PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.Invariants

Let $R$ be a commutative ring and let $\iota$ and $\kappa$ be index sets. Given a point $x: \iota \to R$, a collection of multivariate polynomials $f: \kappa \to R[X_i]_{i \in \iota}$, and a multivariate polynomial $G \in R[Y_k]_{k \in \kappa}$, the evaluation of the substituted polynomial $\text{aeval}_f G$ at the point $x$ is equal to the evaluation of $G$ at the point in $R^\kappa$ defined by the evaluations of each $f_k$ at $x$. Mathematically, this is expressed as: $$\text{eval}_x (\text{aeval}_f G) = \text{eval}_{k \mapsto \text{eval}_x(f_k)} G$$ where $\text{aeval}_f G$ denotes the polynomial substitution of the family $f$ into $G$.

For a unitary complex number $u = a + bi \in \mathbb{C}$ (where $a, b \in \mathbb{R}$ and $a^2 + b^2 = 1$), this function defines the Cartan hypercharge rotation of the slice parameters as a substitution of variables in the polynomial ring $\mathbb{R}[X_0, \dots, X_5]$. Specifically, it maps each index $j \in \{0, \dots, 5\}$ to a polynomial as follows: - $0 \mapsto a X_0 - b X_1$ - $1 \mapsto b X_0 + a X_1$ - $2 \mapsto a X_2 - b X_3$ - $3 \mapsto b X_2 + a X_3$ - $4 \mapsto a X_4 + b X_5$ - $5 \mapsto a X_5 - b X_4$ This substitution represents the action of the Cartan subgroup of the gauge group on the configuration space of the two Higgs doublet model, expressed in real and imaginary parts of the field components.

For a unitary complex number $u \in U(1)$ and a vector of six real parameters $\mathbf{a} = (a_0, \dots, a_5) \in \mathbb{R}^6$, evaluating the Cartan hypercharge rotation substitution polynomials $\text{cartanSubst}(u)$ at the point $\mathbf{a}$ is equivalent to the rotation of the parameters $\mathbf{a}$ by the phase $u$. Mathematically, for each index $k \in \{0, \dots, 5\}$: $$\text{eval}_{\mathbf{a}}(\text{cartanSubst}(u)_k) = \text{cartanRotParam}(u, \mathbf{a})_k$$ where $\text{cartanSubst}(u)$ is the substitution mapping variables to the rotated coordinates, and $\text{cartanRotParam}(u, \mathbf{a})$ is the vector resulting from rotating the components of $\mathbf{a}$ by $u$.

Let $V$ be a gauge-invariant effective potential in the Two Higgs Doublet Model. If the restriction of $V$ to the upper-triangular gauge slice defined by the map $\text{sliceR}: \mathbb{R}^6 \to \text{TwoHiggsDoublet}$ is represented by a multivariate polynomial $P \in \mathbb{R}[X_0, \dots, X_5]$ such that $V(\text{sliceR}(a)) = P(a)$ for all $a \in \mathbb{R}^6$, then for any complex phase $u \in U(1)$, the polynomial $P$ is invariant under the variable substitution $\text{cartanSubst}(u)$ corresponding to the Cartan hypercharge rotation. That is, $\text{aeval}(\text{cartanSubst}(u), P) = P$.

For a unitary complex number $c \in U(1)$, this function defines a substitution for the variables $X_0, \dots, X_5$ of a multivariate polynomial over $\mathbb{R}$. The substitution leaves the first four variables invariant and rotates the last two by the phase of $c^6$: - $X_i \mapsto X_i$ for $i \in \{0, 1, 2, 3\}$ - $X_4 \mapsto \text{Re}(c^6) X_4 - \text{Im}(c^6) X_5$ - $X_5 \mapsto \text{Im}(c^6) X_4 + \text{Re}(c^6) X_5$ This represents the residual $U(1)$ gauge rotation acting on the perpendicular parameters of the two Higgs doublet potential slice.

