Physlib.Particles.StandardModel.AnomalyCancellation.NoGrav.One.LinearParameterization

Let $S$ and $T$ be two instances of the `linearParameters` structure, which parameterize the solutions to the linear anomaly cancellation conditions (ACCs) for one family of fermions in the Standard Model without gravity. If the components $Q'$, $Y$, and $E'$ of $S$ are equal to the corresponding components of $T$ (i.e., $S.Q' = T.Q'$, $S.Y = T.Y$, and $S.E' = T.E'$), then $S$ and $T$ are equal.

For a given set of linear parameters $S$ consisting of rational numbers $Q'$, $Y$, and $E'$, the function `asCharges` constructs a charge configuration $(Q, U, D, L, E) \in \mathbb{Q}^5$ for the one-family Standard Model without gravity. The vector is defined by the following assignments for the five fermion species: - $Q = Q'$ - $U = Y - Q'$ - $D = -(Y + Q')$ - $L = -3Q'$ - $E = E'$ This mapping ensures that the resulting charges satisfy the linear anomaly cancellation conditions for the $SU(2)$ and $SU(3)$ gauge groups.

For any set of linear parameters $S$ (consisting of $Q', Y, E' \in \mathbb{Q}$) in the one-family ($n=1$) Standard Model without gravity, let $q = (Q, U, D, L, E) \in \mathbb{Q}^5$ be the charge configuration assigned by the map `asCharges`, where $Q = Q'$, $U = Y - Q'$, $D = -(Y + Q')$, $L = -3Q'$, and $E = E'$. For any fermion species index $i \in \{0, 1, 2, 3, 4\}$, the charge of the $i$-th species in the single generation (index $0$), obtained by the projection map $\text{toSpecies}_i$ applied to $q$, is equal to the $i$-th component of the charge configuration $q$.

Given a set of linear parameters $S$ represented by rational numbers $(Q', Y, E')$, this function constructs a linear solution to the anomaly cancellation conditions for the one-family ($n=1$) Standard Model without gravity. The resulting charge configuration $(Q, U, D, L, E) \in \mathbb{Q}^5$ is defined by: - $Q = Q'$ - $U = Y - Q'$ - $D = -(Y + Q')$ - $L = -3Q'$ - $E = E'$ This mapping ensures that the charges satisfy the $SU(2)$ anomaly condition $3Q + L = 0$ and the $SU(3)$ anomaly condition $2Q + U + D = 0$.

For any set of linear parameters $S = (Q', Y, E') \in \mathbb{Q}^3$ in the one-family ($n=1$) Standard Model without gravity, the underlying rational charge configuration of the linear solution $S.\text{asLinear}$ is equal to the charge configuration $S.\text{asCharges}$, which is defined by the assignments: - $Q = Q'$ - $U = Y - Q'$ - $D = -(Y + Q')$ - $L = -3Q'$ - $E = E'$

For a set of linear parameters $S$ consisting of rational numbers $(Q', Y, E')$ that define the charges for a one-family Standard Model without gravity as $Q = Q'$, $U = Y - Q'$, $D = -(Y + Q')$, $L = -3Q'$, and $E = E'$, the cubic anomaly cancellation condition evaluates to: $$\text{accCube}(S) = -54 (Q')^3 - 18 Q' Y^2 + (E')^3$$

Let $S = (Q', Y, E') \in \mathbb{Q}^3$ be the linear parameters for the one-family Standard Model without gravity. If the resulting charges satisfy the cubic anomaly cancellation condition $\text{accCube}(S) = 0$ and the parameter $Q'$ is zero, then the parameter $E'$ must also be zero. Given that the cubic condition for these parameters is $-54 (Q')^3 - 18 Q' Y^2 + (E')^3 = 0$, this follows from the fact that $Q' = 0$ implies $(E')^3 = 0$.

Consider the linear parameterization of charges for the one-family Standard Model without gravity, where the charges are determined by the rational parameters $Q'$, $Y$, and $E'$. If a charge configuration $S$ satisfies the cubic anomaly cancellation condition ($\text{accCube}(S) = 0$) and the parameter $E'$ is zero, then the parameter $Q'$ must also be zero.

