Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.Ordinary.Basic

The definition $\text{SMRHN.SM}(n)$ constructs an Anomaly Cancellation Condition (ACC) system for the Standard Model (SM) extended by $n$ right-handed neutrinos (RHN), excluding hypercharge. This system is defined over the charge space $V = \text{SM}\nu\text{Charges}(n)$, which assigns rational charges to the standard fermions and the $n$ additional neutrinos. The system consists of: 1. Three linear anomaly cancellation equations: the gravitational anomaly ($\text{accGrav}$), the $SU(2)$ gauge anomaly ($\text{accSU2}$), and the $SU(3)$ gauge anomaly ($\text{accSU3}$). 2. Zero quadratic anomaly cancellation equations. 3. A single cubic anomaly cancellation equation ($\text{accCube}$) corresponding to the $U(1)$ cubic gauge anomaly.

For any linear solution $S$ of the Anomaly Cancellation Condition (ACC) system for the Standard Model with $n$ right-handed neutrinos (without hypercharge), the gravitational anomaly cancellation condition $\text{accGrav}$ evaluated on $S$ is satisfied, i.e., $\text{accGrav}(S) = 0$.

For any linear solution $S$ of the anomaly cancellation system for the Standard Model with $n$ right-handed neutrinos (without hypercharge), the $SU(2)$ gauge anomaly equation $\text{accSU2}$ evaluated at $S$ is zero.

In the anomaly cancellation condition (ACC) system for the Standard Model with $n$ right-handed neutrinos, let $S$ be a linear solution (a charge vector satisfying the linear anomaly equations). Then the $SU(3)$ gauge anomaly cancellation equation, denoted as $\text{accSU3}$, evaluates to zero for $S$: $$\text{accSU3}(S) = 0$$

In the Anomaly Cancellation Condition (ACC) system for the Standard Model extended with $n$ right-handed neutrinos, let $S$ be a valid solution to the system (i.e., $S \in (\text{SM } n).\text{Sols}$). Then, the $U(1)$ cubic gauge anomaly cancellation equation $\text{accCube}$ evaluated at the charge assignment $S$ is equal to zero.

For the anomaly cancellation system $(SM \ n)$ representing the Standard Model with $n$ right-handed neutrinos, this function takes a charge assignment $S \in (SM \ n).\text{Charges}$ and proofs that $S$ satisfies the three linear anomaly cancellation conditions (ACCs): 1. The gravitational anomaly: $\text{accGrav}(S) = 0$ 2. The $SU(2)$ gauge anomaly: $\text{accSU2}(S) = 0$ 3. The $SU(3)$ gauge anomaly: $\text{accSU3}(S) = 0$ It returns $S$ as an element of $(SM \ n).\text{LinSols}$, the space of charges that satisfy all linear ACCs in the system.

In the context of the Anomaly Cancellation Condition (ACC) system for the Standard Model with $n$ right-handed neutrinos (denoted as $(SM \ n)$), this function maps a linear solution $S \in (SM \ n).\text{LinSols}$ to a quadratic solution $S \in (SM \ n).\text{QuadSols}$. A linear solution is a charge assignment that satisfies the gravitational, $SU(2)$, and $SU(3)$ anomaly cancellation equations. Because the $(SM \ n)$ system is defined with zero quadratic anomaly equations, any charge vector $S$ that satisfies the linear conditions is vacuously a solution to the quadratic conditions.

For the Standard Model with $n$ right-handed neutrinos, let $S$ be a vector of rational charges that satisfies the linear and quadratic anomaly cancellation conditions (ACCs). Given a proof $hc$ that $S$ also satisfies the cubic anomaly cancellation equation, $\text{accCube}(S) = 0$, this function constructs a complete anomaly-free solution $S \in (\text{SM } n).\text{Sols}$.

For the anomaly cancellation system of the Standard Model with $n$ right-handed neutrinos (denoted $(SM \ n)$), this function takes a charge assignment $S \in (SM \ n).\text{Charges}$ and proofs that $S$ satisfies the three linear anomaly cancellation conditions (ACCs): 1. The gravitational anomaly: $\text{accGrav}(S) = 0$ 2. The $SU(2)$ gauge anomaly: $\text{accSU2}(S) = 0$ 3. The $SU(3)$ gauge anomaly: $\text{accSU3}(S) = 0$ It returns $S$ as an element of $(SM \ n).\text{QuadSols}$, the space of charges satisfying both linear and quadratic ACCs. Since the $(SM \ n)$ system is defined with zero quadratic anomaly equations, any charge vector satisfying the linear conditions is vacuously a solution to the quadratic sector.

In the context of the Standard Model with $n$ right-handed neutrinos (without hypercharge), let $S \in (\text{SM } n).\text{Charges}$ be a vector of rational charges. Given proofs that $S$ satisfies the three linear anomaly cancellation conditions—the gravitational anomaly $\text{accGrav}(S) = 0$, the $SU(2)$ anomaly $\text{accSU2}(S) = 0$, and the $SU(3)$ anomaly $\text{accSU3}(S) = 0$—and the cubic anomaly cancellation condition $\text{accCube}(S) = 0$, this function constructs an element of $(\text{SM } n).\text{Sols}$, representing a complete anomaly-free solution. This construction utilizes the fact that the system has no quadratic anomaly constraints, so satisfying the linear and cubic conditions is sufficient for a full solution.

In the context of the Anomaly Cancellation Condition (ACC) system for the Standard Model with $n$ right-handed neutrinos (denoted as $(\text{SM } n)$), let $S$ be a linear solution ($S \in (\text{SM } n).\text{LinSols}$). A linear solution is a charge assignment that satisfies the three linear anomaly equations: the gravitational anomaly, the $SU(2)$ gauge anomaly, and the $SU(3)$ gauge anomaly. Given a proof $hc$ that $S$ also satisfies the cubic anomaly cancellation equation $\text{accCube}(S) = 0$, this function constructs a complete anomaly-free solution $S \in (\text{SM } n).\text{Sols}$. This construction is possible because the $(\text{SM } n)$ system has no quadratic anomaly constraints.

For a given number of right-handed neutrinos $n \in \mathbb{N}$, the definition $\text{SMRHN.SM.perm}(n)$ constructs a group action of the permutation group (typically acting on fermion generations or the $n$ additional neutrinos) on the charge space of the Anomaly Cancellation Condition (ACC) system $\text{SMRHN.SM}(n)$. This action is shown to leave the four primary anomaly equations invariant: the three linear anomalies—gravitational ($\text{accGrav}$), $SU(2)$ ($\text{accSU2}$), and $SU(3)$ ($\text{accSU3}$)—and the cubic $U(1)$ anomaly ($\text{accCube}$). Since there are no quadratic anomalies in this system, the invariance is trivially satisfied for that category.