Physlib.StringTheory.FTheory.SU5.Charges.Viable

Given a codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the multiset $\text{viableCompletions}(I)$ consists of $\mathbb{Z}$-valued $U(1)$ charge spectra that are completions of charge assignments minimally allowing the top Yukawa coupling. To be included in this multiset, a charge spectrum must be phenomenologically viable—meaning it is not phenomenologically constrained and does not regenerate dangerous couplings (such as those leading to rapid proton decay) via a single insertion of a Yukawa coupling. The multiset is explicitly defined via a lookup table for the geometric configurations `same`, `nearestNeighbor`, and `nextToNearestNeighbor`.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the cardinality of the multiset of viable $U(1)$ charge completions, denoted by $|\text{viableCompletions}(I)|$, is given by: $$|\text{viableCompletions}(I)| = \begin{cases} 70 & \text{if } I = \text{same} \\ 48 & \text{if } I = \text{nearestNeighbor} \\ 37 & \text{if } I = \text{nextToNearestNeighbor} \end{cases}$$ The multiset $\text{viableCompletions}(I)$ contains $\mathbb{Z}$-valued charge spectra that are completions of charge assignments minimally allowing the top Yukawa coupling, while being phenomenologically viable (not constrained by dangerous couplings).

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the multiset $\text{viableCompletions}(I)$ of viable $U(1)$ charge completions contains no duplicate elements.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, every charge spectrum $x$ belonging to the multiset $\text{viableCompletions}(I)$ is not phenomenologically constrained. Specifically, such a charge spectrum $x$ does not allow any of the following operators in the superpotential or Kähler potential: $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, every charge spectrum $x$ belonging to the multiset $\text{viableCompletions}(I)$ does not regenerate dangerous phenomenologically constrained couplings with a single insertion of a Yukawa-related singlet. This means the condition $\text{YukawaGeneratesDangerousAtLevel}(x, 1)$ is false for all $x \in \text{viableCompletions}(I)$.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, the multiset $\text{viableCompletions}(I)$ contains all phenomenologically viable completions of charge spectra that minimally allow the top Yukawa coupling. Specifically, let $S_{\bar{\mathbf{5}}}(I)$ and $S_{\mathbf{10}}(I)$ be the finite sets of allowed $U(1)$ charges for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations associated with configuration $I$. For any charge spectrum $x$ that minimally allows the top Yukawa interaction given $S_{\bar{\mathbf{5}}}(I)$ and $S_{\mathbf{10}}(I)$, if $x$ is not phenomenologically constrained, then any completion $y$ of $x$ that is neither phenomenologically constrained nor generates dangerous couplings (such as those leading to rapid proton decay) via a single Yukawa insertion is an element of the multiset $\text{viableCompletions}(I)$.

Given a codimension-one configuration $I$ of sections in an $I_5$ fiber of an $SU(5)$ F-theory model, this definition provides the multiset of "additional" viable $\mathbb{Z}$-valued $U(1)$ charge spectra. A charge spectrum $(Q_{10}, Q_{\bar{5}})$ is included in this multiset if it: 1. Permits a top Yukawa coupling. 2. Is not phenomenologically constrained (i.e., satisfies specific requirements on the number of generations or representations). 3. Does not regenerate dangerous couplings (such as those leading to rapid proton decay) with a single insertion of a Yukawa coupling. 4. Is not already contained in the multiset of minimal completions (`viableCompletions`). The multiset is defined by an explicit enumeration of valid charge assignments for each of the three possible geometric configurations of the sections: `same`, `nearestNeighbor`, and `nextToNearestNeighbor`.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, the multiset of additional viable $U(1)$ charge spectra, denoted as $\text{viableChargesAdditional}(I)$, contains no duplicate elements.

Let $I$ be a codimension-one configuration of sections in an $SU(5)$ F-theory model. For any $U(1)$ charge spectrum $x$ belonging to the multiset of additional viable charge spectra $\text{viableChargesAdditional}(I)$, the spectrum $x$ is not phenomenologically constrained ($\neg\text{IsPhenoConstrained } x$). This implies that $x$ does not allow any of the operators $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$ that lead to proton decay or R-parity violation in the superpotential or Kähler potential.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, every $U(1)$ charge spectrum $x$ in the multiset of additional viable charge spectra $\text{viableChargesAdditional}(I)$ does not regenerate dangerous couplings with one insertion of a Yukawa coupling. Mathematically, this is denoted as $\neg\text{YukawaGeneratesDangerousAtLevel } x\ 1$. This implies that a single insertion of a Yukawa-related coupling into the theory's operators cannot result in a phenomenologically constrained term (such as those leading to proton decay).