There is a bijection (equivalence) between the type of linear parameters $(Q', Y, E') \in \mathbb{Q}^3$ and the set of linear solutions $(Q, U, D, L, E) \in \mathbb{Q}^5$ to the anomaly cancellation conditions for the 1-family Standard Model without gravity. The bijection maps a parameter triplet $(Q', Y, E')$ to a solution configuration defined by: - $Q = Q'$ - $U = Y - Q'$ - $D = -(Y + Q')$ - $L = -3Q'$ - $E = E'$ Conversely, any linear solution $(Q, U, D, L, E)$ satisfying the conditions $3Q + L = 0$ and $2Q + U + D = 0$ uniquely determines the parameters as $Q' = Q$, $Y = \frac{U - D}{2}$, and $E' = E$.

Consider the space of linear solutions to the anomaly cancellation conditions (ACCs) for a single family ($n=1$) of the Standard Model without gravity, characterized by the linear equations $3Q + L = 0$ and $2Q + U + D = 0$. There exists a bijection between: 1. The set of linear parameters $(Q', Y, E') \in \mathbb{Q}^3$ such that $Q' \neq 0$ and $E' \neq 0$. 2. The set of linear solutions $(Q, U, D, L, E) \in \mathbb{Q}^5$ such that the charges for the left-handed quark doublet $Q$ and the right-handed charged lepton $E$ are both non-zero. This bijection maps the parameters to the solution via $Q = Q'$, $U = Y - Q'$, $D = -(Y + Q')$, $L = -3Q'$, and $E = E'$.

For any set of linear parameters $S$ (consisting of rational numbers $Q', Y, E'$) representing a solution to the linear anomaly cancellation conditions for a single generation ($n=1$) of the Standard Model, let the corresponding charges be $(Q, U, D, L, E) = (Q', Y - Q', -(Y + Q'), -3Q', E')$. The gravitational anomaly $\text{accGrav}$, defined as $6Q + 3U + 3D + 2L + E$, vanishes (equals $0$) if and only if $E' = 6Q'$.

Consider the parameterization $(x, v, w)$ of solutions to the linear Anomaly Cancellation Conditions (ACCs) for a single Standard Model family (without gravity) where the charges $Q$ and $E$ are non-zero. For any two such parameter sets $S$ and $T$, if their components are equal, i.e., $S.x = T.x$, $S.v = T.v$, and $S.w = T.w$, then $S = T$.

This function defines a mapping from the parameterization $(x, v, w)$ of the linear Anomaly Cancellation Conditions (ACCs) for a single Standard Model family (without gravity) to the general structure of linear parameters. Given an input $S$ with parameters $x, v$, and $w$, the map produces a triplet of values corresponding to the charge parameters $(Q', L', E')$ defined as: $$Q' = x, \quad L' = \frac{3x(v - w)}{v + w}, \quad E' = -\frac{6x}{v + w}$$ The definition further ensures that the resulting values satisfy the constraints $Q' \neq 0$ and $E' \neq 0$.

This function defines a mapping from the `linearParameters` of the linear Anomaly Cancellation Conditions (ACCs) for a single family to the $(x, v, w)$ parameterization (represented by `linearParametersQENeqZero`), specifically for the case where the charges $Q'$ and $E'$ are non-zero. Given a set of linear parameters $S$, the corresponding $(x, v, w)$ values are defined as: $$x = Q', \quad v = -\frac{3Q' + Y}{E'}, \quad w = -\frac{3Q' - Y}{E'}$$ where $Q'$, $E'$, and $Y$ are the charge and hypercharge parameters associated with $S$.

This definition establishes a bijection (equivalence) between the parameterization $(x, v, w)$ for solutions to the linear Anomaly Cancellation Conditions (ACCs) for a single family and the set of linear parameters $(Q', L', E')$ constrained by $Q' \neq 0$ and $E' \neq 0$. The bijection identifies a triple $(x, v, w)$ with a triple $(Q', L', E')$ using the following coordinate transformations: The forward map is given by: $$Q' = x, \quad L' = \frac{3x(v - w)}{v + w}, \quad E' = -\frac{6x}{v + w}$$ The inverse map is given by: $$x = Q', \quad v = -\frac{3Q' + Y}{E'}, \quad w = -\frac{3Q' - Y}{E'}$$ where $Y$ is the hypercharge parameter associated with the linear parameters.

