Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.Permutations

The group of family permutations for the Minimal Supersymmetric Standard Model (MSSM) is defined as the product group $S_3^6$. It is represented as the type of functions from an index set of size 6 (corresponding to the six types of MSSM charge multiplets) to the symmetric group $S_3$ (the group of permutations of the three generations or families).

The type $\text{PermGroup}$, representing the family permutations of the Minimal Supersymmetric Standard Model (MSSM), is equipped with a group structure. This group structure is the product group structure of $S_3^6$, where the group operations (identity, multiplication, and inverse) are defined pointwise for each of the six charge multiplets.

For an element $f$ of the family permutation group $\text{PermGroup} \cong S_3^6$, the function returns a $\mathbb{Q}$-linear map from the space of MSSM charges to itself. This map acts by permuting the three generations (families) of each of the six fermion species according to the permutations specified by $f$. Specifically, for each species $i \in \{1, \dots, 6\}$, the charges assigned to the three generations are reordered by the permutation $f(i) \in S_3$.

Let $S$ be a configuration of rational charges for the MSSM, and let $f$ be an element of the family permutation group $\text{PermGroup} \cong S_3^6$, where $f_j \in S_3$ denotes the permutation acting on the three generations of the $j$-th fermion species for $j \in \{0, \dots, 5\}$. Let $\text{toSMSpecies}_j(S)$ denote the vector of charges (or the map from family indices to $\mathbb{Q}$) associated with the three generations of the $j$-th species. Then, the charges of species $j$ after applying the permutation map $\text{chargeMap}(f)$ to the total charge configuration $S$ are given by the composition: $$\text{toSMSpecies}_j(\text{chargeMap}(f, S)) = \text{toSMSpecies}_j(S) \circ f_j$$ This shows that the global charge map acts species-wise by permuting family indices according to $f_j$.

The representation $\rho$ of the MSSM family permutation group $G = S_3^6$ on the vector space of rational charges $V = \text{MSSMCharges.Charges}$. For any permutation $f \in G$, the linear map $\rho(f): V \to V$ is defined as the $\mathbb{Q}$-linear map that permutes the three generations of each of the six fermion species. Specifically, the action of $f$ on a charge vector $S$ is given by the linear map corresponding to the inverse permutation $f^{-1}$, ensuring the group homomorphism property $\rho(f \cdot g) = \rho(f) \circ \rho(g)$.

Let $S \in \text{MSSMCharges.Charges}$ be a configuration of rational charges for the Minimal Supersymmetric Standard Model (MSSM), and let $f \in \text{PermGroup} \cong S_3^6$ be a family permutation, where $f_j \in S_3$ is the permutation acting on the three generations of the $j$-th fermion species for $j \in \{0, \dots, 5\}$. Let $\rho(f)$ denote the representation of the permutation group on the space of charges. Then, the charges associated with the $j$-th species after applying the representation $\rho(f)$ to $S$ are given by the composition of the original charges for that species with the inverse permutation $f_j^{-1}$: $$\text{toSMSpecies}_j(\rho(f) S) = \text{toSMSpecies}_j(S) \circ f_j^{-1}$$

For any natural number $m$, any family permutation $f \in \text{PermGroup} \cong S_3^6$, any MSSM charge configuration $S$, and any fermion species index $j \in \{0, \dots, 5\}$, the sum of the $m$-th powers of the charges of the three generations of the $j$-th species remains invariant under the action of the permutation: $$\sum_i \left( (\text{toSMSpecies}_j(\rho(f) S))_i \right)^m = \sum_i \left( (\text{toSMSpecies}_j(S))_i \right)^m$$ where $\rho(f)$ is the representation of the permutation $f$ on the space of charges and $\text{toSMSpecies}_j$ extracts the charges for the generations of the $j$-th species.

For any family permutation $f$ in the MSSM permutation group $S_3^6$ and any vector of rational charges $S$ in the MSSM charge space, the down-type Higgs charge $H_d$ remains invariant under the action of the permutation, such that $H_d(\rho(f)S) = H_d(S)$, where $\rho(f)S$ denotes the charge vector after permuting the generations of the fermion species according to $f$.

Let $G$ be the MSSM family permutation group $S_3^6$ and $V$ be the space of rational MSSM charges. For any family permutation $f \in G$ and any charge configuration $S \in V$, the charge associated with the up-type Higgs doublet $H_u$ remains invariant under the action of the permutation, such that $H_u(\rho(f)S) = H_u(S)$, where $\rho(f)$ denotes the representation of the permutation on the charge space.

For any family permutation $f$ in the MSSM permutation group $S_3^6$ and any vector of rational charges $S$ in the MSSM charge space, the gravitational anomaly cancellation term $\text{accGrav}$ remains invariant under the action of the permutation: $$\text{accGrav}(\rho(f) S) = \text{accGrav}(S)$$ where $\rho(f)$ denotes the representation of the permutation $f$ on the space of charges.

Let $G \cong S_3^6$ be the MSSM family permutation group and $V$ be the space of rational MSSM charges. For any family permutation $f \in G$ and any charge configuration $S \in V$, the $SU(2)$ anomaly cancellation condition $\text{accSU2}$ remains invariant under the action of the permutation, such that $\text{accSU2}(\rho(f)S) = \text{accSU2}(S)$, where $\rho(f)$ denotes the representation of the permutation on the space of charges.

For any family permutation $f$ in the MSSM family permutation group $\text{PermGroup} \cong S_3^6$ and any MSSM charge configuration $S$, the $SU(3)$ anomaly cancellation condition $\text{accSU3}$ is invariant under the action of the permutation: $$\text{accSU3}(\rho(f) S) = \text{accSU3}(S)$$ where $\rho(f)$ is the representation of the permutation $f$ on the space of charges.

For any family permutation $f \in S_3^6$ and any MSSM charge configuration $S \in \text{MSSMCharges.Charges}$, the anomaly cancellation value $\text{accYY}$ remains invariant under the action of the permutation, such that: $$\text{accYY}(\rho(f)S) = \text{accYY}(S)$$ where $\rho(f)$ is the representation of the permutation group on the space of charges that permutes the three generations of each fermion species.

For any family permutation $f$ in the MSSM permutation group $S_3^6$ and any configuration of rational MSSM charges $S$, the quadratic anomaly cancellation condition $accQuad$ remains invariant under the action of the permutation, such that $accQuad(\rho(f)S) = accQuad(S)$, where $\rho(f)$ denotes the representation of the permutation $f$ on the space of charges.

Let $G$ be the MSSM family permutation group $S_3^6$ and $V$ be the space of rational MSSM charges. For any family permutation $f \in G$ and any charge configuration $S \in V$, the cubic anomaly cancellation map $\text{accCube}: V \to \mathbb{Q}$ is invariant under the action of the permutation, such that $\text{accCube}(\rho(f)S) = \text{accCube}(S)$, where $\rho(f)$ denotes the representation of the permutation on the charge space.