Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.MinimallyAllowsTerm.OfFinset

Given a finite set $s$ of elements in $\mathcal{Z}$ (representing charges), this function returns the multiset containing all singleton multisets $\{x\}$ such that $x \in s$. Mathematically, for a finite set $s = \{x_1, x_2, \dots, x_n\}$, the function returns the multiset collection $\{\{x_1\}, \{x_2\}, \dots, \{x_n\}\}$.

Let $s$ be a finite set of charges in $\mathcal{Z}$. For any multiset $X$ of charges, $X$ is an element of the collection of singleton multisets derived from $s$ (denoted `toMultisetsOne s`) if and only if the set of elements in $X$ is a subset of $s$ and the cardinality of $X$ is $1$.

Given a finite set of charges $s \subseteq \mathcal{Z}$, this function constructs the multiset of all multisets $X$ of cardinality 2 whose elements are contained in $s$. Specifically, the resulting collection includes multisets of the form $\{x, x\}$ for every $x \in s$, and multisets $\{x, y\}$ for every distinct pair $x, y \in s$.

Let $\mathcal{Z}$ be a type of charges with decidable equality and $s$ be a finite set of charges in $\mathcal{Z}$. A multiset $X$ of charges is an element of the collection `toMultisetsTwo s` if and only if the set of its elements is a subset of $s$ (i.e., $X.\text{toFinset} \subseteq s$) and its cardinality is exactly 2.

Given a finite set $s$ of charges in $\mathcal{Z}$, this function constructs the multiset of all multisets $X$ of cardinality 3 such that every element of $X$ is an element of $s$. This collection includes multisets with repeated elements, specifically those of the forms $\{x, x, x\}$, $\{x, y, y\}$, and $\{x, y, z\}$ for distinct $x, y, z \in s$.

Let $\mathcal{Z}$ be a type of charges and $s \subseteq \mathcal{Z}$ be a finite set. A multiset $X$ is an element of the collection `toMultisetsThree s` if and only if every element contained in $X$ is also an element of $s$ and the cardinality of $X$ (the total number of elements counting multiplicities) is 3.

Given finite sets of charges $S_5$ and $S_{10}$ in an additive group $\mathcal{Z}$ (representing the available charges for fields in the $\mathbf{5}$ and $\mathbf{10}$ representations of $SU(5)$), this function constructs the multiset of all charge spectra $(q_{\bar{5}H_d}, q_{5H_u}, Q_{\bar{5}M}, Q_{10})$ that minimally allow a specific potential term $T$. A spectrum is "minimal" if it contains exactly the number of fields required by the interaction $T$ and satisfies the $U(1)$ selection rule (the weighted sum of charges is zero). The selection rules for each potential term $T$ are: - $\mu$: $q_{\bar{5}H_d}, q_{5H_u} \in S_5$ such that $-q_{\bar{5}H_d} + q_{5H_u} = 0$. - $K^2$: $q_{\bar{5}H_d}, q_{5H_u} \in S_5$ and $q_{10} \in S_{10}$ such that $q_{\bar{5}H_d} + q_{5H_u} + q_{10} = 0$. - $K^1$: $q_5 \in S_5$ and $q_{10,1}, q_{10,2} \in S_{10}$ such that $-q_5 + q_{10,1} + q_{10,2} = 0$. - $W^4$: $q_{\bar{5}H_d}, q_{5H_u} \in S_5$ and $q_5 \in S_5$ such that $q_{\bar{5}H_d} - 2q_{5H_u} + q_5 = 0$. - $W^3$: $q_{5H_u} \in S_5$ and $q_{5,1}, q_{5,2} \in S_5$ such that $-2q_{5H_u} + q_{5,1} + q_{5,2} = 0$. - $W^2$: $q_{\bar{5}H_d} \in S_5$ and $q_{10,1}, q_{10,2}, q_{10,3} \in S_{10}$ such that $q_{\bar{5}H_d} + q_{10,1} + q_{10,2} + q_{10,3} = 0$. - $W^1$: $q_5 \in S_5$ and $q_{10,1}, q_{10,2}, q_{10,3} \in S_{10}$ such that $q_5 + q_{10,1} + q_{10,2} + q_{10,3} = 0$. - $\Lambda$: $q_{5,1}, q_{5,2} \in S_5$ and $q_{10} \in S_{10}$ such that $q_{5,1} + q_{5,2} + q_{10} = 0$. - $\beta$: $q_{5H_u} \in S_5$ and $q_5 \in S_5$ such that $-q_{5H_u} + q_5 = 0$. - $\text{topYukawa}$: $q_{5H_u} \in S_5$ and $q_{10,1}, q_{10,2} \in S_{10}$ such that $-q_{5H_u} + q_{10,1} + q_{10,2} = 0$. - $\text{bottomYukawa}$: $q_{\bar{5}H_d} \in S_5, q_5 \in S_5$, and $q_{10} \in S_{10}$ such that $q_{\bar{5}H_d} + q_5 + q_{10} = 0$. In each case, fields not mentioned in the selection rule are assigned empty charge sets or `none`.

