Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.B3

The definition $B_3$ represents a specific element in the space of rational charges for the MSSM, $\text{MSSMACC.Charges} \cong \mathbb{Q}^{20}$. This charge assignment corresponds to the $B-L$ symmetry (scaled by a factor of 3 to yield integers) for the first two generations of fermions, while the signs are reversed for the third generation. Specifically, for the matter fermion species $(Q, u^c, d^c, L, e^c, \nu^c)$ and the three generations $i \in \{1, 2, 3\}$, the charges are: - For $i=1$ and $i=2$: $(1, -1, -1, -3, 3, 3)$. - For $i=3$: $(-1, 1, 1, 3, -3, -3)$. Additionally, the Higgsinos $(\tilde{H}_d, \tilde{H}_u)$ are assigned the charges $(-3, 3)$.

The definition $B_3$ represents a specific anomaly-free solution within the anomaly cancellation system for the Minimal Supersymmetric Standard Model (MSSM) with three generations and right-handed neutrinos. It is constructed from the charge assignment $B_{3\text{AsCharge}} \in \mathbb{Q}^{20}$ (which corresponds to a $B-L$ symmetry with a sign flip for the third generation) and provides the necessary proofs that this assignment satisfies all six anomaly cancellation conditions: 1. The gravitational anomaly condition $\text{accGrav}(B_{3\text{AsCharge}}) = 0$. 2. The $SU(2)$ gauge anomaly condition $\text{accSU2}(B_{3\text{AsCharge}}) = 0$. 3. The $SU(3)$ gauge anomaly condition $\text{accSU3}(B_{3\text{AsCharge}}) = 0$. 4. The hypercharge anomaly condition $\text{accYY}(B_{3\text{AsCharge}}) = 0$. 5. The quadratic anomaly condition $\text{accQuad}(B_{3\text{AsCharge}}) = 0$. 6. The cubic anomaly condition $\text{accCube}(B_{3\text{AsCharge}}) = 0$.

The underlying charge assignment of the anomaly-free solution $B_3$ for the MSSM is equal to $B_{3\text{AsCharge}}$.

Let $B_3 \in \mathbb{Q}^{20}$ be the specific anomaly-free charge assignment for the MSSM. For any charge assignment $R \in \mathbb{Q}^{20}$ that satisfies the four linear anomaly cancellation conditions of the MSSM (the gravitational, $SU(2)$, $SU(3)$, and hypercharge anomalies), the symmetric trilinear form $f$ (denoted by `cubeTriLin`) associated with the cubic anomaly cancellation condition satisfies $f(B_3, B_3, R) = 0$.