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the multiset of viable $U(1)$ charge completions $\text{viableCompletions}(I)$ and the multiset of additional viable $U(1)$ charge spectra $\text{viableChargesAdditional}(I)$ are disjoint.

For a given codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the multiset $\text{viableCharges}(I)$ consists of all $\mathbb{Z}$-valued $U(1)$ charge spectra $(Q_{10}, Q_{\bar{5}})$ that: 1. Permit a top Yukawa coupling. 2. Are not phenomenologically constrained. 3. Do not regenerate dangerous couplings (such as those leading to rapid proton decay) with a single insertion of a Yukawa coupling. The multiset is defined as the sum of the multiset of viable completions $\text{viableCompletions}(I)$ and the multiset of additional viable charges $\text{viableChargesAdditional}(I)$.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the multiset of viable $U(1)$ charge spectra, denoted as $\text{viableCharges}(I)$, contains no duplicate elements.

For a given codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, the cardinality of the multiset of viable $U(1)$ charge spectra, denoted as $|\text{viableCharges}(I)|$, is determined by the specific geometry of the configuration as follows: $$|\text{viableCharges}(I)| = \begin{cases} 102 & \text{if } I = \text{same} \\ 71 & \text{if } I = \text{nearestNeighbor} \\ 51 & \text{if } I = \text{nextToNearestNeighbor} \end{cases}$$ The multiset $\text{viableCharges}(I)$ contains the charge spectra $(Q_{10}, Q_{\bar{5}})$ that permit a top Yukawa coupling, are not phenomenologically constrained, and do not regenerate dangerous couplings.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, every charge spectrum $x$ in the multiset of viable $U(1)$ charges $\text{viableCharges}(I)$ is an element of the finite set of charge spectra whose constituent charges for the $\mathbf{\bar{5}}$ and $\mathbf{10}$ representations are contained within the sets of allowed charges $\text{allowedBarFiveCharges}(I)$ and $\text{allowedTenCharges}(I)$, respectively.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, every $U(1)$ charge spectrum $x$ that belongs to the multiset of viable charges $\text{viableCharges}(I)$ is complete ($\text{IsComplete } x$). A charge spectrum is considered complete if it contains a down-type Higgs charge $qH_d$, an up-type Higgs charge $qH_u$, and the sets of charges for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations are both non-empty.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, every $U(1)$ charge spectrum $x$ belonging to the multiset of viable charges $\text{viableCharges}(I)$ allows the top Yukawa coupling term.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, every $U(1)$ charge spectrum $x$ that belongs to the multiset of viable charges $\text{viableCharges}(I)$ is not phenomenologically constrained ($\neg\text{IsPhenoConstrained } x$). This implies that the spectrum $x$ does not allow any of the operators $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$ in the superpotential or Kähler potential that are associated with rapid proton decay or R-parity violation.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, every $U(1)$ charge spectrum $x$ belonging to the multiset of viable charges $\text{viableCharges}(I)$ does not regenerate dangerous phenomenologically constrained couplings (such as those associated with rapid proton decay) with a single insertion of a Yukawa-related singlet. This is expressed as the proposition $\neg\text{YukawaGeneratesDangerousAtLevel}(x, 1)$.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, every $U(1)$ charge spectrum $x$ that is an element of the multiset $\text{viableCharges}(I)$ contains at most two charges in its $\bar{5}$ representation, which is expressed as the cardinality of its $\bar{5}$ charge multiset being less than or equal to 2 ($|x.Q_{\bar{5}}| \le 2$).