For any finite sets of charges $S_5, S_{10} \subset \mathcal{Z}$ and any potential term $T$ of the $SU(5)$ SUSY GUT, if a charge spectrum $x$ is an element of the multiset of spectra that minimally allow $T$ (denoted `minimallyAllowsTermsOfFinset` $S_5 S_{10} T$), then $x$ is an element of the set of all charge spectra whose component charges are drawn from $S_5$ and $S_{10}$ (denoted `ofFinset` $S_5 S_{10}$).

Let $\mathcal{Z}$ be an additive group of charges. For any finite sets of charges $S_5, S_{10} \subseteq \mathcal{Z}$ and any potential term $T$ of the $SU(5)$ SUSY GUT, the multiset of charge spectra that minimally allow the term $T$ (constructed using charges from $S_5$ and $S_{10}$), denoted as $\text{minimallyAllowsTermsOfFinset}(S_5, S_{10}, T)$, is a subset of the set of all charge spectra whose component charges are drawn from $S_5$ and $S_{10}$, denoted as $\text{ofFinset}(S_5, S_{10})$.

Let $S_5$ and $S_{10}$ be finite sets of charges in an abelian group $\mathcal{Z}$, and let $T$ be a potential term in the $SU(5)$ theory. If a charge spectrum $x$ is an element of the multiset $\text{minimallyAllowsTermsOfFinset}(S_5, S_{10}, T)$, then $x$ allows the potential term $T$.

Let $S_5$ and $S_{10}$ be finite sets of charges in an additive group $\mathcal{Z}$. For any potential term $T$ in the $SU(5)$ supersymmetric theory, if a charge spectrum $x$ is an element of the multiset $\text{minimallyAllowsTermsOfFinset}(S_5, S_{10}, T)$, then $x$ minimally allows the potential term $T$. This means the spectrum $x$ contains only the fields required for the interaction $T$ and satisfies the corresponding $U(1)$ selection rule.

Let $S_5$ and $S_{10}$ be finite sets of charges in an additive group $\mathcal{Z}$, and let $T$ be a potential term in the $SU(5)$ theory. If a charge spectrum $x$ minimally allows the potential term $T$ and its constituent charges are contained within the sets $S_5$ and $S_{10}$ (i.e., $x \in \text{ofFinset } S_5 S_{10}$), then $x$ is an element of the multiset $\text{minimallyAllowsTermsOfFinset}(S_5, S_{10}, T)$.

Let $S_5$ and $S_{10}$ be finite sets of charges in an additive group $\mathcal{Z}$, and let $T$ be a potential term in the $SU(5)$ supersymmetric theory. For any charge spectrum $x$ whose constituent charges are contained within $S_5$ and $S_{10}$ (i.e., $x \in \text{ofFinset } S_5 S_{10}$), $x$ minimally allows the potential term $T$ if and only if $x$ is an element of the multiset $\text{minimallyAllowsTermsOfFinset}(S_5, S_{10}, T)$.

Let $\mathcal{Z}$ be an additive group of charges. For any potential term $T$ and finite sets of charges $S_5, S'_5, S_{10}, S'_{10} \subseteq \mathcal{Z}$, if $S'_5 \subseteq S_5$ and $S'_{10} \subseteq S_{10}$, then the multiset of charge spectra that minimally allow the term $T$ constructed from the smaller sets, $\text{minimallyAllowsTermsOfFinset}(S'_5, S'_{10}, T)$, is a sub-multiset of the multiset constructed from the larger sets, $\text{minimallyAllowsTermsOfFinset}(S_5, S_{10}, T)$.

Let $\mathcal{Z}$ be an abelian group of charges, and let $S_5, S_{10} \subset \mathcal{Z}$ be finite sets of available charges for fields in the $\mathbf{5}$ and $\mathbf{10}$ representations of $SU(5)$. If a charge spectrum $x$ is an element of the multiset of spectra that minimally allow the top Yukawa coupling (meaning $x$ contains only the charges $q_{5H_u} \in S_5$ and $q_{10,1}, q_{10,2} \in S_{10}$ required to satisfy the selection rule $-q_{5H_u} + q_{10,1} + q_{10,2} = 0$), then $x$ is not phenomenologically constrained. That is, $x$ does not allow any of the potentially dangerous terms $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$.