Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.NoGrav.Basic

For a given natural number $n$, let $n$ denote the number of right-handed neutrinos added to the Standard Model. The anomaly cancellation condition (ACC) system for the Standard Model with $n$ right-handed neutrinos, excluding the gravitational anomaly, is defined by: 1. A charge space $\mathbb{Q}^{15+n}$ (or the appropriate dimension for the SM fermion content plus $n$ neutrinos) representing the rational hypercharges of the particles. 2. Two linear anomaly equations: the $SU(2)^2 U(1)$ condition ($\text{accSU2}$) and the $SU(3)^2 U(1)$ condition ($\text{accSU3}$). 3. Zero quadratic anomaly equations. 4. One cubic anomaly equation: the $U(1)^3$ condition ($\text{accCube}$).

For an anomaly cancellation system representing the Standard Model with $n$ right-handed neutrinos (excluding the gravitational anomaly), let $S$ be a linear solution to the system's equations. Then the $SU(2)^2 U(1)$ anomaly condition, denoted by $\text{accSU2}$, evaluated at $S$, is equal to zero.

For any linear solution $S$ in the anomaly cancellation condition (ACC) system for the Standard Model with $n$ right-handed neutrinos (excluding the gravitational anomaly), the $SU(3)^2 U(1)$ anomaly cancellation condition is satisfied, such that $\text{accSU3}(S) = 0$.

Consider the anomaly cancellation condition (ACC) system for the Standard Model with $n$ right-handed neutrinos, excluding the gravitational anomaly. For any charge assignment $S$ that is a solution to this system, the cubic anomaly equation $\text{accCube}$ (representing the $U(1)^3$ anomaly condition) evaluates to zero.

Given a vector of rational charges $S$ for the Standard Model with $n$ right-handed neutrinos (represented as an element of the charge space $\mathbb{Q}^{15+n}$), if $S$ satisfies the $SU(2)^2 U(1)$ anomaly cancellation condition ($\text{accSU2}(S) = 0$) and the $SU(3)^2 U(1)$ anomaly cancellation condition ($\text{accSU3}(S) = 0$), then $S$ constitutes an element of the space of linear solutions $(SMNoGrav\ n).LinSols$.

For the Anomaly Cancellation Condition (ACC) system of the Standard Model with $n$ right-handed neutrinos (excluding the gravitational anomaly), this function maps a solution $S$ of the linear anomaly equations to a solution of the quadratic anomaly equations. Since this specific system defines zero quadratic anomaly equations, any charge assignment $S$ that satisfies the linear conditions (the $SU(2)^2 U(1)$ and $SU(3)^2 U(1)$ conditions) trivially satisfies the quadratic conditions.

In the context of the Standard Model with $n$ right-handed neutrinos (excluding gravitational anomalies), let $S$ be an element of the space of quadratic solutions $\text{QuadSols}$ (which, in this system, consists of charges satisfying the $SU(2)^2 U(1)$ and $SU(3)^2 U(1)$ linear conditions). If $S$ also satisfies the cubic $U(1)^3$ anomaly cancellation condition, denoted as $\text{accCube}(S) = 0$, then it is a complete solution to the anomaly cancellation system.

Given a vector of rational charges $S$ for the Standard Model with $n$ right-handed neutrinos (represented as an element of the charge space $\mathbb{Q}^{15+n}$), if $S$ satisfies the $SU(2)^2 U(1)$ anomaly cancellation condition ($\text{accSU2}(S) = 0$) and the $SU(3)^2 U(1)$ anomaly cancellation condition ($\text{accSU3}(S) = 0$), then $S$ constitutes an element of the space of quadratic solutions $(\text{SMNoGrav } n).\text{QuadSols}$. In this specific system, which defines zero quadratic anomaly equations, any charge assignment satisfying these linear conditions trivially satisfies the quadratic requirements.

In the context of the Standard Model with $n$ right-handed neutrinos (excluding gravitational anomalies), let $S$ be a vector of rational charges in the charge space $\mathbb{Q}^{15+n}$. If $S$ satisfies the linear $SU(2)^2 U(1)$ anomaly cancellation condition ($\text{accSU2}(S) = 0$), the linear $SU(3)^2 U(1)$ anomaly cancellation condition ($\text{accSU3}(S) = 0$), and the cubic $U(1)^3$ anomaly cancellation condition ($\text{accCube}(S) = 0$), then $S$ constitutes a complete solution to the anomaly cancellation system.

For the Anomaly Cancellation Condition (ACC) system of the Standard Model with $n$ right-handed neutrinos (excluding the gravitational anomaly), let $S$ be a solution to the linear anomaly equations (the $SU(2)^2 U(1)$ and $SU(3)^2 U(1)$ conditions). If $S$ additionally satisfies the cubic $U(1)^3$ anomaly equation, denoted as $\text{accCube}(S) = 0$, then $S$ is a complete solution to the anomaly cancellation system.

For a natural number $n$, let $SMNoGrav(n)$ be the Anomaly Cancellation Condition (ACC) system for the Standard Model with $n$ right-handed neutrinos and no gravitational anomaly. The definition `SMRHN.SMNoGrav.perm` defines a group action of permutations on this system. This action is characterized by a group of permutations acting on the charge space $\mathbb{Q}^{15+n}$ such that the linear anomaly equations (the $SU(2)^2 U(1)$ and $SU(3)^2 U(1)$ conditions) and the cubic anomaly equation (the $U(1)^3$ condition) remain invariant.