDM) potential in the Gram vector formalism. Given the potential parameters $P$ and a vector $\mathbf{k} \in \mathbb{R}^3$, the function computes the value: $$J_4(P, \mathbf{k}) = \eta_{00} + 2 \sum_{i=1}^3 k_i \eta_{0i} + \sum_{i=1}^3 \sum_{j=1}^3 k_i k_j \eta_{ij}$$ where $\eta_{\mu\nu}$ is the symmetric $4 \times 4$ matrix of Gram parameters associated with $P$. Here, the index $0$ corresponds to the scalar component (`Sum.inl 0`) and indices $i, j \in \{1, 2, 3\}$ correspond to the vector components (`Sum.inr`). This function is used to determine the stability (boundedness from below) of the 2HDM potential.

If the potential parameters $P$ of the Two-Higgs-Doublet Model (2HDM) are all zero (i.e., all mass squared and quartic coupling parameters are zero), then the reduced quartic term $J_4(P, \mathbf{k})$ is zero for any vector $\mathbf{k} \in \mathbb{R}^3$.

For any vector $\mathbf{k} \in \mathbb{R}^3$, the reduced quartic term $J_4$ of the Two-Higgs-Doublet Model (2HDM) potential, evaluated at the specific parameters $P_{CE}$ designated as the stability counterexample, is given by: $$J_4(P_{CE}, \mathbf{k}) = (1 - k_1)^2$$ where $k_1$ denotes the first component of the vector $\mathbf{k}$.

For any vector $\mathbf{k} \in \mathbb{R}^3$, the reduced quartic term $J_4$ of the Two-Higgs-Doublet Model (2HDM) potential, evaluated at the stability counterexample parameters $P_{CE}$, is non-negative: $$0 \leq J_4(P_{CE}, \mathbf{k})$$ where $J_4$ (represented by `quarticTermReduced`) is the function evaluating the quartic part of the potential in the Gram vector formalism, and $P_{CE}$ (represented by `stabilityCounterExample`) is a specific set of parameters used to test stability conditions.

The potential of the Two Higgs Doublet Model (2HDM) with parameters $P$ is stable (i.e., bounded from below) if and only if there exists a real constant $c$ such that for all 4-vectors $K = (K_0, K_1, K_2, K_3) \in \mathbb{R}^4$ satisfying $K_0 \geq 0$ and $\sum_{i=1}^3 K_i^2 \leq K_0^2$, the following inequality holds: \[ c \leq \sum_{\mu=0}^3 \xi_\mu K_\mu + \sum_{a,b=0}^3 K_a K_b \eta_{ab} \] where $\xi_\mu$ are the mass-term parameters and $\eta_{ab}$ is the symmetric $4 \times 4$ matrix of quartic parameters derived from $P$.

The potential of the Two Higgs Doublet Model (2HDM) with parameters $P$ is stable (i.e., bounded from below) if and only if there exists a real constant $c$ such that for all $K_0 \in \mathbb{R}$ and all vectors $\mathbf{K} \in \mathbb{R}^3$ satisfying $K_0 \geq 0$ and $\|\mathbf{K}\|^2 \leq K_0^2$, the following inequality holds: \[ c \leq \xi_0 K_0 + \sum_{i=1}^3 \xi_i K_i + \eta_{00} K_0^2 + 2 K_0 \sum_{i=1}^3 \eta_{0i} K_i + \sum_{i=1}^3 \sum_{j=1}^3 \eta_{ij} K_i K_j \] where $\xi_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) are the mass-term parameters and $\eta_{\mu\nu}$ are the entries of the symmetric $4 \times 4$ matrix of quartic parameters derived from $P$.

