Physlib.Particles.StandardModel.HiggsBoson.Basic

The space $\text{HiggsVec}$ is defined as the 2-dimensional complex Euclidean space $\mathbb{C}^2$. This space serves as the target space for the Higgs field, meaning that at any given spacetime point, the value of a Higgs field is an element of $\text{HiggsVec}$.

The continuous linear map over $\mathbb{R}$ from the Higgs vector space $\text{HiggsVec}$ to the space of functions from $\{0, 1\}$ to $\mathbb{C}$ (denoted as $\text{Fin } 2 \to \mathbb{C}$), which maps a vector $x \in \text{HiggsVec}$ to its representation as a complex-valued function via casting.

The map $\text{toFin2ℂ} : \text{HiggsVec} \to (\text{Fin } 2 \to \mathbb{C})$, which casts vectors from the 2-dimensional complex Higgs vector space $\text{HiggsVec}$ to the space of complex-valued functions on $\{0, 1\}$, is smooth (infinitely differentiable).

The definition provides the standard orthonormal basis $\{e_i\}_{i \in \{0, 1\}}$ for the Higgs vector space $\text{HiggsVec} \cong \mathbb{C}^2$. This basis consists of vectors $e_0 = (1, 0)$ and $e_1 = (0, 1)$ such that their inner product satisfies $\langle e_i, e_j \rangle = \delta_{ij}$.

For any real number $a$, the function returns a vector in the Higgs vector space $\text{HiggsVec}$ (isomorphic to $\mathbb{C}^2$) defined by the components $(\sqrt{a}, 0)$. This vector is constructed such that its squared norm $\|(\sqrt{a}, 0)\|^2$ is equal to $a$ (for $a \ge 0$).

For any non-negative real number $a \in \mathbb{R}$ ($a \ge 0$), let $\text{ofReal}(a)$ be the vector in the Higgs vector space $\text{HiggsVec} \cong \mathbb{C}^2$ defined by the components $(\sqrt{a}, 0)$. The squared norm of this vector satisfies $\|\text{ofReal}(a)\|^2 = a$.

This definition establishes the group action (scalar multiplication) of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the Higgs vector space $\text{HiggsVec} \cong \mathbb{C}^2$. For an element $g = (g_3, g_2, g_1) \in \mathcal{G}$ and a Higgs vector $\phi \in \mathbb{C}^2$, the action is defined as: $$g \cdot \phi = g_1^3 (g_2 \phi)$$ where $g_2 \in SU(2)$ acts on $\phi$ via standard matrix-vector multiplication, $g_1 \in U(1)$ acts as a complex phase multiplication raised to the third power, and the $SU(3)$ component $g_3$ acts trivially.

For any element $g$ of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ and any vector $\phi$ in the Higgs vector space $\text{HiggsVec} \cong \mathbb{C}^2$, the action of $g$ on $\phi$ is given by $g \cdot \phi = g_1^3 (g_2 \phi)$, where $g_1 \in U(1)$ and $g_2 \in SU(2)$ are the respective projections of $g$ onto its $U(1)$ and $SU(2)$ components. Here, $g_2 \phi$ denotes matrix-vector multiplication.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and $\text{HiggsVec} \cong \mathbb{C}^2$ be the Higgs vector space. For any element $g \in \mathcal{G}$ and any Higgs vector $\phi \in \text{HiggsVec}$, the action of $g$ on $\phi$ satisfies $g \cdot \phi = g_2 (g_1^3 \phi)$, where $g_1 \in U(1)$ and $g_2 \in SU(2)$ are the projections of $g$ onto its $U(1)$ and $SU(2)$ components, respectively. Here, $g_1^3$ acts on the vector $\phi$ via scalar multiplication, and $g_2$ acts on the resulting vector via matrix-vector multiplication.

For any element $g$ of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ and any vector $\phi$ in the Higgs vector space $\text{HiggsVec} \cong \mathbb{C}^2$, the action of $g$ on $\phi$ is given by $g \cdot \phi = (g_1^3 g_2) \phi$. Here, $g_1 \in U(1)$ and $g_2 \in SU(2)$ are the projections of $g$ onto its respective components, and $(g_1^3 g_2) \phi$ denotes the application of the scalar-multiplied matrix $g_1^3 g_2$ to the vector $\phi$.

