Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.Completions

Let $x$ be an $SU(5)$ charge spectrum. We say that $x$ is **complete** if the following conditions are met: 1. The down-type Higgs charge $qH_d$ is present. 2. The up-type Higgs charge $qH_u$ is present. 3. The set of $\bar{\mathbf{5}}$ charges $Q_{\bar{5}}$ is non-empty. 4. The set of $\mathbf{10}$ charges $Q_{10}$ is non-empty.

For an $SU(5)$ charge spectrum $x$ defined over a type of charges $\mathcal{Z}$ with decidable equality, it is decidable whether $x$ is complete. A charge spectrum $x$ is considered complete if it contains a down-type Higgs charge $qH_d$, an up-type Higgs charge $qH_u$, and the sets of $\bar{\mathbf{5}}$ charges $Q_{\bar{5}}$ and $\mathbf{10}$ charges $Q_{10}$ are both non-empty.

The empty $SU(5)$ charge spectrum $\emptyset$ is not complete. A charge spectrum is defined as complete if it contains both the down-type and up-type Higgs charges ($qH_d$ and $qH_u$) and has non-empty sets of charges for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ matter representations ($Q_{\bar{5}}$ and $Q_{10}$).

Let $x$ and $y$ be two $SU(5)$ charge spectra. If $x$ is a subset of $y$ ($x \subseteq y$) and $x$ is complete, then $y$ is also complete. A charge spectrum is defined as complete if it contains the down-type Higgs charge $qH_d$, the up-type Higgs charge $qH_u$, and the sets of $\bar{\mathbf{5}}$ and $\mathbf{10}$ matter charges are both non-empty. The subset relation $x \subseteq y$ denotes the component-wise inclusion of these charges.

Let $x = (q_{H_d}, q_{H_u}, Q_{\bar{5}}, Q_{10})$ be a charge spectrum over a set of charges $\mathcal{Z}$, where $q_{H_d}, q_{H_u} \in \text{Option } \mathcal{Z}$ represent optional Higgs charges and $Q_{\bar{5}}, Q_{10} \in \text{Finset } \mathcal{Z}$ are finite sets of charges for the $\mathbf{\bar{5}}$ and $\mathbf{10}$ matter representations. Given finite sets of allowed charges $S_5, S_{10} \subseteq \mathcal{Z}$, the function `completions` returns the multiset of all charge spectra $y = (q'_{H_d}, q'_{H_u}, Q'_{\bar{5}}, Q'_{10})$ that are super sets of $x$ and are "complete." A spectrum $y$ in this multiset is formed by replacing any missing components in $x$ with a single charge from the allowed sets as follows: - $q'_{H_d} = q_{H_d}$ if $q_{H_d}$ is present; otherwise, $q'_{H_d} \in \{ \text{some } z \mid z \in S_5 \}$. - $q'_{H_u} = q_{H_u}$ if $q_{H_u}$ is present; otherwise, $q'_{H_u} \in \{ \text{some } z \mid z \in S_5 \}$. - $Q'_{\bar{5}} = Q_{\bar{5}}$ if $Q_{\bar{5}}$ is non-empty; otherwise, $Q'_{\bar{5}} \in \{ \{z\} \mid z \in S_5 \}$. - $Q'_{10} = Q_{10}$ if $Q_{10}$ is non-empty; otherwise, $Q'_{10} \in \{ \{z\} \mid z \in S_{10} \}$. The resulting multiset is the Cartesian product of these possible choices for each component, mapped back into the charge spectrum type.

Let $x = (q_{H_d}, q_{H_u}, Q_{\bar{5}}, Q_{10})$ and $y = (q'_{H_d}, q'_{H_u}, Q'_{\bar{5}}, Q'_{10})$ be charge spectra over a set of charges $\mathcal{Z}$, where $q_{H_d}, q_{H_u} \in \text{Option } \mathcal{Z}$ and $Q_{\bar{5}}, Q_{10} \in \text{Finset } \mathcal{Z}$. For any finite sets of allowed charges $S_5, S_{10} \subseteq \mathcal{Z}$, $y$ is a member of the multiset of completions of $x$ (denoted `completions S5 S10 x`) if and only if the following four conditions hold: 1. The Higgs charge $q'_{H_d}$ is equal to $q_{H_d}$ if $q_{H_d}$ is present; otherwise, $q'_{H_d} = \text{some } z$ for some $z \in S_5$. 2. The Higgs charge $q'_{H_u}$ is equal to $q_{H_u}$ if $q_{H_u}$ is present; otherwise, $q'_{H_u} = \text{some } z$ for some $z \in S_5$. 3. The set of $\mathbf{\bar{5}}$ matter charges $Q'_{\bar{5}}$ is equal to $Q_{\bar{5}}$ if $Q_{\bar{5}}$ is non-empty; otherwise, $Q'_{\bar{5}} = \{z\}$ for some $z \in S_5$. 4. The set of $\mathbf{10}$ matter charges $Q'_{10}$ is equal to $Q_{10}$ if $Q_{10}$ is non-empty; otherwise, $Q'_{10} = \{z\}$ for some $z \in S_{10}$.