For a phase $c \in U(1)$ and a vector of six real parameters $a = (a_0, \dots, a_5) \in \mathbb{R}^6$, the evaluation of the six polynomials defined by the residual $U(1)$ rotation substitution $\text{residualSubst}(c)$ at the point $a$ is equal to the rotated parameter vector $\text{resRotParam}(c, a)$. Specifically, this identity holds for each component $k \in \{0, \dots, 5\}$, where the substitution leaves the first four components invariant and rotates the complex combination of the last two components by the phase $c^6$, resulting in the vector $(a_0, a_1, a_2, a_3, \text{Re}(c^6(a_4 + i a_5)), \text{Im}(c^6(a_4 + i a_5)))$.

Let $V$ be a gauge-invariant effective potential in the two-Higgs-doublet model. Let $P \in \mathbb{R}[X_0, X_1, X_2, X_3, X_4, X_5]$ be a real multivariate polynomial that represents the potential on the upper-triangular gauge slice, satisfying $V(\text{sliceR}(a)) = P(a_0, a_1, a_2, a_3, a_4, a_5)$ for any parameter vector $a \in \mathbb{R}^6$. Then, for any complex phase $c \in U(1)$, the polynomial $P$ is invariant under the substitution defined by the residual $U(1)$ rotation $\text{residualSubst}(c)$, which is expressed as $\text{aeval}(\text{residualSubst}(c)) P = P$.

This function defines the transformation of the six real coordinates $a_k$ (for $k \in \{0, \dots, 5\}$) into polynomials in the complex hypercharge eigen-coordinates $z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1$, represented by the variables $X_0, \dots, X_5$. The mapping is defined as: - $a_0 = \frac{1}{2}(X_0 + X_1)$ - $a_1 = \frac{1}{2i}(X_0 - X_1)$ - $a_2 = \frac{1}{2}(X_2 + X_3)$ - $a_3 = \frac{1}{2i}(X_2 - X_3)$ - $a_4 = \frac{1}{2}(X_4 + X_5)$ - $a_5 = \frac{1}{2i}(X_4 - X_5)$ This coordinate change diagonalizes the gauge-torus rotation into a scaling.

Given a unitary complex number $u$, this function defines the diagonal action of the Cartan gauge subgroup on the six hypercharge eigen-coordinates $X_0, \dots, X_5$ of the multivariate polynomial ring $\mathbb{C}[X_0, \dots, X_5]$. It maps the indices to the polynomials $(u X_0, \bar{u} X_1, u X_2, \bar{u} X_3, \bar{u} X_4, u X_5)$ respectively, where $\bar{u}$ is the complex conjugate of $u$, corresponding to the Cartan charges $(1, -1, 1, -1, -1, 1)$.

Given a unitary complex number $c \in U(1)$, this definition describes the diagonal action of the residual $U(1)$ gauge transformation on the six hypercharge eigen-coordinates represented by the variables $X_0, \dots, X_5$ in the multivariate polynomial ring $\mathbb{C}[X_0, \dots, X_5]$. The transformation leaves the first four coordinates $X_0, X_1, X_2, X_3$ invariant, while the "perpendicular pair" $X_4$ and $X_5$ are scaled by $c^6$ and its conjugate $\bar{c}^6$ respectively: $$(X_0, X_1, X_2, X_3, X_4, X_5) \mapsto (X_0, X_1, X_2, X_3, c^6 X_4, \bar{c}^6 X_5)$$

For a unitary complex number $u \in \mathbb{C}$ (representing an element of the gauge torus), let $H_k \in \mathbb{C}[X_0, \dots, X_5]$ be the polynomial defined by `hyperchargeEigen` that expresses the $k$-th real coordinate $a_k$ (for $k \in \{0, \dots, 5\}$) in terms of the complex hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$. Let $D_u$ be the diagonal Cartan transformation that scales these eigen-coordinates by their respective hypercharges, and let $R_u$ be the Cartan rotation substitution acting on the real coordinates. The theorem states that applying the diagonal scaling $D_u$ to the eigen-coordinates within the transformation $H_k$ is equivalent to first applying the real Cartan rotation $R_u$ and then converting the resulting real polynomials into the eigen-basis: $$\text{bind}_1 (D_u) (H_k) = \text{bind}_1 (H) (R_u)_k$$ In other words, the diagonal Cartan scaling on the eigen-coordinates is the pullback of the Cartan rotation on the real coordinates through the eigen-coordinate transformation.