The definition establishes that the action of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the Higgs vector space $\text{HiggsVec} \cong \mathbb{C}^2$ satisfies the axioms of a group action. Specifically, for any Higgs vector $\phi \in \text{HiggsVec}$ and gauge group elements $g, h \in \mathcal{G}$, the action satisfies the identity property $1 \cdot \phi = \phi$ and the associativity property $(g \cdot h) \cdot \phi = g \cdot (h \cdot \phi)$.

This definition establishes that the action of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the Higgs vector space $\text{HiggsVec} \cong \mathbb{C}^2$ is a distributive multiplicative action. This means that for any gauge transformation $g \in \mathcal{G}$ and any Higgs vectors $\phi, \psi \in \text{HiggsVec}$, the action distributes over addition, $g \cdot (\phi + \psi) = g \cdot \phi + g \cdot \psi$, and the action on the zero vector results in the zero vector, $g \cdot 0 = 0$.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and let $\text{HiggsVec} \cong \mathbb{C}^2$ be the Higgs vector space. For any gauge group element $g \in \mathcal{G}$ and any Higgs vectors $\phi, \psi \in \text{HiggsVec}$, the complex inner product is invariant under the action of the gauge group, such that: $$\langle g \cdot \phi, g \cdot \psi \rangle_{\mathbb{C}} = \langle \phi, \psi \rangle_{\mathbb{C}}$$ where $g \cdot \phi$ denotes the group action of $g$ on the vector $\phi$.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and let $\text{HiggsVec} \cong \mathbb{C}^2$ be the Higgs vector space. For any gauge group element $g \in \mathcal{G}$ and any Higgs vector $\phi \in \text{HiggsVec}$, the norm of the vector is invariant under the action of the gauge group, such that: $$\|g \cdot \phi\| = \|\phi\|$$ where $g \cdot \phi$ denotes the group action of $g$ on the vector $\phi$.

Given a Higgs vector $\phi = (\phi_0, \phi_1) \in \mathbb{C}^2$, this function returns an element of the Standard Model gauge group $G = SU(3) \times SU(2) \times U(1)$. For a non-zero vector $\phi$, the $SU(3)$ and $U(1)$ components of the returned element are identity matrices, while the $SU(2)$ component is defined by the matrix $$M = \frac{1}{\|\phi\|} \begin{pmatrix} \bar{\phi}_0 & \bar{\phi}_1 \\ -\phi_1 & \phi_0 \end{pmatrix}$$ where $\bar{z}$ denotes the complex conjugate of $z$. This group element $g$ has the property that its action on $\phi$ rotates the vector to the form $(\|\phi\|, 0)$. If $\phi = 0$, the function returns the identity element of the gauge group.

For any Higgs vector $\phi$ in the 2-dimensional complex vector space $\text{HiggsVec} \cong \mathbb{C}^2$, the action of the gauge group element $g_\phi = \text{toRealGroupElem}(\phi)$ on the vector $\phi$ results in the vector $(\|\phi\|, 0)$. In the formalization, this is expressed as $g_\phi \cdot \phi = \text{ofReal}(\|\phi\|^2)$, where $\text{ofReal}(a)$ is defined as the Higgs vector $(\sqrt{a}, 0)$.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and let $\text{HiggsVec} \cong \mathbb{C}^2$ be the Higgs vector space. For any two Higgs vectors $\phi, \psi \in \text{HiggsVec}$, $\psi$ belongs to the orbit of $\phi$ under the action of $\mathcal{G}$ if and only if their norms are equal: $$\psi \in \text{orb}_{\mathcal{G}}(\phi) \iff \|\psi\| = \|\phi\|$$

The stability group of a non-zero Higgs vector represented by the element $\begin{pmatrix} 0 \\ \|\phi\| \end{pmatrix}$ under the action of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ is the $SU(3) \times U(1)$ subgroup. An element $(g, e^{i\theta}) \in SU(3) \times U(1)$ is embedded into the gauge group via the mapping $(g, \text{diag}(e^{3i\theta}, e^{-3i\theta}), e^{i\theta})$.

