Physlib.Particles.StandardModel.HiggsBoson.EffectivePotential

The type of general effective potentials for the Higgs field, defined as the set of real-valued functions $V: \text{HiggsVec} \to \mathbb{R}$, where $\text{HiggsVec}$ is the 2-dimensional complex vector space $\mathbb{C}^2$.

The property `IsInvariant` defines the gauge symmetry of an effective potential $V: \mathbb{C}^2 \to \mathbb{R}$. It states that for every element $g$ in the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ and every Higgs field value $\phi \in \mathbb{C}^2$, the potential remains unchanged under the gauge action: $$V(g \cdot \phi) = V(\phi)$$ where the action $g \cdot \phi$ is defined by the transformation $g_1^3 (g_2 \phi)$ with $g_2 \in SU(2)$ and $g_1 \in U(1)$.

Let $V: \mathbb{C}^2 \to \mathbb{R}$ be an effective potential of the Higgs field. If $V$ is gauge invariant (meaning $V(g \cdot \phi) = V(\phi)$ for all elements $g$ of the Standard Model gauge group $\mathcal{G}$), then there exists a function $f: \mathbb{R} \to \mathbb{R}$ such that $V = f \circ \|\cdot\|$, or equivalently $V(\phi) = f(\|\phi\|)$ for all $\phi \in \mathbb{C}^2$, where $\|\cdot\|$ denotes the standard Euclidean norm on $\mathbb{C}^2$.

The effective potential $V: \text{HiggsVec} \to \mathbb{R}$ is said to have a maximum mass dimension less than or equal to $n$ if there exists a real multivariate polynomial $p$ in four variables such that for every $\phi \in \text{HiggsVec}$, the potential $V(\phi)$ is equal to the evaluation of $p$ on the four real components of $\phi$, and the total degree of $p$ is at most $n$. Here, $\text{HiggsVec}$ is the 2-dimensional complex vector space $\mathbb{C}^2$ representing the Higgs doublet.

Given an effective potential $V: \text{HiggsVec} \to \mathbb{R}$ that has a maximum mass dimension at most $n$, where $\text{HiggsVec} \cong \mathbb{C}^2$ is the 2-dimensional complex vector space of the Higgs field, this definition provides the corresponding multivariate polynomial $p \in \mathbb{R}[x_1, x_2, x_3, x_4]$ in four real variables. This polynomial represents the potential such that $V(\phi) = p(\text{Re}(\phi_1), \text{Im}(\phi_1), \text{Re}(\phi_2), \text{Im}(\phi_2))$, and its total degree is at most $n$.

Let $V$ be an effective potential for the Higgs field $\phi \in \text{HiggsVec} \cong \mathbb{C}^2$. If $V$ has a maximum mass dimension at most $n \in \mathbb{N}$, then the total degree of its corresponding real multivariate polynomial $p(x_1, x_2, x_3, x_4)$ satisfies $\text{deg}(p) \le n$.

Let $V: \text{HiggsVec} \to \mathbb{R}$ be an effective potential of the Higgs field with maximum mass dimension at most $n \in \mathbb{N}$, and let $p$ be its associated multivariate polynomial in four real variables. For any Higgs doublet $\phi \in \text{HiggsVec}$, the value of the potential $V(\phi)$ is equal to the evaluation of the polynomial $p$ on the four real components of $\phi$, namely $(\text{Re}(\phi_1), \text{Im}(\phi_1), \text{Re}(\phi_2), \text{Im}(\phi_2))$.

Given an effective potential $V: \text{HiggsVec} \to \mathbb{R}$ with a maximum mass dimension at most $n$, where $\text{HiggsVec} \cong \mathbb{C}^2$ is the space of the Higgs doublet, the function `termOfMassDim V h m` represents the part of the potential with exactly mass dimension $m \in \mathbb{N}$. It is defined by taking the $m$-th homogeneous component of the multivariate polynomial $p \in \mathbb{R}[x_1, x_2, x_3, x_4]$ associated with $V$ and evaluating it on the four real components of the Higgs field $\phi = (\phi_1, \phi_2) \in \mathbb{C}^2$.

Let $V: \text{HiggsVec} \to \mathbb{R}$ be an effective potential of the Higgs field with maximum mass dimension at most $n \in \mathbb{N}$, where $\text{HiggsVec} \cong \mathbb{C}^2$. If $m$ is a natural number such that $n < m$, then the term of mass dimension $m$ of the potential $V$ (denoted as `termOfMassDim V h m`) is zero for any Higgs doublet $\phi \in \text{HiggsVec}$.

