Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.PlusU1.BMinusL

The definition `SMRHN.PlusU1.BL₁` represents the $B - L$ (Baryon number minus Lepton number) charge assignment for a single family ($n=1$) of fermions in the Standard Model with Right-Handed Neutrinos. It is defined as a solution to the $U(1)$ anomaly cancellation conditions (linear, quadratic, and cubic) where the charges for the six fermion representations are given by the vector $(1, -1, -1, -3, 3, 3)$. Specifically, the value for each index $i \in \{0, \dots, 5\}$ is: - $val(0) = 1$ - $val(1) = -1$ - $val(2) = -1$ - $val(3) = -3$ - $val(4) = 3$ - $val(5) = 3$

For a given number of fermion families $n \in \mathbb{N}$, the definition `SMRHN.PlusU1.BL n` represents the $B - L$ (Baryon number minus Lepton number) charge assignment in the Standard Model with Right-Handed Neutrinos. This assignment is a solution to the $U(1)$ anomaly cancellation conditions for $n$ families. It is constructed as a "family-universal" solution, where the charges for each of the $n$ families are identical to the single-family case `SMRHN.PlusU1.BL₁`. Specifically, for each family, the charges for the six fermion representations are given by the vector $(1, -1, -1, -3, 3, 3)$.

For any charge configuration $S$ in the $n$-generation Standard Model with right-handed neutrinos, the evaluation of the symmetric bilinear form $B$ (defined by `quadBiLin`) on the $B-L$ charge assignment $BL$ and $S$ is given by the following linear combination of the $Y^2$, $SU(2)$, and $SU(3)$ anomaly cancellation conditions: $$B(BL, S) = \frac{1}{2} \text{accYY}(S) + \frac{3}{2} \text{accSU2}(S) - 2 \text{accSU3}(S)$$ where $\text{accYY}(S)$, $\text{accSU2}(S)$, and $\text{accSU3}(S)$ denote the evaluations of the $Y^2$, $SU(2)$, and $SU(3)$ anomaly cancellation conditions on the charge configuration $S$, respectively.

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration that satisfies the linear anomaly cancellation conditions (specifically the $SU(2)$, $SU(3)$, and $Y^2$ anomalies). Then the symmetric bilinear form $B$ associated with the quadratic anomaly cancellation condition, evaluated on the $B-L$ charge assignment $BL$ and the configuration $S$, is zero: $$B(BL, S) = 0.$$

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration that satisfies the linear anomaly cancellation conditions, and let $BL$ denote the $B-L$ (Baryon minus Lepton number) charge assignment. For any rational numbers $a$ and $b$, the evaluation of the quadratic anomaly cancellation condition $Q$ (represented by `accQuad`) on the linear combination $aS + bBL$ satisfies: $$Q(a S + b BL) = a^2 Q(S)$$

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration that satisfies the quadratic anomaly cancellation condition $Q$ (represented by `accQuad`), and let $BL$ denote the $B-L$ (Baryon minus Lepton number) charge assignment. For any rational numbers $a$ and $b$, the evaluation of the quadratic anomaly cancellation condition on the linear combination $aS + bBL$ is zero: $$Q(a S + b BL) = 0$$

For an $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration that satisfies the quadratic anomaly cancellation conditions. Given rational scalars $a, b \in \mathbb{Q}$, this function returns a new solution to the quadratic anomaly cancellation conditions formed by the linear combination $a S + b(B-L)_n$, where $(B-L)_n$ denotes the $B-L$ (baryon minus lepton number) charge assignment.

In the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a charge configuration that satisfies the quadratic anomaly cancellation conditions. For any rational number $a$, the linear combination $aS + b(B-L)_n$ (defined by the function `addQuad`) with $b=0$ is equal to the scalar multiplication of $S$ by $a$: $$\text{addQuad}(S, a, 0) = a \cdot S$$

For an $n$-generation Standard Model with right-handed neutrinos, let $(B-L)_n$ be the baryon minus lepton number charge assignment, where the charges for each generation are given by the vector $(Q, u, d, L, e, \nu) = (1, -1, -1, -3, 3, 3)$. For any rational charge configuration $S$, the symmetric trilinear form $\text{cubeTriLin}$ associated with the cubic anomaly satisfies the following relationship with the gravitational anomaly $\text{accGrav}(S)$ and the $SU(3)$ anomaly $\text{accSU3}(S)$: $$\text{cubeTriLin}((B-L)_n, (B-L)_n, S) = 9 \cdot \text{accGrav}(S) - 24 \cdot \text{accSU3}(S)$$

For the $n$-generation Standard Model with right-handed neutrinos, let $(B-L)_n$ be the baryon minus lepton number charge assignment. For any charge configuration $S$ that is a linear solution to the anomaly cancellation conditions (satisfying $\text{accGrav}(S) = 0$ and $\text{accSU3}(S) = 0$), the symmetric trilinear form $\text{cubeTriLin}$ associated with the cubic anomaly satisfies: $$\text{cubeTriLin}((B-L)_n, (B-L)_n, S) = 0$$

For the $n$-generation Standard Model with right-handed neutrinos, let $(B-L)_n$ be the baryon minus lepton number charge assignment. Let $S$ be a charge configuration that is a linear solution to the anomaly cancellation conditions. For any rational scalars $a, b \in \mathbb{Q}$, the cubic anomaly $\mathcal{A}_{\text{cube}}$ evaluated on the linear combination $a S + b (B-L)_n$ is given by: \[ \mathcal{A}_{\text{cube}}(a S + b (B-L)_n) = a^3 \mathcal{A}_{\text{cube}}(S) + 3 a^2 b \tau(S, S, (B-L)_n) \] where $\tau$ is the symmetric trilinear form associated with the cubic anomaly.