The stability group of the Higgs vector space $V_{\text{Higgs}}$ under the action of the Standard Model gauge group $G = SU(3) \times SU(2) \times U(1)$ is given by the subgroup $SU(3) \times \mathbb{Z}_6$. Here, $\mathbb{Z}_6$ is the subgroup of $SU(2) \times U(1)$ consisting of elements of the form $(\alpha^{-3} I_2, \alpha)$, where $I_2$ is the $2 \times 2$ identity matrix and $\alpha$ is a sixth root of unity (i.e., $\alpha^6 = 1$). This subgroup represents the set of all gauge transformations that act as the identity on every Higgs vector $\phi \in V_{\text{Higgs}}$.

For any unitary complex number $u \in U(1)$ and any Higgs vector $\phi \in \text{HiggsVec} \cong \mathbb{C}^2$, the action of the gauge group element $(I_3, \text{diag}(\bar{u}^3, u^3), u)$—which represents the inclusion of $u$ into the Standard Model gauge group—on $\phi$ is given by the matrix multiplication: $$u \cdot \phi = \begin{pmatrix} 1 & 0 \\ 0 & u^6 \end{pmatrix} \begin{pmatrix} \phi_0 \\ \phi_1 \end{pmatrix}$$ where $\phi_0$ and $\phi_1$ are the complex components of the Higgs vector $\phi$.

For any Higgs vector $\phi = (\phi_0, \phi_1) \in \mathbb{C}^2$, there exists an element $g$ of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ such that: 1. The second component of the transformed vector $g \cdot \phi$ is equal to the norm of the original second component, i.e., $(g \cdot \phi)_1 = |\phi_1|$, which is a non-negative real number. 2. The action of $g$ preserves the first component of any Higgs vector $\psi \in \mathbb{C}^2$, i.e., $(g \cdot \psi)_0 = \psi_0$ for all $\psi$. 3. For any real number $a$, the Higgs vector $(a, 0)$ is fixed under the action of $g$, i.e., $g \cdot (a, 0) = (a, 0)$.

The Higgs bundle is defined as the trivial vector bundle over the spacetime manifold $M$ with the fiber being the Higgs vector space $V \cong \mathbb{C}^2$. As a smooth manifold, it is represented as the Cartesian product $M \times V$, which corresponds to $\mathbb{R}^4 \times \mathbb{C}^2$.

The Higgs bundle $\text{HiggsBundle}$ is a smooth vector bundle of class $C^\infty$ over the spacetime manifold $M$ (modeled as the manifold of Lorentz vectors with spatial dimension 3), where the fiber at each point is the 2-dimensional complex vector space $\text{HiggsVec} \cong \mathbb{C}^2$.

The type `HiggsField` represents the space of smooth ($C^\infty$) sections of the Higgs bundle. Since the Higgs bundle is a trivial vector bundle over the spacetime manifold $M$ (modeled as $\mathbb{R}^4$) with fiber $V \cong \mathbb{C}^2$, a Higgs field is equivalent to a smooth map $\phi: M \to \mathbb{C}^2$ that assigns a 2-dimensional complex vector to each point in spacetime.

The function is an $\mathbb{R}$-linear map that takes a vector $v \in \text{HiggsVec} \cong \mathbb{C}^2$ and returns a constant Higgs field $\phi \in \text{HiggsField}$. For any point $x$ in spacetime, the value of this field is $\phi(x) = v$. This construction defines the constant section of the Higgs bundle associated with the vector $v$.

For any Higgs vector $\phi \in \text{HiggsVec}$ (the 2-dimensional complex vector space $\mathbb{C}^2$) and any point $x$ in spacetime, the constant Higgs field $\text{const}(\phi)$ evaluated at $x$ returns the vector $\phi$. That is, $\text{const}(\phi)(x) = \phi$.

For a Higgs field $\phi$, which is defined as a smooth section of the Higgs bundle, the function $\text{toHiggsVec}(\phi)$ represents the underlying mapping $\phi: \text{SpaceTime} \to \text{HiggsVec}$ that assigns a 2-dimensional complex vector to each point in spacetime.

For any Higgs field $\phi \in \text{HiggsField}$, the corresponding map $\phi: \text{SpaceTime} \to \text{HiggsVec}$ that assigns a Higgs vector to each point in spacetime is smooth ($C^\infty$) with respect to the standard manifold structures on $\text{SpaceTime}$ and $\text{HiggsVec}$.

