Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.HyperCharge

The hypercharge assignment $Y$ is defined as an element of the rational charge space $\mathbb{Q}^{20}$ for the Minimal Supersymmetric Standard Model (MSSM). Under the representation defined by `toSpecies`, which partitions the 20 charges into matter fermions and Higgsinos, the charges are assigned as follows: - For the three generations of matter fermions ($i \in \{1, 2, 3\}$): - Quark doublets $Q_i$ are assigned the value $1$. - Up-type singlets $u^c_i$ are assigned the value $-4$. - Down-type singlets $d^c_i$ are assigned the value $2$. - Lepton doublets $L_i$ are assigned the value $-3$. - Charged lepton singlets $e^c_i$ are assigned the value $6$. - Right-handed neutrinos $\nu^c_i$ are assigned the value $0$. - For the Higgsino sector: - The down-type Higgsino $\tilde{H}_d$ is assigned the value $-3$. - The up-type Higgsino $\tilde{H}_u$ is assigned the value $3$. These values correspond to the standard $U(1)_Y$ hypercharges scaled by a factor of 6.

The hypercharge solution $Y$ is an element of the solution space $\text{MSSMACC.Sols}$ for the anomaly cancellation system of the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos. It is constructed from the charge vector $Y \in \mathbb{Q}^{20}$ (defined by `YAsCharge`), which assigns the following values to the fermion species: - Quark doublets $Q_i = 1$ - Up-type singlets $u^c_i = -4$ - Down-type singlets $d^c_i = 2$ - Lepton doublets $L_i = -3$ - Charged lepton singlets $e^c_i = 6$ - Right-handed neutrinos $\nu^c_i = 0$ - Down-type Higgsino $\tilde{H}_d = -3$ - Up-type Higgsino $\tilde{H}_u = 3$ This assignment is proven to satisfy all six anomaly cancellation conditions: the gravitational anomaly ($\text{accGrav}$), the $SU(2)$ gauge anomaly ($\text{accSU2}$), the $SU(3)$ gauge anomaly ($\text{accSU3}$), the hypercharge anomaly ($\text{accYY}$), and the quadratic ($\text{accQuad}$) and cubic ($\text{accCube}$) conditions.