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, every charge spectrum $x$ in the multiset of viable charges $\text{viableCharges}(I)$ contains at most two charges for matter fields in the $\mathbf{10}$ representation. That is, for any $x \in \text{viableCharges}(I)$, the cardinality of the multiset of charges $x.Q_{10}$ satisfies $|x.Q_{10}| \leq 2$.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, the multiset of viable $U(1)$ charge spectra $\text{viableCharges}(I)$ is phenomenologically closed under the addition of $\bar{5}$ charges from the set of allowed charges $I.\text{allowedBarFiveCharges}$. Specifically, for every $q_5 \in I.\text{allowedBarFiveCharges}$ and every viable charge spectrum $x \in \text{viableCharges}(I)$, inserting the charge $q_5$ into $x$ results in a new charge spectrum that either: 1. is also an element of $\text{viableCharges}(I)$, or 2. is phenomenologically constrained (allowing terms that lead to proton decay or R-parity violation) or regenerates dangerous couplings with the Yukawa couplings.

For any codimension-one configuration $I$ of sections $\sigma_0$ and $\sigma_1$ in an $SU(5)$ F-theory model, the multiset of viable $U(1)$ charge spectra $\text{viableCharges}(I)$ is phenomenologically closed under the addition of $10$-representation charges from the set of allowed charges $S_{10}(I)$. This means that for any spectrum $x \in \text{viableCharges}(I)$ and any charge $q_{10} \in S_{10}(I)$, the new charge spectrum $y$ (formed by adding $q_{10}$ to the multiset of $10$-dimensional charges $Q_{10}$ in $x$) satisfies at least one of the following conditions: 1. $y$ is already an element of $\text{viableCharges}(I)$. 2. $y$ is phenomenologically constrained (i.e., it allows dangerous superpotential or Kähler potential terms). 3. $y$ regenerates dangerous couplings via the insertion of one singlet.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, every charge spectrum $x$ that is an element of the multiset of viable completions $\text{viableCompletions}(I)$ is also an element of the multiset of all viable charges $\text{viableCharges}(I)$.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, let $x$ be a $U(1)$ charge spectrum whose charges for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ representations are members of the sets of allowed charges defined by $I$ (specifically, $x \in \text{ofFinset}(I.\text{allowedBarFiveCharges}, I.\text{allowedTenCharges})$). Then $x$ is an element of the multiset of viable charges $\text{viableCharges}(I)$ if and only if it satisfies the following four conditions: 1. $x$ allows the top Yukawa coupling interaction ($\text{AllowsTerm } x \text{ topYukawa}$). 2. $x$ is not phenomenologically constrained ($\neg \text{IsPhenoConstrained } x$), meaning it does not allow operators in the superpotential or Kähler potential that lead to rapid proton decay or R-parity violation (such as $\mu, \beta, \Lambda, W_i,$ or $K_i$). 3. $x$ does not regenerate dangerous couplings with a single insertion of a Yukawa-related singlet ($\neg \text{YukawaGeneratesDangerousAtLevel } x \, 1$). 4. $x$ is complete ($\text{IsComplete } x$), meaning it contains a down-type Higgs charge $qH_d$, an up-type Higgs charge $qH_u$, and non-empty sets of charges for the $\bar{\mathbf{5}}$ and $\mathbf{10}$ matter representations.

For any codimension-one configuration $I$ of sections in an $SU(5)$ F-theory model, a $U(1)$ charge spectrum $x$ is an element of the multiset $\text{viableCharges}(I)$ if and only if it satisfies the following conditions: 1. The charges in $x$ for the $\mathbf{\bar{5}}$ and $\mathbf{10}$ representations are contained within the sets of allowed charges $\text{allowedBarFiveCharges}(I)$ and $\text{allowedTenCharges}(I)$, respectively (i.e., $x \in \text{ofFinset}(I.\text{allowedBarFiveCharges}, I.\text{allowedTenCharges})$). 2. $x$ allows the top Yukawa coupling interaction ($\text{AllowsTerm } x \text{ topYukawa}$). 3. $x$ is not phenomenologically constrained ($\neg \text{IsPhenoConstrained } x$), meaning it does not allow operators leading to rapid proton decay or R-parity violation. 4. $x$ does not regenerate dangerous couplings with a single insertion of a Yukawa-related singlet ($\neg \text{YukawaGeneratesDangerousAtLevel } x \, 1$). 5. $x$ is complete ($\text{IsComplete } x$), meaning it contains the required Higgs and matter representation charges.