The potential of the Two Higgs Doublet Model (2HDM) with parameters $P$ is stable (i.e., bounded from below) if and only if there exists a non-positive real constant $c \leq 0$ such that for all $K_0 \in \mathbb{R}$ and all vectors $\mathbf{K} \in \mathbb{R}^3$ satisfying $K_0 > 0$ and $\|\mathbf{K}\|^2 \leq K_0^2$, the following inequality holds: \[ c \leq \xi_0 K_0 + \sum_{i=1}^3 \xi_i K_i + \eta_{00} K_0^2 + 2 K_0 \sum_{i=1}^3 \eta_{0i} K_i + \sum_{i=1}^3 \sum_{j=1}^3 \eta_{ij} K_i K_j \] where $\xi_\mu$ (for $\mu \in \{0, 1, 2, 3\}$) are the mass-term parameters and $\eta_{\mu\nu}$ are the entries of the symmetric $4 \times 4$ matrix of quartic parameters derived from $P$.

The potential of the Two Higgs Doublet Model (2HDM) with parameters $P$ is stable (i.e., bounded from below) if and only if there exists a non-positive real constant $c \leq 0$ such that for all $K_0 \in \mathbb{R}$ and all vectors $\mathbf{k} \in \mathbb{R}^3$ satisfying $K_0 > 0$ and $\|\mathbf{k}\|^2 \leq 1$, the following inequality holds: \[ c \leq K_0 J_2(P, \mathbf{k}) + K_0^2 J_4(P, \mathbf{k}) \] where $J_2(P, \mathbf{k})$ is the reduced mass term and $J_4(P, \mathbf{k})$ is the reduced quartic term of the potential.

For a set of potential parameters $P$ in the Two Higgs Doublet Model (2HDM), if the potential is stable (i.e., bounded from below), then for any vector $\mathbf{k} \in \mathbb{R}^3$ satisfying $\|\mathbf{k}\|^2 \leq 1$, the reduced quartic term $J_4(P, \mathbf{k})$ is non-negative, satisfying: \[ J_4(P, \mathbf{k}) \geq 0 \] where $J_4(P, \mathbf{k})$ is defined in the Gram vector formalism as $J_4(P, \mathbf{k}) = \eta_{00} + 2 \sum_{i=1}^3 k_i \eta_{0i} + \sum_{i=1}^3 \sum_{j=1}^3 k_i k_j \eta_{ij}$ for the quartic parameters $\eta_{\mu\nu}$.

The potential of the Two Higgs Doublet Model (2HDM) with parameters $P$ is stable (i.e., bounded from below) if and only if there exists a non-negative real constant $c \geq 0$ such that for every vector $\mathbf{k} \in \mathbb{R}^3$ satisfying $\|\mathbf{k}\|^2 \leq 1$, the reduced quartic term $J_4(P, \mathbf{k})$ is non-negative and, whenever the reduced mass term $J_2(P, \mathbf{k})$ is negative, the following inequality holds: \[ J_2(P, \mathbf{k})^2 \leq 4 c J_4(P, \mathbf{k}) \]

For any set of potential parameters $P$ in the Two Higgs Doublet Model (2HDM), if the potential is stable (i.e., bounded from below), then for any vector $\mathbf{k} \in \mathbb{R}^3$ satisfying $\|\mathbf{k}\|^2 \leq 1$, the condition that the reduced quartic term vanishes ($J_4(P, \mathbf{k}) = 0$) implies that the reduced mass term must be non-negative ($J_2(P, \mathbf{k}) \geq 0$).

For any set of potential parameters $P$ in the Two Higgs Doublet Model (2HDM), if the reduced quartic term $J_4(P, \mathbf{k})$ is strictly positive for all vectors $\mathbf{k} \in \mathbb{R}^3$ such that $\|\mathbf{k}\|^2 \leq 1$, then the potential is stable (i.e., it is bounded from below).

There exists a set of potential parameters $P$ for the Two Higgs Doublet Model (2HDM) such that the potential is not stable (i.e., it is not bounded from below), even though for all vectors $\mathbf{k} \in \mathbb{R}^3$ in the unit ball ($\|\mathbf{k}\|^2 \leq 1$), the following conditions are satisfied: 1. The reduced quartic term $J_4(P, \mathbf{k})$ is non-negative: $J_4(P, \mathbf{k}) \geq 0$. 2. If the reduced quartic term is zero ($J_4(P, \mathbf{k}) = 0$), then the reduced mass term $J_2(P, \mathbf{k})$ is non-negative: $J_2(P, \mathbf{k}) \geq 0$.