For any unitary complex number $c \in U(1)$ and any coordinate index $k \in \{0, \dots, 5\}$, the following identity holds regarding the transformation of the two-Higgs-doublet parameters into hypercharge eigen-coordinates: $$\text{bind}_1 (\text{diagRes}(c)) (\text{hyperchargeEigen}(k)) = \text{bind}_1 (\text{hyperchargeEigen}) (\text{residualSubst}(c, k))$$ Here, $\text{hyperchargeEigen}$ is the mapping that expresses the real coordinates $a_k$ as polynomials in the complex eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$. The term $\text{diagRes}(c)$ denotes the diagonal action of the residual $U(1)$ gauge group on these eigen-coordinates (scaling the perpendicular pair $w_1, \bar{w}_1$ by $c^6$ and $\bar{c}^6$ respectively), while $\text{residualSubst}(c)$ represents the rotation of the real parameters $a_4$ and $a_5$ by the phase of $c^6$. The identity states that applying the gauge transformation directly to the eigen-coordinate representation is equivalent to transforming the rotated real parameters.

The function $\text{cartanCharge}: \{0, 1, 2, 3, 4, 5\} \to \mathbb{Z}$ assigns a Cartan charge to each of the six hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$. The charges are given by the vector $(1, -1, 1, -1, -1, 1)$.

The residual $U(1)$ hypercharges assigned to the six hypercharge eigen-coordinates, indexed by $k \in \{0, 1, \dots, 5\}$. The charges are defined by the sequence $(0, 0, 0, 0, 1, -1)$, where only the final two coordinates (the "perpendicular pair") carry non-zero charges.

This function takes a real multivariate polynomial $P \in \mathbb{R}[a_0, \dots, a_5]$, representing the effective potential on the configuration slice of the two Higgs doublet model, and returns a complex multivariate polynomial in the hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$. The transformation is performed by complexifying the coefficients of $P$ and substituting each real variable $a_k$ with its representation in terms of the eigen-coordinates $X_0, \dots, X_5$ as defined by the `hyperchargeEigen` coordinate change.

For any unitary complex number $u \in U(1)$, the diagonal Cartan transformation $\text{diagCartan}(u)$ is defined as the map that sends each coordinate index $i \in \{0, \dots, 5\}$ to the polynomial: $$\text{diagCartan}(u)(i) = u^{\text{cartanCharge}(i)} X_i$$ where $X_i$ is the $i$-th variable of the multivariate polynomial ring $\mathbb{C}[X_0, \dots, X_5]$, and $\text{cartanCharge}(i)$ is the $i$-th element of the Cartan charge vector $(1, -1, 1, -1, -1, 1)$.

For any unit complex number $c \in U(1)$, the residual diagonal gauge transformation $\text{diagRes}(c)$ on the hypercharge eigen-coordinates $X_i$ (for $i \in \{0, \dots, 5\}$) is given by the scaling $X_i \mapsto (c^6)^{y_i} X_i$, where $y_i$ is the $i$-th residual $U(1)$ hypercharge.

Let $V$ be a gauge-invariant effective potential for the two-Higgs-doublet model. Suppose $P \in \mathbb{R}[a_0, \dots, a_5]$ is a real multivariate polynomial that represents $V$ on the upper-triangular Higgs slice, such that $V(\text{sliceR}(a)) = P(a)$ for all $a \in \mathbb{R}^6$. For any unitary complex number $u \in U(1)$, the complexified slice potential $Q_{\text{slice}}(P)$ in the hypercharge eigen-coordinates is invariant under the diagonal Cartan transformation $\text{diagCartan}(u)$, which scales the $i$-th coordinate by $u^{\text{cartanCharge}(i)}$.

