Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.Permutations

For a given natural number $n$ (representing the number of generations), this definition describes the group of permutations acting on the six species of particles in the Standard Model with right-handed neutrinos. It is defined as the set of functions from the set $\{0, 1, 2, 3, 4, 5\}$ (representing the six species) to the symmetric group $S_n$, which consists of all permutations of $n$ elements. This corresponds to the direct product of six copies of the symmetric group, $(S_n)^6$.

For a given number of generations $n \in \mathbb{N}$, the space `PermGroup n`, which corresponds to the direct product of six copies of the symmetric group $S_n$ (one for each particle species in the Standard Model with right-handed neutrinos), is equipped with a group structure. The group operation is defined component-wise, where the product of two elements is the point-wise composition of permutations for each particle species.

For a given natural number $n$ representing the number of fermion generations and an element $f = (\sigma_0, \dots, \sigma_5)$ in the permutation group $(S_n)^6$, the function `chargeMap f` is a $\mathbb{Q}$-linear map from the space of charges $\mathbb{Q}^{6n}$ to itself. This map transforms a charge configuration $S$—which assigns a rational charge $q_{i,j}$ to the $j$-th generation of the $i$-th fermion species—by permuting the generation indices within each species. Specifically, the resulting charge for the $j$-th generation of species $i$ is given by $q_{i, \sigma_i(j)}$.

For $n$ fermion generations, this defines the $\mathbb{Q}$-linear representation of the permutation group $(S_n)^6$ on the vector space of charges $\mathbb{Q}^{6n}$ for the Standard Model with right-handed neutrinos. For an element $f = (\sigma_0, \sigma_1, \dots, \sigma_5) \in (S_n)^6$ and a charge configuration $S \in \mathbb{Q}^{6n}$, the representation acts by permuting the generation indices of each fermion species $i$ by the inverse permutation $\sigma_i^{-1}$.

For any natural number $n$, let $S \in \mathbb{Q}^{6n}$ be a charge configuration for the $n$-generation Standard Model with right-handed neutrinos. Let $f = (\sigma_0, \sigma_1, \dots, \sigma_5)$ be an element of the permutation group $(S_n)^6$, where each $\sigma_i \in S_n$ is a permutation of the $n$ generations for the $i$-th fermion species. Let $\text{repCharges}(f, S)$ be the $\mathbb{Q}$-linear representation of the group element $f$ on the charge vector $S$. For any species index $j \in \{0, 1, \dots, 5\}$, the projection of the permuted charges onto the $j$-th species, denoted by $\text{toSpecies}_j$, satisfies $$\text{toSpecies}_j(\text{repCharges}(f, S)) = \text{toSpecies}_j(S) \circ \sigma_j^{-1},$$ where $\circ$ denotes function composition.

For an $n$-generation Standard Model with right-handed neutrinos, let $S \in \mathbb{Q}^{6n}$ be a configuration of fermion charges and $f = (\sigma_0, \sigma_1, \dots, \sigma_5)$ be an element of the permutation group $(S_n)^6$, which acts on $S$ by permuting the generation indices of each fermion species. For any species index $j \in \{0, 1, \dots, 5\}$ and any natural number $m$, the sum of the $m$-th powers of the charges of the $j$-th species is invariant under the action of $f$: $$\sum_{i=0}^{n-1} \left( \text{toSpecies}_j(\text{repCharges}(f, S))_i \right)^m = \sum_{i=0}^{n-1} \left( \text{toSpecies}_j(S)_i \right)^m,$$ where $\text{toSpecies}_j$ is the projection of the total charge vector onto the $n$ charges of the $j$-th fermion species.

