Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.Y3

The charge assignment $Y_3$ is a vector in the rational charge space $\mathbb{Q}^{20}$ of the MSSM anomaly cancellation system. It is defined by assigning charges to the three generations of fermions and the two Higgsinos. For the first two generations, the charges match the standard hypercharge assignments (normalized such that $Q=1$); for the third generation, the charges are the negatives of the standard hypercharge assignments. The Higgsino charges are set to their standard hypercharge values. Specifically, the charges for each species $s$ and generation $i \in \{1, 2, 3\}$ are: - **Quark doublets ($Q_i$):** $Y_3(Q_1) = 1, Y_3(Q_2) = 1, Y_3(Q_3) = -1$ - **Up-type singlets ($u^c_i$):** $Y_3(u^c_1) = -4, Y_3(u^c_2) = -4, Y_3(u^c_3) = 4$ - **Down-type singlets ($d^c_i$):** $Y_3(d^c_1) = 2, Y_3(d^c_2) = 2, Y_3(d^c_3) = -2$ - **Lepton doublets ($L_i$):** $Y_3(L_1) = -3, Y_3(L_2) = -3, Y_3(L_3) = 3$ - **Charged lepton singlets ($e^c_i$):** $Y_3(e^c_1) = 6, Y_3(e^c_2) = 6, Y_3(e^c_3) = -6$ - **Right-handed neutrinos ($\nu^c_i$):** $Y_3(\nu^c_1) = 0, Y_3(\nu^c_2) = 0, Y_3(\nu^c_3) = 0$ - **Higgsinos:** $Y_3(\tilde{H}_d) = -3, Y_3(\tilde{H}_u) = 3$

The object $Y_3$ is the anomaly-free solution for the MSSM anomaly cancellation system corresponding to the specific charge assignment vector $Y_3 \in \mathbb{Q}^{20}$ (given by `MSSMACC.Y₃AsCharge`). It is constructed by providing proofs that this vector satisfies all six anomaly cancellation conditions: the gravitational, $SU(2)$, $SU(3)$, and hypercharge linear conditions, the quadratic condition, and the cubic condition.

The underlying charge vector of the $Y_3$ anomaly-free solution is exactly the $Y_3$ charge assignment vector.

Let $f: \mathbb{Q}^{20} \times \mathbb{Q}^{20} \times \mathbb{Q}^{20} \to \mathbb{Q}$ be the symmetric trilinear form `cubeTriLin` corresponding to the cubic anomaly cancellation condition for the MSSM with three generations and right-handed neutrinos. Let $Y_3 \in \mathbb{Q}^{20}$ be the specific anomaly-free charge assignment vector. For any charge vector $R \in \mathbb{Q}^{20}$ that satisfies the four linear anomaly cancellation conditions (the gravitational, $SU(2)$, $SU(3)$, and hypercharge anomalies), it holds that $f(Y_3, Y_3, R) = 0$.