This definition establishes a bijection between the parameterization $(x, v, w) \in \mathbb{Q}^3$ (given by `linearParametersQENeqZero`) and the set of linear solutions to the anomaly cancellation conditions (ACCs) for a single family ($n=1$) of the Standard Model without gravity, specifically those where the charges of the left-handed quark doublet $Q$ and the right-handed charged lepton $E$ are non-zero. The linear solutions $(Q, U, D, L, E) \in \mathbb{Q}^5$ satisfy the equations: 1. $3Q + L = 0$ 2. $2Q + U + D = 0$ The bijection is formed by composing the mapping from $(x, v, w)$ to the intermediate linear parameters $(Q', Y, E')$ with the mapping from those parameters to the physical charge assignments.

For a set of rational parameters $S = (x, v, w)$ in the `linearParametersQENeqZero` parameterization of the one-family Standard Model without gravity, let $(Q, U, D, L, E)$ be the resulting charge configuration determined by the bijection. The cubic anomaly cancellation condition (ACC) is satisfied, i.e., $$\sum (6 Q^3 + 3 U^3 + 3 D^3 + 2 L^3 + E^3) = 0$$ if and only if the parameters $v$ and $w$ satisfy the condition: $$v^3 + w^3 = -1$$ where $v$ and $w$ are rational numbers related to the charges $Q$, $E$, and the hypercharge parameter $Y$ by $v = -\frac{3Q + Y}{E}$ and $w = -\frac{3Q - Y}{E}$.

For a set of rational parameters $S$ in the `linearParametersQENeqZero` parameterization of the one-family Standard Model without gravity, if the resulting charge configuration determined by the bijection satisfies the cubic anomaly cancellation condition, then $S.v = 0$ or $S.w = 0$.

For a parameter $S = (x, v, w)$ in the `linearParametersQENeqZero` parameterization of the one-family Standard Model without gravity, if the charge configuration determined by $S$ satisfies the cubic anomaly cancellation condition (ACC) and the parameter component $v = 0$, then $w = -1$.

For a set of rational parameters $S = (x, v, w)$ in the `linearParametersQENeqZero` parameterization of the one-family Standard Model without gravity, if the resulting charge configuration determined by the bijection satisfies the cubic anomaly cancellation condition $\text{accCube} = 0$, and the parameter $w = 0$, then $v = -1$.

For a set of rational parameters $S = (x, v, w)$ in the `linearParametersQENeqZero` parameterization of the one-family ($n=1$) Standard Model without gravity (where charges $Q$ and $E$ are non-zero), if the resulting charge configuration satisfies the cubic anomaly cancellation condition $\text{accCube} = 0$, then either $(v = -1 \text{ and } w = 0)$ or $(v = 0 \text{ and } w = -1)$.

For a single-family ($n=1$) Standard Model without right-handed neutrinos, let $S = (x, v, w) \in \mathbb{Q}^3$ be the parameters defining a solution to the linear anomaly cancellation conditions where the charges of the left-handed quark doublet $Q$ and the right-handed charged lepton $E$ are non-zero. The gravitational anomaly condition, defined as $\text{accGrav} = 6Q + 3U + 3D + 2L + E = 0$, holds if and only if the parameters $v$ and $w$ satisfy $v + w = -1$.

For a single-family ($n=1$) Standard Model without right-handed neutrinos, let $S = (x, v, w) \in \mathbb{Q}^3$ be parameters defining a solution to the linear anomaly cancellation conditions where the charges $Q$ and $E$ are non-zero. If the resulting charge configuration satisfies the cubic anomaly cancellation condition $\text{accCube} = 0$, then it also satisfies the gravitational anomaly cancellation condition $\text{accGrav} = 0$.