For any charge spectrum $x$ and any finite sets of charges $S_5$ and $S_{10}$, the multiset of completions `completions S5 S10 x` contains no duplicate elements.

Let $x$ be an $SU(5)$ charge spectrum and $S_5, S_{10}$ be finite sets of charges. If $x$ is complete—meaning that its up-type Higgs charge $qH_u$ and down-type Higgs charge $qH_d$ are present, and its sets of $\bar{\mathbf{5}}$ and $\mathbf{10}$ charges are non-empty—then the multiset of completions of $x$ with respect to the allowed charge sets $S_5$ and $S_{10}$ is the singleton multiset $\{x\}$.

Let $x$ be an $SU(5)$ charge spectrum and $S_5, S_{10}$ be finite sets of charges. The charge spectrum $x$ is a member of its own multiset of completions $\text{completions}(S_5, S_{10}, x)$ if and only if $x$ is complete.

Let $S_5$ and $S_{10}$ be finite sets of charges, and let $x$ and $y$ be $SU(5)$ charge spectra. If $y$ is an element of the multiset of completions of $x$ given the allowed charges $S_5$ and $S_{10}$ (denoted `completions S5 S10 x`), then $y$ is complete. A charge spectrum is defined as **complete** if it satisfies the following four conditions: 1. The down-type Higgs charge $qH_d$ is present. 2. The up-type Higgs charge $qH_u$ is present. 3. The set of $\bar{\mathbf{5}}$ charges $Q_{\bar{5}}$ is non-empty. 4. The set of $\mathbf{10}$ charges $Q_{10}$ is non-empty.

Let $x$ and $y$ be $SU(5)$ charge spectra and let $S_5, S_{10}$ be finite sets of allowed charges. If $y$ is an element of the multiset of completions of $x$ (constructed by filling missing charges in $x$ using elements from $S_5$ and $S_{10}$), then $x$ is a subset of $y$ ($x \subseteq y$). This means that the charges associated with the Higgs fields ($H_u, H_d$) and the matter representations ($\mathbf{\bar{5}}, \mathbf{10}$) in $x$ are component-wise contained within those of $y$.

Let $S_5$ and $S_{10}$ be finite sets of charges. Let $x$ and $y$ be $SU(5)$ charge spectra such that $x \subseteq y$. If $y$ is complete and all charges in $y$ are contained in $S_5$ and $S_{10}$ (i.e., $y \in \text{ofFinset}(S_5, S_{10})$), then there exists a completion $z$ of $x$ relative to $S_5$ and $S_{10}$ (i.e., $z \in \text{completions}(S_5, S_{10}, x)$) such that $z \subseteq y$.

Given a finite set of charges $S_5 \subset \mathcal{Z}$ and a charge spectrum $x$, the function `completionsTopYukawa` constructs a multiset of charge spectra. Each spectrum in this multiset is formed by taking the $H_u$ charge ($x.q_{H_u}$) and the set of $10$-representation charges ($x.Q_{10}$) from the input spectrum $x$, and completing them with an $H_d$ charge $q_{H_d} \in S_5$ and a single $\bar{5}$-representation charge $q_{\bar{5}} \in S_5$. Specifically, the function maps every pair $(q_{H_d}, q_{\bar{5}}) \in S_5 \times S_5$ to a new charge spectrum $\langle q_{H_d}, x.q_{H_u}, \{q_{\bar{5}}\}, x.Q_{10} \rangle$.

For any finite set of charges $S_5 \subset \mathcal{Z}$ and any charge spectrum $x$, the multiset $\text{completionsTopYukawa}(S_5, x)$, which consists of charge spectra formed by supplementing $x$ with $H_d$ and $\bar{5}$-representation charges from $S_5$, contains no duplicate elements.

Let $\mathcal{Z}$ be an additive abelian group of charges. For any finite sets of charges $S_5, S_{10} \subset \mathcal{Z}$ and any charge spectrum $x$, if $x$ is a spectrum that minimally allows the top Yukawa coupling (which implies $x$ contains an $H_u$ charge and a non-empty set of $\mathbf{10}$ charges, but lacks an $H_d$ charge and $\mathbf{\bar{5}}$ charges), then the multiset of all completions of $x$ using $S_5$ and $S_{10}$ is equal to the multiset of top Yukawa completions of $x$ using $S_5$. That is, $$\text{completions}(S_5, S_{10}, x) = \text{completionsTopYukawa}(S_5, x)$$