Let $V$ be a gauge-invariant effective potential in the Two-Higgs-Doublet Model. Suppose $P \in \mathbb{R}[a_0, \dots, a_5]$ is a real multivariate polynomial that represents the potential on the configuration slice, such that $V(\text{sliceR}(a)) = P(a)$ for all $a \in \mathbb{R}^6$. Then for any unitary complex number $c \in U(1)$, the complexified slice potential $Q_{\text{slice}}(P)$ expressed in hypercharge eigen-coordinates is invariant under the residual $U(1)$ gauge transformation $\text{diagRes}(c)$, which is to say that the operation of $\text{diagRes}(c)$ on the polynomial leaves it unchanged.

There exists a unitary complex number $\omega \in U(1)$ (that is, $\omega \in \mathbb{C}$ such that $|\omega| = 1$) with infinite order. Specifically, for any integer $n \in \mathbb{Z}$, if $\omega^n = 1$, then $n = 0$.

Consider a gauge-invariant effective potential $V$ for the two-Higgs-doublet model. Let $P \in \mathbb{R}[a_0, \dots, a_5]$ be a real multivariate polynomial such that $V(\text{sliceR}(a)) = P(a)$ for all $a \in \mathbb{R}^6$, where $\text{sliceR}$ is the map to the upper-triangular gauge slice. Let $Q_{\text{slice}}(P)$ be the complexified polynomial expressed in the hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$. For any monomial $m$ with exponents $m_i \in \mathbb{N}$ (for $i = 0, \dots, 5$), if the total Cartan charge $\sum_{i=0}^5 m_i \cdot \text{cartanCharge}_i \neq 0$ or the total residual hypercharge $\sum_{i=0}^5 m_i \cdot \text{hyperCharge}_i \neq 0$, then the coefficient of $m$ in $Q_{\text{slice}}(P)$ is zero.

This definition provides a sequence of five quadratic polynomials in six variables over the complex numbers $\mathbb{C}$. These polynomials represent the hypercharge-neutral bilinears expressed in hypercharge eigen-coordinates $(X_0, X_1, X_2, X_3, X_4, X_5)$, which correspond to $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$. The five polynomials are: 1. $X_0 X_1$ (representing $z \bar{z}$) 2. $X_2 X_3$ (representing $w_0 \bar{w}_0$) 3. $X_0 X_3$ (representing $z \bar{w}_0$) 4. $X_1 X_2$ (representing $\bar{z} w_0$) 5. $X_4 X_5$ (representing $w_1 \bar{w}_1$)

Let $w: \{0, \dots, 5\} \to \mathbb{Z}$ be an assignment of charges to the indices of a polynomial, and let $m$ be a monomial represented by its exponents $m_i \in \mathbb{N}$. The total charge of the monomial, calculated by summing the products of the exponents and charges over all indices, is equal to the sum restricted to the support of the monomial (the indices where $m_i \neq 0$): $$\sum_{i=0}^5 m_i w_i = \sum_{i \in \text{supp}(m)} m_i w_i$$

Let $w \in \mathbb{Z}^6$ be a vector of weights (representing charges) associated with six variables. For any two exponent vectors $a, b \in \mathbb{N}^6$ representing monomials, the total charge of the product monomial is the sum of the total charges of the individual monomials: $$\sum_{k=0}^5 (a_k + b_k) w_k = \left( \sum_{k=0}^5 a_k w_k \right) + \sum_{k=0}^5 b_k w_k$$

Given a weight vector $w: \{0, \dots, 5\} \to \mathbb{Z}$ and an index $i \in \{0, \dots, 5\}$, the total charge of a monomial consisting of a single power of the $i$-th variable—represented by the multi-index $m = \text{single}(i, 1)$—is equal to the charge of that variable: $$\sum_{k=0}^5 m_k w_k = w_i$$ where $m_k = 1$ if $k = i$ and $m_k = 0$ otherwise.