For any Higgs field $\phi$ and any point $x$ in spacetime, let $\phi(x) \in \text{HiggsVec}$ be the value of the field at that point. If $\text{const}(v)$ denotes the constant Higgs field that assigns the vector $v$ to every point in spacetime, then evaluating the constant field generated by the value $\phi(x)$ at the point $x$ yields $(\text{const}(\phi(x)))(x) = \phi(x)$.

For any Higgs field $\phi$, which is defined as a smooth section of the Higgs bundle over spacetime, the underlying map $\phi: \text{SpaceTime} \to \text{HiggsVec}$ that assigns a 2-dimensional complex vector to each point in spacetime is equal to $\phi$.

For any Higgs field $\phi$, the underlying map $\phi: \text{SpaceTime} \to \text{HiggsVec}$ is an infinitely differentiable ($C^\infty$) function with respect to the real numbers $\mathbb{R}$.

For any Higgs field $\phi$, the map $\phi : \text{SpaceTime} \to \mathbb{C}^2$ is smooth ($C^\infty$) with respect to the standard manifold structures on spacetime and the 2-dimensional complex Euclidean space $\mathbb{C}^2$.

For any Higgs field $\phi$, which is a smooth section of the Higgs bundle, each component map $\phi_i: \text{SpaceTime} \to \mathbb{C}$ (where $i \in \{0, 1\}$) is infinitely differentiable ($C^\infty$) with respect to the standard smooth manifold structures on spacetime and the complex numbers.

For any Higgs field $\phi$ and any index $i \in \{0, 1\}$, the function $x \mapsto \text{Re}(\phi_i(x))$, which maps each point $x$ in spacetime to the real part of the $i$-th complex component of the Higgs field, is smooth ($C^\infty$) from spacetime to the real numbers $\mathbb{R}$.

For any Higgs field $\phi$ and any index $i \in \{0, 1\}$, the function $x \mapsto \text{Im}(\phi_i(x))$, which maps each point $x$ in spacetime to the imaginary part of the $i$-th complex component of the Higgs field, is smooth ($C^\infty$) from spacetime to the real numbers $\mathbb{R}$.

The inner product of two Higgs fields $\phi_1, \phi_2$ is defined as a complex-valued function on spacetime, where for each point $x \in \text{SpaceTime}$, the value is given by the standard Hermitian inner product $\langle \phi_1(x), \phi_2(x) \rangle_{\mathbb{C}^2}$ of the field values in the Higgs vector space $\mathbb{C}^2$.

For any two Higgs fields $\phi_1, \phi_2$ and any point $x$ in spacetime, the pointwise inner product $\langle \phi_1, \phi_2 \rangle$ evaluated at $x$ is equal to the Hermitian inner product of the vectors $\phi_1(x)$ and $\phi_2(x)$ in the Higgs vector space $\mathbb{C}^2$: $$\langle \phi_1, \phi_2 \rangle(x) = \langle \phi_1(x), \phi_2(x) \rangle_{\mathbb{C}^2}$$

For any two Higgs fields $\phi_1$ and $\phi_2$, their pointwise inner product $\langle \phi_1, \phi_2 \rangle: \text{SpaceTime} \to \mathbb{C}$ is given by the expansion: $$\langle \phi_1, \phi_2 \rangle(x) = \sum_{j=0}^1 \left( \text{Re}(\phi_{1,j}(x)) \text{Re}(\phi_{2,j}(x)) + \text{Im}(\phi_{1,j}(x)) \text{Im}(\phi_{2,j}(x)) \right) + i \sum_{j=0}^1 \left( \text{Re}(\phi_{1,j}(x)) \text{Im}(\phi_{2,j}(x)) - \text{Im}(\phi_{1,j}(x)) \text{Re}(\phi_{2,j}(x)) \right)$$ where $x$ is a point in spacetime, $\phi_{i,j}(x)$ denotes the $j$-th complex component of the Higgs field $\phi_i$ at that point, and $\text{Re}(\cdot)$ and $\text{Im}(\cdot)$ denote the real and imaginary parts of a complex number, respectively.

