Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.PlusU1.HyperCharge

The definition `SMRHN.PlusU1.Y₁` specifies the hypercharge values for a single generation (family) of fermions in the Standard Model including a right-handed neutrino. This assignment is given by a vector of charges $(1, -4, 2, -3, 6, 0)$ corresponding to the six fermion species. The definition also includes proofs that these values satisfy the linear, quadratic, and cubic anomaly cancellation conditions required for a consistent $U(1)$ gauge symmetry.

For a natural number $n$ representing the number of fermion families, the function $Y(n)$ defines the hypercharge assignment for the Standard Model with right-handed neutrinos. It is constructed as a "family-universal" assignment, meaning it repeats the single-family hypercharge vector $Y_1 = (1, -4, 2, -3, 6, 0)$ for each of the $n$ generations. The resulting assignment is an element of the space of solutions to the anomaly cancellation conditions for $n$ families, denoted as $(\text{PlusU1 } n).\text{Sols}$.

For an $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the standard family-universal hypercharge assignment. For any charge configuration $S \in \mathbb{Q}^{6n}$, the symmetric bilinear form $B(Y, S)$ (associated with the quadratic anomaly cancellation conditions) is equal to the linear $Y^2$ anomaly cancellation condition $\text{accYY}(S)$: $$B(Y, S) = \text{accYY}(S)$$ where $B$ is defined by the sum over generations $i$: $$B(S, T) = \sum_{i=0}^{n-1} \left( Q_i(S)Q_i(T) - 2u_i(S)u_i(T) + d_i(S)d_i(T) - L_i(S)L_i(T) + e_i(S)e_i(T) \right)$$ and $\text{accYY}$ is defined as: $$\text{accYY}(S) = \sum_{i=0}^{n-1} \left( Q_i(S) + 8u_i(S) + 2d_i(S) + 3L_i(S) + 6e_i(S) \right)$$

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the standard family-universal hypercharge assignment. For any charge configuration $S$ that satisfies the linear anomaly cancellation conditions (i.e., $S$ is an element of the linear solution space $\text{LinSols}$), the symmetric bilinear form $B(Y, S)$ associated with the quadratic anomaly cancellation conditions is zero: $$B(Y, S) = 0$$ where the bilinear form $B$ is defined by the sum over generations $i \in \{0, \dots, n-1\}$: $$B(S, T) = \sum_{i=0}^{n-1} \left( Q_i(S)Q_i(T) - 2u_i(S)u_i(T) + d_i(S)d_i(T) - L_i(S)L_i(T) + e_i(S)e_i(T) \right)$$ Here, $Q_i, u_i, d_i, L_i,$ and $e_i$ represent the rational charges assigned to the left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton for the $i$-th generation, respectively.

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration satisfying the linear anomaly cancellation conditions, and let $Y$ be the standard family-universal hypercharge assignment. For any rational scalars $a, b \in \mathbb{Q}$, the quadratic anomaly cancellation condition $Q$ satisfies: $$Q(a \cdot S + b \cdot Y) = a^2 Q(S)$$ where $Q$ is the homogeneous quadratic map defined by the sum over generations $i \in \{0, \dots, n-1\}$: $$Q(S) = \sum_{i=0}^{n-1} \left( Q_i(S)^2 - 2u_i(S)^2 + d_i(S)^2 - L_i(S)^2 + e_i(S)^2 \right)$$ Here, $Q_i, u_i, d_i, L_i,$ and $e_i$ represent the rational charges assigned to the left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton for the $i$-th generation, respectively.

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration satisfying the quadratic anomaly cancellation conditions, and let $Y$ be the standard family-universal hypercharge assignment. For any rational scalars $a, b \in \mathbb{Q}$, the quadratic anomaly cancellation condition $Q$ satisfies: $$Q(a \cdot S + b \cdot Y) = 0$$ where $Q$ is the homogeneous quadratic map defined by the sum over generations $i \in \{0, \dots, n-1\}$: $$Q(S) = \sum_{i=0}^{n-1} \left( Q_i(S)^2 - 2u_i(S)^2 + d_i(S)^2 - L_i(S)^2 + e_i(S)^2 \right)$$ Here, $Q_i, u_i, d_i, L_i,$ and $e_i$ represent the rational charges assigned to the left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton for the $i$-th generation, respectively.

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration satisfying the quadratic anomaly cancellation conditions and $Y$ be the standard family-universal hypercharge assignment. For any rational scalars $a, b \in \mathbb{Q}$, this definition constructs a new quadratic solution by forming the linear combination $a \cdot S + b \cdot Y$. The resulting configuration is guaranteed to satisfy the quadratic anomaly cancellation conditions.

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration that satisfies the quadratic anomaly cancellation conditions, and let $a \in \mathbb{Q}$ be a rational scalar. Let $\text{addQuad}(S, a, b)$ denote the quadratic solution formed by the linear combination $a \cdot S + b \cdot Y$, where $Y$ is the standard family-universal hypercharge assignment. Then, for $b = 0$, this construction satisfies: $$\text{addQuad}(S, a, 0) = a \cdot S$$ where $a \cdot S$ is the scalar multiplication of the charge configuration $S$ by $a$.