In the context of the two Higgs doublet model, let $(X_0, X_1, X_2, X_3, X_4, X_5)$ represent the hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$. A monomial $M = \prod_{i=0}^5 X_i^{m_i}$ is considered hypercharge-neutral if it satisfies two conditions: 1. The total Cartan charge vanishes: $\sum_{i=0}^5 m_i Q^C_i = 0$, where $Q^C = (1, -1, 1, -1, -1, 1)$. 2. The total residual $U(1)$ hypercharge vanishes: $\sum_{i=0}^5 m_i Q^H_i = 0$, where $Q^H = (0, 0, 0, 0, 1, -1)$. The theorem states that every such hypercharge-neutral monomial belongs to the subalgebra over $\mathbb{C}$ generated by the five neutral quadratic bilinears: - $B_1 = z \bar{z}$ - $B_2 = w_0 \bar{w}_0$ - $B_3 = z \bar{w}_0$ - $B_4 = \bar{z} w_0$ - $B_5 = w_1 \bar{w}_1$ Effectively, this means every hypercharge-neutral monomial can be written as a product of these five bilinears.

Let $V$ be a gauge-invariant effective potential in the two-Higgs-doublet model. Suppose that on the upper-triangular gauge slice, the potential is given by a real multivariate polynomial $P \in \mathbb{R}[a_0, \dots, a_5]$, such that $V(\text{sliceR}(a)) = P(a)$ for all $a \in \mathbb{R}^6$. Then the complexified polynomial $Q_{\text{slice}}(P)$ expressed in the hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$ belongs to the complex subalgebra generated by the five hypercharge-neutral quadratic bilinears: - $B_1 = z \bar{z}$ - $B_2 = w_0 \bar{w}_0$ - $B_3 = z \bar{w}_0$ - $B_4 = \bar{z} w_0$ - $B_5 = w_1 \bar{w}_1$

Let $V$ be a gauge-invariant effective potential in the two-Higgs-doublet model. Suppose there exists a real polynomial $P \in \mathbb{R}[a_0, \dots, a_5]$ such that for all $a \in \mathbb{R}^6$, the potential evaluated on the gauge slice $\text{sliceR}(a)$ is given by $V(\text{sliceR}(a)) = P(a)$. Then the complexified slice potential $Q_{\text{slice}}(P)$, expressed in hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$, can be written as a complex polynomial in the five neutral quadratic bilinears. Specifically, there exists a polynomial $G \in \mathbb{C}[X_1, \dots, X_5]$ such that $Q_{\text{slice}}(P) = G(B_1, \dots, B_5)$, where the bilinears are defined as $B_1 = z \bar{z}$, $B_2 = w_0 \bar{w}_0$, $B_3 = z \bar{w}_0$, $B_4 = \bar{z} w_0$, and $B_5 = w_1 \bar{w}_1$.

For a given vector $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$ representing the gauge-invariant bilinears (the Gram vector), this function returns the 6-tuple of real slice parameters $(X_0, 0, X_1, X_2, X_3, 0)$. These parameters are used to define the specific Higgs field configuration $\text{repHiggs}(X)$ that acts as a representative for the gauge orbit corresponding to $X$.

For any vector of four real parameters $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$, the Higgs configuration $\text{repHiggs}(X)$ is identical to the configuration obtained by applying the real-linear slice map $\text{sliceR}$ to the six-tuple of parameters $\text{aRep}(X) = (X_0, 0, X_1, X_2, X_3, 0)$. This shows that the polynomial family of orbit representatives used to describe the gauge orbits in the Two-Higgs-Doublet Model is a subset of the general upper-triangular gauge slice.

For a given vector $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$ representing the parameters of a Higgs orbit representative, the hypercharge eigen-point is a vector in $\mathbb{C}^6$ defined by the coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$, where $z = X_0$ (real), $w_0 = X_1 + i X_2$, and $w_1 = X_3$ (real). Explicitly, the mapping is: $$X \mapsto (X_0, \ X_0, \ X_1 + i X_2, \ X_1 - i X_2, \ X_3, \ X_3)$$

For any vector $X \in \mathbb{R}^4$ representing the gauge-invariant parameters, let $\text{aRep}(X) \in \mathbb{R}^6$ be the vector of slice parameters $(X_0, 0, X_1, X_2, X_3, 0)$ and $\text{eigenPoint}(X) \in \mathbb{C}^6$ be the hypercharge eigen-point $(X_0, X_0, X_1 + i X_2, X_1 - i X_2, X_3, X_3)$. Let $\text{hyperchargeEigen}_k$ be the polynomial that expresses the $k$-th real coordinate $a_k$ in terms of the complex eigen-coordinates $z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1$. Then, evaluating the $k$-th polynomial $\text{hyperchargeEigen}_k$ at the point $\text{eigenPoint}(X)$ yields the $k$-th component of the slice parameters, i.e., $$\text{eval}(\text{eigenPoint}(X), \text{hyperchargeEigen}_k) = \text{aRep}(X)_k$$ for all $k \in \{0, \dots, 5\}$.