For any two Higgs fields $\phi_1$ and $\phi_2$ and any point $x$ in spacetime, the pointwise inner product $\langle \phi_1, \phi_2 \rangle$ evaluated at $x$ is given by the sum of the products of the complex conjugates of the components of $\phi_1(x)$ and the components of $\phi_2(x)$: $$\langle \phi_1, \phi_2 \rangle(x) = \overline{\phi_1(x)_0} \phi_2(x)_0 + \overline{\phi_1(x)_1} \phi_2(x)_1$$ where $\phi_i(x)_j$ denotes the $j$-th complex component of the Higgs field $\phi_i$ at point $x$, and $\overline{z}$ denotes the complex conjugate of $z$.

For any two Higgs fields $\phi_1$ and $\phi_2$, the pointwise complex-valued inner product satisfies the conjugate symmetry property: $$\overline{\langle \phi_2, \phi_1 \rangle} = \langle \phi_1, \phi_2 \rangle$$ where $\langle \phi_1, \phi_2 \rangle$ denotes the function mapping each point $x$ in spacetime to the Hermitian inner product of the field values $\phi_1(x)$ and $\phi_2(x)$ in the Higgs vector space $\mathbb{C}^2$.

For any three Higgs fields $\phi_1, \phi_2, \phi_3$, the pointwise inner product of the sum $\phi_1 + \phi_2$ with $\phi_3$ is equal to the sum of their individual pointwise inner products: $$\langle \phi_1 + \phi_2, \phi_3 \rangle = \langle \phi_1, \phi_3 \rangle + \langle \phi_2, \phi_3 \rangle$$ where each term represents a complex-valued function on spacetime, defined by the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$ at each point $x \in \text{SpaceTime}$.

For any three Higgs fields $\phi_1, \phi_2, \phi_3$ (smooth sections of the Higgs bundle over spacetime), the pointwise inner product satisfies the additivity property in the second argument: $$\langle \phi_1, \phi_2 + \phi_3 \rangle = \langle \phi_1, \phi_2 \rangle + \langle \phi_1, \phi_3 \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the pointwise Hermitian inner product in the complex vector space $\mathbb{C}^2$, resulting in a complex-valued function on spacetime.

For any Higgs field $\phi$, the pointwise inner product of the zero Higgs field with $\phi$ is the zero function on spacetime: $\langle 0, \phi \rangle = 0$. Here, the zero Higgs field is the smooth section that is zero at every point in spacetime, and the inner product $\langle \cdot, \cdot \rangle$ is defined such that at each point $x \in \text{SpaceTime}$, $\langle 0, \phi \rangle(x) = \langle 0, \phi(x) \rangle_{\mathbb{C}^2}$.

For any Higgs field $\phi$, the pointwise inner product of $\phi$ with the zero Higgs field $0$ is the zero function from spacetime to the complex numbers, denoted as $\langle \phi, 0 \rangle = 0$. Here, the inner product is defined such that at each point $x$ in spacetime, $\langle \phi, 0 \rangle(x) = \langle \phi(x), 0 \rangle_{\mathbb{C}^2} = 0$.

For any two Higgs fields $\phi_1$ and $\phi_2$, the pointwise inner product of $-\phi_1$ and $\phi_2$ is equal to the negation of the pointwise inner product of $\phi_1$ and $\phi_2$. That is, $$\langle -\phi_1, \phi_2 \rangle = -\langle \phi_1, \phi_2 \rangle$$ where $\phi_1$ and $\phi_2$ are smooth sections of the Higgs bundle, and the result is a complex-valued function on spacetime.

For any two Higgs fields $\phi_1$ and $\phi_2$, the pointwise inner product of $\phi_1$ and the negative of $\phi_2$ is equal to the negative of the pointwise inner product of $\phi_1$ and $\phi_2$, which is expressed as $\langle \phi_1, -\phi_2 \rangle = -\langle \phi_1, \phi_2 \rangle$.

For any two Higgs fields $\phi_1$ and $\phi_2$, their pointwise Hermitian inner product $\langle \phi_1, \phi_2 \rangle: \text{SpaceTime} \to \mathbb{C}$ is a smooth ($C^\infty$) function from spacetime to the complex numbers.