Let $V$ be an effective potential of the Higgs field with maximum mass dimension at most $n$. For any natural number $m$, the term of mass dimension $m$ of the potential, denoted as $V_m$, is homogeneous of degree $m$. That is, for any Higgs field vector $\phi \in \text{HiggsVec}$ and real scalar $t \in \mathbb{R}$, $$V_m(t \phi) = t^m V_m(\phi).$$

Let $V: \text{HiggsVec} \to \mathbb{R}$ be an effective potential for the Higgs field, where $\text{HiggsVec} \cong \mathbb{C}^2$ is the 2-dimensional complex vector space of the Higgs doublet. If $V$ has a maximum mass dimension at most $n \in \mathbb{N}$ (i.e., it can be represented as a polynomial of degree at most $n$ in the real components of the field), then for any $\phi \in \text{HiggsVec}$, the potential is equal to the sum of its terms of mass dimension $m$ for $m \in \{0, \dots, n\}$: $$V(\phi) = \sum_{m=0}^n V_m(\phi)$$ where $V_m$ denotes the component of the potential with exactly mass dimension $m$.

Let $V: \text{HiggsVec} \to \mathbb{R}$ be an effective potential of the Higgs field with maximum mass dimension less than or equal to $n$, where $\text{HiggsVec} \cong \mathbb{C}^2$. For any Higgs field vector $\phi \in \text{HiggsVec}$ and any real scalar $t \in \mathbb{R}$, the potential evaluated at the scaled field $t \phi$ satisfies $$V(t \phi) = \sum_{m=0}^{n} t^m V_m(\phi)$$ where $V_m$ denotes the $m$-th mass dimension term of the potential $V$.

Suppose $V: \mathbb{C}^2 \to \mathbb{R}$ is an effective Higgs potential with a maximum mass dimension at most $n$. If $V$ is gauge invariant, meaning $V(g \cdot \phi) = V(\phi)$ for all elements $g$ of the gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ and Higgs fields $\phi \in \mathbb{C}^2$, then for any $m \in \mathbb{N}$, the term of the potential with mass dimension $m$ is also gauge invariant.

Let $V: \mathbb{C}^2 \to \mathbb{R}$ be an effective potential for the Higgs field with a maximum mass dimension at most $n$. If $V$ is gauge invariant under the Standard Model gauge group, then for any $m \in \mathbb{N}$, there exists a constant $c \in \mathbb{R}$ such that for every Higgs field value $\phi \in \mathbb{C}^2$, the term of mass dimension $m$ in the potential, denoted $V_m(\phi)$, is given by $$V_m(\phi) = c \|\phi\|^m$$ where $\|\phi\|$ is the norm of the Higgs doublet in the 2-dimensional complex vector space $\mathbb{C}^2$.

Let $V: \mathbb{C}^2 \to \mathbb{R}$ be an effective potential of the Higgs field that is gauge-invariant and has maximum mass dimension at most $n$. For any odd natural number $m$, the term of mass dimension $m$ in the potential vanishes for all $\phi \in \mathbb{C}^2$, i.e., $$V_m(\phi) = 0.$$

Let $V: \mathbb{C}^2 \to \mathbb{R}$ be an effective Higgs potential. Suppose $V$ is invariant under the gauge group action (i.e., $V(g \cdot \phi) = V(\phi)$ for all $g \in SU(2) \times U(1)$) and has a maximum mass dimension less than or equal to $n$ (meaning $V$ is a polynomial in the real components of the Higgs field $\phi$ with total degree at most $n$). Then, for any $\phi \in \mathbb{C}^2$, the potential $V(\phi)$ is equal to the sum of its homogeneous components of even mass dimension $2m$: $$V(\phi) = \sum_{m=0}^{\lfloor n/2 \rfloor} V_{2m}(\phi)$$ where $V_{2m}(\phi)$ denotes the part of the potential with exactly mass dimension $2m$.

Let $V: \mathbb{C}^2 \to \mathbb{R}$ be an effective potential for the Higgs field. Suppose $V$ is invariant under the gauge group action and has a maximum mass dimension at most $n$. For any $\phi \in \mathbb{C}^2$, the potential $V(\phi)$ can be expressed as the sum of its terms with even mass dimension $2m$ for $m \in \{0, 1, \dots, \lfloor n/2 \rfloor\}$: $$V(\phi) = \sum_{m=0}^{\lfloor n/2 \rfloor} V_{2m}(\phi)$$ where $V_{2m}$ denotes the part of the potential with exactly mass dimension $2m$.

Let $V: \mathbb{C}^2 \to \mathbb{R}$ be an effective potential for the Higgs field. Suppose $V$ is invariant under the gauge group action (i.e., $V(g \cdot \phi) = V(\phi)$ for all $g \in SU(2) \times U(1)$) and has a maximum mass dimension less than or equal to $n$ (meaning $V$ is a polynomial in the real components of the Higgs field $\phi$ with total degree at most $n$). Then, for any $\phi \in \mathbb{C}^2$, there exist real coefficients $c_m$ for $m \in \{0, 1, \dots, \lfloor n/2 \rfloor\}$ such that the potential can be expressed as a sum of even powers of the norm $\|\phi\|$: $$V(\phi) = \sum_{m=0}^{\lfloor n/2 \rfloor} c_m \|\phi\|^{2m}$$