Let $P \in \mathbb{R}[a_0, \dots, a_5]$ be a real multivariate polynomial representing the potential on the configuration slice. For any vector $X \in \mathbb{R}^4$ representing the gauge-invariant bilinears, let $\text{aRep}(X) \in \mathbb{R}^6$ be the vector of slice parameters and $\text{eigenPoint}(X) \in \mathbb{C}^6$ be the hypercharge eigen-point. Let $Q_{\text{slice}} P$ be the complexification of $P$ expressed in hypercharge eigen-coordinates. Then, evaluating $Q_{\text{slice}} P$ at the point $\text{eigenPoint}(X)$ yields the same value as evaluating $P$ at the point $\text{aRep}(X)$, specifically: $$\text{eval}(\text{eigenPoint}(X), Q_{\text{slice}} P) = P(\text{aRep}(X))$$ where the real result of $P(\text{aRep}(X))$ is embedded into the complex numbers.

Given a multivariate polynomial $H$ in 5 variables with complex coefficients, its real part is the polynomial obtained by replacing each coefficient $c \in \mathbb{C}$ with its real part $\text{Re}(c) \in \mathbb{R}$. If $H = \sum_{\alpha} c_{\alpha} X^{\alpha}$, where $\alpha$ is a multi-index, then $\text{realPart}(H) = \sum_{\alpha} \text{Re}(c_{\alpha}) X^{\alpha}$.

Let $H$ be a multivariate polynomial in 5 variables with complex coefficients, and let $m \in \mathbb{N}^5$ be a multi-index. The coefficient of the monomial $X^m$ in the polynomial $\text{realPart}(H)$ is equal to the real part of the coefficient of the monomial $X^m$ in $H$. That is, $$\text{coeff}_m(\text{realPart}(H)) = \text{Re}(\text{coeff}_m(H)).$$ The polynomial $\text{realPart}(H)$ is the polynomial obtained by replacing each coefficient $c \in \mathbb{C}$ of $H$ with its real part $\text{Re}(c)$.

For any complex number $a \in \mathbb{C}$, the real part of the constant polynomial $C(a)$ is equal to the constant polynomial $C(\text{Re}(a))$, where $\text{Re}(a)$ denotes the real part of $a$.

For any two multivariate polynomials $p$ and $q$ in five variables with complex coefficients, the real part of their sum is equal to the sum of their individual real parts: $$\text{realPart}(p + q) = \text{realPart}(p) + \text{realPart}(q)$$ where the operator $\text{realPart}$ is defined as taking the real part of each coefficient of the polynomial.

Let $P \in \mathbb{C}[X_0, \dots, X_4]$ be a multivariate polynomial in five variables with complex coefficients. Let $\text{realPart}(P)$ be the polynomial in $\mathbb{R}[X_0, \dots, X_4]$ obtained by replacing each coefficient $c \in \mathbb{C}$ of $P$ with its real part $\text{Re}(c)$. For any variable $X_i$ where $i \in \{0, \dots, 4\}$, the real part of the product of $P$ and $X_i$ is equal to the product of the real part of $P$ and $X_i$: $$\text{realPart}(P \cdot X_i) = \text{realPart}(P) \cdot X_i$$

Let $H \in \mathbb{C}[X_0, \dots, X_4]$ be a multivariate polynomial in five variables with complex coefficients. Let $\text{realPart}(H) \in \mathbb{R}[X_0, \dots, X_4]$ be the polynomial obtained by taking the real part of each coefficient of $H$. For any vector of real numbers $y = (y_0, \dots, y_4) \in \mathbb{R}^5$, the real part of the complex value $H(y)$ is equal to the evaluation of the polynomial $\text{realPart}(H)$ at $y$: $$\text{Re}\left( H(y_0, y_1, y_2, y_3, y_4) \right) = (\text{realPart } H)(y_0, y_1, y_2, y_3, y_4)$$