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the family-universal hypercharge assignment and $S$ be any configuration of rational charges. The symmetric trilinear form $\text{cubeTriLin}$ (which corresponds to the cubic anomaly cancellation condition) evaluated at the arguments $(Y, Y, S)$ satisfies the identity: $$\text{cubeTriLin}(Y, Y, S) = 6 \cdot \text{accYY}(S)$$ where $\text{accYY}(S)$ is the $Y^2$ anomaly cancellation condition for the charge configuration $S$, defined as: $$\text{accYY}(S) = \sum_{i=0}^{n-1} \left( Q_i(S) + 8u_i(S) + 2d_i(S) + 3L_i(S) + 6e_i(S) \right)$$ Here, $Q_i, u_i, d_i, L_i,$ and $e_i$ denote the rational charges of the $i$-th generation left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton, respectively.

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the family-universal hypercharge assignment. For any configuration of rational charges $S$ that satisfies the linear anomaly cancellation conditions (i.e., $S \in (\text{PlusU1 } n).\text{LinSols}$), the symmetric trilinear form $\text{cubeTriLin}$ (representing the cubic anomaly cancellation condition) evaluated at $(Y, Y, S)$ is zero: $$\text{cubeTriLin}(Y, Y, S) = 0$$

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the family-universal hypercharge assignment and $S$ be any configuration of rational charges. The symmetric trilinear form $\text{cubeTriLin}$ (associated with the cubic anomaly cancellation condition) evaluated at the arguments $(Y, S, S)$ satisfies the identity: $$\text{cubeTriLin}(Y, S, S) = 6 \cdot \text{accQuad}(S)$$ where $\text{accQuad}(S)$ is the quadratic anomaly cancellation condition for the charge configuration $S$, defined as: $$\text{accQuad}(S) = \sum_{i=0}^{n-1} \left( Q_i(S)^2 - 2U_i(S)^2 + D_i(S)^2 - L_i(S)^2 + E_i(S)^2 \right)$$ Here, $Q_i, U_i, D_i, L_i,$ and $E_i$ denote the rational charges of the $i$-th generation left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton, respectively.

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the family-universal hypercharge assignment. For any configuration of rational charges $S$ that satisfies the quadratic anomaly cancellation condition (i.e., $S$ is an element of the space of quadratic solutions $(\text{PlusU1 } n).\text{QuadSols}$), the symmetric trilinear form $\text{cubeTriLin}$ associated with the cubic anomaly cancellation condition satisfies: $$\text{cubeTriLin}(Y, S, S) = 0$$

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the family-universal hypercharge assignment and let $S$ be a configuration of rational charges that satisfies the linear anomaly cancellation conditions. For any rational scalars $a, b \in \mathbb{Q}$, the cubic anomaly cancellation condition $\mathcal{A}_{\text{cube}}$ evaluated on the linear combination $a S + b Y$ is given by: $$\mathcal{A}_{\text{cube}}(a S + b Y) = a^2 (a \mathcal{A}_{\text{cube}}(S) + 3 b \cdot \tau(S, S, Y))$$ where $\tau$ is the symmetric trilinear form associated with the cubic anomaly cancellation condition. This identity utilizes the facts that $Y$ is a solution to the cubic condition ($\mathcal{A}_{\text{cube}}(Y) = 0$) and that the trilinear form vanishes when two of its arguments are the hypercharge $Y$ and the third is a linear solution $S$ ($\tau(Y, Y, S) = 0$).

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the family-universal hypercharge assignment and let $S$ be a configuration of rational charges that satisfies the quadratic anomaly cancellation conditions (i.e., $S$ is an element of the space of quadratic solutions $(\text{PlusU1 } n).\text{QuadSols}$). For any rational scalars $a, b \in \mathbb{Q}$, the cubic anomaly cancellation condition $\mathcal{A}_{\text{cube}}$ evaluated on the linear combination $a S + b Y$ satisfies: $$\mathcal{A}_{\text{cube}}(a S + b Y) = a^3 \mathcal{A}_{\text{cube}}(S)$$ This identity arises because $Y$ is a solution to the cubic ACC ($\mathcal{A}_{\text{cube}}(Y) = 0$) and the symmetric trilinear form $\tau$ associated with the cubic condition vanishes when at least one argument is $Y$ and the remaining arguments are the quadratic solution $S$ (specifically, $\tau(S, S, Y) = 0$ and $\tau(S, Y, Y) = 0$).

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the family-universal hypercharge assignment and let $S$ be a configuration of rational charges that satisfies all anomaly cancellation conditions (i.e., $S$ is an element of the space of solutions $(\text{PlusU1 } n).\text{Sols}$). For any rational scalars $a, b \in \mathbb{Q}$, the cubic anomaly cancellation condition $\mathcal{A}_{\text{cube}}$ evaluated on the linear combination of $S$ and $Y$ vanishes: $$\mathcal{A}_{\text{cube}}(a S + b Y) = 0$$ This result follows from the fact that both $S$ and $Y$ are solutions to the full set of anomaly cancellation conditions, which ensures that not only do their individual cubic sums vanish, but their mutual trilinear cross-terms also vanish.

In the $n$-generation Standard Model with right-handed neutrinos, let $Y$ be the standard family-universal hypercharge assignment and $S$ be a charge configuration satisfying all anomaly cancellation conditions (an element of the space of solutions $(\text{PlusU1 } n).\text{Sols}$). For any rational scalars $a, b \in \mathbb{Q}$, this definition constructs a new solution by forming the linear combination $a S + b Y$. This combination is shown to satisfy all anomaly cancellation conditions (linear, quadratic, and cubic) and is returned as an element of the solution space $(\text{PlusU1 } n).\text{Sols}$.