For an $n$-generation Standard Model with right-handed neutrinos, let $S \in \mathbb{Q}^{6n}$ be a configuration of fermion charges and $f \in (S_n)^6$ be an element of the permutation group that acts on $S$ by permuting the generation indices of each of the six fermion species. The gravitational anomaly $\text{accGrav}$ is invariant under this action: $$\text{accGrav}(\text{repCharges}(f, S)) = \text{accGrav}(S)$$

For the $n$-generation Standard Model with right-handed neutrinos, let $S \in \mathbb{Q}^{6n}$ be a configuration of rational charges and $f \in (S_n)^6$ be an element of the permutation group that acts on $S$ by permuting the generation indices of each fermion species. The $SU(2)$ anomaly cancellation condition, which is defined as the sum over generations $\text{accSU2}(S) = \sum_{i=1}^{n} (3 Q_i + L_i)$, is invariant under this action: $$\text{accSU2}(\text{repCharges}(f, S)) = \text{accSU2}(S).$$

In the Standard Model with $n$ generations of fermions and right-handed neutrinos, let $S$ be a configuration of rational charges and $f$ be an element of the permutation group $(S_n)^6$ that permutes the generation indices of each of the six fermion species. The $SU(3)$ anomaly cancellation condition (ACC) is invariant under the action of these permutations: $$\text{accSU3}(\text{repCharges}(f, S)) = \text{accSU3}(S),$$ where $\text{repCharges}$ is the $\mathbb{Q}$-linear representation of $(S_n)^6$ on the space of charges $\mathbb{Q}^{6n}$, and $\text{accSU3}$ is the linear map calculating the $SU(3)$ anomaly value $\sum_{i=0}^{n-1} (2 Q_i + U_i + D_i)$.

For an $n$-generation Standard Model with right-handed neutrinos, let $S$ be a configuration of rational charges and let $f \in (S_n)^6$ be an element of the permutation group that acts independently on the $n$ generations of each of the six fermion species (the left-handed quark doublet $Q$, the right-handed up-type quark $u$, the right-handed down-type quark $d$, the left-handed lepton doublet $L$, the right-handed charged lepton $e$, and the right-handed neutrino $\nu$). The $Y^2$ anomaly cancellation condition $\text{accYY}$ is invariant under the action of $f$: $$\text{accYY}(\text{repCharges}(f, S)) = \text{accYY}(S).$$

In the Standard Model with $n$ generations of fermions and right-handed neutrinos, let $S \in \mathbb{Q}^{6n}$ be a configuration of rational charges and $f$ be an element of the permutation group $(S_n)^6$ that independently permutes the generation indices of each of the six fermion species (the left-handed quark doublet $Q$, the right-handed up-type quark $u$, the right-handed down-type quark $d$, the left-handed lepton doublet $L$, the right-handed charged lepton $e$, and the right-handed neutrino $\nu$). The quadratic anomaly cancellation condition (ACC), $\text{accQuad}$, is invariant under the action of these permutations: $$\text{accQuad}(\text{repCharges}(f, S)) = \text{accQuad}(S),$$ where $\text{repCharges}$ is the $\mathbb{Q}$-linear representation of $(S_n)^6$ on the space of charges $\mathbb{Q}^{6n}$, and $\text{accQuad}$ is the quadratic form defined by: $$\text{accQuad}(S) = \sum_{i=0}^{n-1} \left( Q_i(S)^2 - 2u_i(S)^2 + d_i(S)^2 - L_i(S)^2 + e_i(S)^2 \right).$$

For the $n$-generation Standard Model with right-handed neutrinos, let $S \in \mathbb{Q}^{6n}$ be a configuration of rational charges and let $f \in (S_n)^6$ be an element of the permutation group that acts on the generation indices of each of the six fermion species. The cubic anomaly cancellation condition $\mathcal{A}_{\text{cube}}$, defined by \[ \mathcal{A}_{\text{cube}}(S) = \sum_{i=0}^{n-1} \left( 6 Q_i(S)^3 + 3 U_i(S)^3 + 3 D_i(S)^3 + 2 L_i(S)^3 + E_i(S)^3 + N_i(S)^3 \right), \] is invariant under the action of $f$: \[ \mathcal{A}_{\text{cube}}(\text{repCharges}(f, S)) = \mathcal{A}_{\text{cube}}(S). \]