For a vector $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$ representing a point on the gauge-fixing slice for two Higgs doublets, this function calculates the five real gauge-invariant bilinears. The result is a vector in $\mathbb{R}^5$ defined as: $$(X_0^2, X_0 X_1, X_0 X_2, X_1^2 + X_2^2, X_3^2)$$ Physically, these components correspond to the squared norm $\|\Phi_1\|^2$, the real part of the inner product $\text{Re}\langle \Phi_1, \Phi_2 \rangle$, the imaginary part of the inner product $\text{Im}\langle \Phi_1, \Phi_2 \rangle$, and the squared magnitudes of the components of the second doublet $|\Phi_{2,0}|^2$ and $|\Phi_{2,1}|^2$, all evaluated at the representative configuration $\text{repHiggs } X$.

The function defines a substitution that maps the five complex neutral bilinears to polynomials in five variables $X_0, \dots, X_4$ over the complex field $\mathbb{C}$. The mapping is given by the vector: $$[X_0, X_3, X_1 - i X_2, X_1 + i X_2, X_4]$$ where $i$ is the imaginary unit. This substitution expresses each complex neutral bilinear (evaluated at the eigen-point) in terms of the real gauge-invariant generators, where the off-diagonal pair $z \bar{w}_0$ and $\bar{z} w_0$ are expressed as linear combinations of the real and imaginary parts of the doublet inner product.

Let $X \in \mathbb{R}^4$ be a vector parameterizing a configuration on the gauge-fixing slice of the two Higgs doublet model. Let $\text{eigenPoint}(X) \in \mathbb{C}^6$ be the vector of hypercharge eigen-coordinates $(z, \bar{z}, w_0, \bar{w}_0, w_1, \bar{w}_1)$ associated with $X$. Let $\text{neutralBilinear}_i$ (for $i=0, \dots, 4$) be the five complex hypercharge-neutral quadratic bilinears: $$(z \bar{z}, \ w_0 \bar{w}_0, \ z \bar{w}_0, \ \bar{z} w_0, \ w_1 \bar{w}_1).$$ Let $\text{sliceBilinear}(X) \in \mathbb{R}^5$ be the five real gauge-invariant bilinears evaluated at the representative: $$(\|\Phi_1\|^2, \ \text{Re}\langle \Phi_1, \Phi_2 \rangle, \ \text{Im}\langle \Phi_1, \Phi_2 \rangle, \ |\Phi_{2,0}|^2, \ |\Phi_{2,1}|^2).$$ Then, the value of each complex neutral bilinear at the eigen-point is equal to the evaluation of the corresponding transformation polynomial $\text{transf}_i$ at the real gauge-invariant bilinears: $$\text{eval}(\text{eigenPoint}(X), \text{neutralBilinear}_i) = \text{eval}(\text{sliceBilinear}(X), \text{transf}_i).$$ This equality confirms that on the gauge slice, the complex neutral bilinears are polynomials in the real gauge-invariant generators.

Let $V$ be a gauge-invariant effective potential for the two-Higgs-doublet model with maximum mass dimension at most $n$ (meaning $V$ is a polynomial in the field components of total degree $\le n$). Then there exists a multivariate polynomial $p$ in five variables over $\mathbb{R}$ such that for any parameter vector $X = (X_0, X_1, X_2, X_3) \in \mathbb{R}^4$ defining a representative configuration $\text{repHiggs } X$, the potential satisfies: $$V(\text{repHiggs } X) = p(\|\Phi_1\|^2, \text{Re}\langle\Phi_1, \Phi_2\rangle, \text{Im}\langle\Phi_1, \Phi_2\rangle, |\Phi_{2,0}|^2, |\Phi_{2,1}|^2)$$ where the five arguments of $p$ are the real gauge-invariant bilinears evaluated at the representative configuration $\text{repHiggs } X$.