For a Higgs field $\phi$, the function $\text{normSq } \phi: \text{SpaceTime} \to \mathbb{R}$ is defined as the pointwise squared norm of the field, given by $x \mapsto \|\phi(x)\|^2$. At each point $x$ in spacetime, the value is the square of the Euclidean norm of the Higgs vector $\phi(x) \in \mathbb{C}^2$. This function is denoted by the notation $\|\phi\|_H^2$.

This notation denotes the pointwise squared norm of a Higgs field $\phi$, represented as $\|\phi\|_H^2$. It maps each point in spacetime to the square of the norm of the Higgs vector at that point, defined by the function $\text{normSq}(\phi)$.

Let $\phi$ be a Higgs field, which is a smooth section of the Higgs bundle over spacetime $M$. For any point $x \in M$, the pointwise Hermitian inner product of the field with itself at $x$, denoted by $\langle \phi, \phi \rangle(x)$, is equal to the squared norm of the field at that point, denoted by $\|\phi\|_H^2(x) = \|\phi(x)\|^2$.

For any Higgs field $\phi$, which is a smooth section of the Higgs bundle over spacetime $M$, and for any point $x \in M$, the pointwise squared norm of the field $\|\phi\|_H^2(x)$ is equal to the real part of the pointwise Hermitian inner product of the field with itself $\langle \phi, \phi \rangle(x)$: $$\|\phi\|_H^2(x) = \text{Re}(\langle \phi, \phi \rangle(x))$$ where the inner product is the standard Hermitian inner product on the Higgs vector space $\mathbb{C}^2$.

For any Higgs field $\phi$ and any point $x$ in spacetime, the pointwise squared norm $\|\phi\|_H^2(x)$ is given by the real part of the sum of the products of the complex conjugates of the field components and the components themselves: $$\|\phi\|_H^2(x) = \text{Re}\left( \overline{\phi(x)_0} \phi(x)_0 + \overline{\phi(x)_1} \phi(x)_1 \right)$$ where $\phi(x)_i$ denotes the $i$-th complex component of the Higgs field at the point $x$, and $\overline{z}$ denotes the complex conjugate of $z$.

For any Higgs field $\phi$ and any point $x$ in spacetime, the pointwise squared norm of the field, denoted by $\|\phi(x)\|_H^2$, is non-negative: $$0 \leq \|\phi(x)\|_H^2$$ where $\phi(x) \in \mathbb{C}^2$ is the value of the Higgs field at $x$.

The pointwise squared norm $\|0\|_H^2$ of the zero Higgs field is equal to zero at every point in spacetime.

For any Higgs field $\phi$, the pointwise squared norm function $\|\phi\|_H^2: \text{SpaceTime} \to \mathbb{R}$, which maps each point $x$ in spacetime to the squared Euclidean norm of the field $\|\phi(x)\|^2$, is a smooth ($C^\infty$) function.

For any Higgs vector $\phi \in \text{HiggsVec}$ and any spacetime point $x \in \text{SpaceTime}$, the pointwise squared norm of the constant Higgs field $\text{const } \phi$ evaluated at $x$ is equal to the squared norm of the vector $\phi$: $$\|\text{const } \phi\|_H^2(x) = \|\phi\|^2$$

The gauge action on a Higgs field $\phi$ is defined pointwise using the representation of the gauge group on the Higgs vector space. For a gauge transformation $g$ and a Higgs field $\phi$, the resulting field $g \cdot \phi$ is given by $(g \cdot \phi)(x) = \rho(g)(\phi(x))$ for every point $x$ in $\text{SpaceTime}$, where $\rho$ is the representation of the gauge group on $\text{HiggsVec}$.

Two Higgs fields $\phi$ and $\phi'$ are in the same gauge orbit—meaning there exists a gauge transformation $g$ such that $(g \cdot \phi)(x) = \phi'(x)$—if and only if their pointwise norm-squared values are equal for all $x \in \text{SpaceTime}$, that is, $\phi(x)^\dagger \phi(x) = \phi'(x)^\dagger \phi'(x)$.

For every smooth map $f : \text{SpaceTime} \to \mathbb{R}$ that is positive semidefinite (i.e., $f(x) \geq 0$ for all $x \in \text{SpaceTime}$), there exists a Higgs field $\phi$ (a smooth section of the Higgs bundle) such that $f = \phi^\dagger \phi$ holds pointwise, where $\phi^\dagger \phi$ denotes the squared norm of the Higgs field.