Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.ZMod

For a positive integer $n$, the set of $\mathbb{Z}_n$ charges is the finite set of all $SU(5)$ charge spectra $x$ with charge values in the cyclic group $\mathbb{Z}_n$ that satisfy the following three conditions: 1. $x$ is **complete** ($\text{IsComplete } x$), meaning it contains the required Higgs and matter charges. 2. $x$ is **not phenomenologically constrained** ($\neg \text{IsPhenoConstrained } x$), meaning it does not allow terms that lead to proton decay or R-parity violation. 3. $x$ **does not regenerate dangerous couplings** via insertions of up to 4 Yukawa-related singlets ($\neg \text{YukawaGeneratesDangerousAtLevel } x \ 4$).

The finite set of $SU(5)$ charge spectra with values in the cyclic group $\mathbb{Z}_1$ that are complete, not phenomenologically constrained, and do not regenerate dangerous couplings via insertions of up to 4 Yukawa-related singlets is empty. Symbolically, this is expressed as: $$\text{ZModCharges } 1 = \emptyset$$ where $\text{ZModCharges } n$ denotes the set of $SU(5)$ charge spectra with charge values in $\mathbb{Z}_n$ that satisfy these physical viability criteria.

The set of $SU(5)$ charge spectra with values in the cyclic group $\mathbb{Z}_2$ that are complete, not phenomenologically constrained, and do not regenerate dangerous couplings via insertions of up to 4 Yukawa-related singlets is empty.

The set of viable $SU(5)$ charge spectra with values in the cyclic group $\mathbb{Z}_3$, denoted as $\text{ZModCharges}(3)$, is empty. Here, a charge spectrum is defined as viable if it is complete, is not phenomenologically constrained, and does not regenerate dangerous couplings via insertions of up to 4 Yukawa-related singlets.

The set of viable $SU(5)$ charge spectra with values in the cyclic group $\mathbb{Z}_4$, denoted as $\text{ZModCharges } 4$, consists of exactly four elements: \begin{align*} \{ &\langle 0, 2, \{1\}, \{3\} \rangle, \langle 0, 2, \{3\}, \{1\} \rangle, \\ &\langle 1, 2, \{0\}, \{3\} \rangle, \langle 3, 2, \{0\}, \{1\} \rangle \} \end{align*} where each spectrum is represented as a tuple $\langle qH_d, qH_u, Q_{\bar{\mathbf{5}}}, Q_{\mathbf{10}} \rangle$. In this notation, $qH_d$ and $qH_u$ are the charges of the down-type and up-type Higgs fields in $\mathbb{Z}_4$, while $Q_{\bar{\mathbf{5}}}$ and $Q_{\mathbf{10}}$ are the sets of charges for the matter fields in the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations, respectively. A charge spectrum is defined as viable if it satisfies three conditions: it is complete (contains the required Higgs and matter charges), it is not phenomenologically constrained (prevents terms leading to proton decay or R-parity violation), and it does not regenerate dangerous couplings via the insertion of up to four Yukawa-related singlets.

The set of viable $SU(5)$ charge spectra with values in the cyclic group $\mathbb{Z}_5$, denoted as $\text{ZModCharges } 5$, is empty. A charge spectrum $x$ is considered viable if it satisfies three conditions: it is complete (contains required Higgs and matter charges), it is not phenomenologically constrained (forbids terms leading to proton decay or R-parity violation), and it does not regenerate dangerous couplings via the insertion of up to 4 Yukawa-related singlets.

The finite set of viable $SU(5)$ charge spectra with values in the cyclic group $\mathbb{Z}_6$, denoted as $\text{ZModCharges } 6$, is equal to the following set of 16 spectra: \begin{align*} \{ &\langle 0, 2, \{5\}, \{1\} \rangle, \langle 0, 4, \{1\}, \{5\} \rangle, \langle 1, 0, \{2\}, \{3\} \rangle, \langle 1, 2, \{4\}, \{1\} \rangle, \\ &\langle 1, 4, \{0\}, \{5\} \rangle, \langle 1, 4, \{3\}, \{2\} \rangle, \langle 2, 0, \{1\}, \{3\} \rangle, \langle 2, 4, \{5\}, \{5\} \rangle, \\ &\langle 3, 2, \{5\}, \{4\} \rangle, \langle 3, 4, \{1\}, \{2\} \rangle, \langle 4, 0, \{5\}, \{3\} \rangle, \langle 4, 2, \{1\}, \{1\} \rangle, \\ &\langle 5, 0, \{4\}, \{3\} \rangle, \langle 5, 2, \{0\}, \{1\} \rangle, \langle 5, 2, \{3\}, \{4\} \rangle, \langle 5, 4, \{2\}, \{5\} \rangle \} \end{align*} where each spectrum is represented as $\langle qH_d, qH_u, Q_{\bar{\mathbf{5}}}, Q_{\mathbf{10}} \rangle$, with $qH_d, qH_u$ being the down-type and up-type Higgs charges in $\mathbb{Z}_6$, and $Q_{\bar{\mathbf{5}}}, Q_{\mathbf{10}}$ being the sets of matter charges in $\mathbb{Z}_6$. A spectrum is viable if it is complete, satisfies phenomenological constraints against proton decay and R-parity violation, and does not regenerate dangerous couplings via up to four Yukawa-related singlet insertions.

For non-zero natural numbers $n$ and $m$, this defines the finite set of $SU(5)$ charge spectra $x$ with charges in the symmetry group $\mathbb{Z}_n \times \mathbb{Z}_m$ that are phenomenologically viable. A charge spectrum $x$ is an element of this set if it satisfies the following three conditions: 1. **Completeness**: $x$ is complete, meaning it contains the Higgs charges $qH_d$ and $qH_u$, and has non-empty sets of matter charges $Q_{\bar{\mathbf{5}}}$ and $Q_{\mathbf{10}}$. 2. **No phenomenological constraints**: The symmetry forbids the "dangerous" terms $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1,$ and $K_2$ which lead to proton decay or R-parity violation. 3. **No Yukawa regeneration at level 4**: The insertion of up to 4 Yukawa-related singlets cannot regenerate any of the phenomenologically constrained terms. Mathematically, if $Y_4(x)$ is the multiset of charges formed by summing up to 4 Yukawa-associated charges and $P(x)$ is the multiset of dangerous superpotential charges, then $Y_4(x) \cap P(x) = \emptyset$.