Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.PhenoConstrained

Let $x$ be a charge spectrum over a group $\mathcal{Z}$. The predicate `IsPhenoConstrained` is true for $x$ if $x$ allows any of the following terms: $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$. Mathematically, this is defined as the disjunction: \[ \text{AllowsTerm}(x, \mu) \lor \text{AllowsTerm}(x, \beta) \lor \text{AllowsTerm}(x, \Lambda) \lor \text{AllowsTerm}(x, W_2) \lor \text{AllowsTerm}(x, W_4) \lor \text{AllowsTerm}(x, K_1) \lor \text{AllowsTerm}(x, K_2) \lor \text{AllowsTerm}(x, W_1) \] In the context of SU(5) supersymmetry, these terms correspond to operators in the superpotential or Kähler potential that lead to proton decay or R-parity violation.

Let $\mathcal{Z}$ be a group with decidable equality and $x$ be a charge spectrum over $\mathcal{Z}$. The property that $x$ is phenomenologically constrained, denoted as $\text{IsPhenoConstrained}(x)$, is decidable. This predicate is defined as the disjunction of whether the spectrum allows any of the following terms: \[ \text{AllowsTerm}(x, \mu) \lor \text{AllowsTerm}(x, \beta) \lor \text{AllowsTerm}(x, \Lambda) \lor \text{AllowsTerm}(x, W_2) \lor \text{AllowsTerm}(x, W_4) \lor \text{AllowsTerm}(x, K_1) \lor \text{AllowsTerm}(x, K_2) \lor \text{AllowsTerm}(x, W_1) \] In the context of SU(5) supersymmetry, these terms correspond to operators in the superpotential or Kähler potential that lead to proton decay or R-parity violation.

The empty charge spectrum $\emptyset$ over a group $\mathcal{Z}$ is not phenomenologically constrained. That is, the predicate `IsPhenoConstrained` is false for the spectrum containing no charges.

Let $x$ and $y$ be charge spectra over a group $\mathcal{Z}$. If $x$ is a subset of $y$ ($x \subseteq y$) and $x$ is phenomenologically constrained, then $y$ is also phenomenologically constrained. A charge spectrum is phenomenologically constrained if it allows any of the terms $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$, which correspond to superpotential or Kähler potential operators leading to proton decay or R-parity violation.

For a given charge spectrum $x$ with charges in a group $\mathcal{Z}$, this function computes the multiset of charges associated with specific superpotential terms that lead to a phenomenologically constrained model (one allowing proton decay or R-parity violation). The resulting multiset is the sum of the charges of the potential terms $\mu, \beta, \Lambda, W_2, W_4,$ and $W_1$ as determined by the spectrum $x$.

For a given group of charges $\mathcal{Z}$, the multiset of charges associated with pheno-constraining superpotential terms (such as $\mu, \beta, \Lambda, W_1, W_2$, and $W_4$) for the empty charge spectrum $\emptyset$ is the empty multiset $\emptyset$.

Let $\mathcal{Z}$ be a group of charges. Given two charge spectra $x$ and $y$ such that $x \subseteq y$, the multiset of charges of the pheno-constraining superpotential terms (the charges of terms $\mu, \beta, \Lambda, W_1, W_2, W_4$ as determined by the spectrum) for $x$ is a sub-multiset of those for $y$: $$x \subseteq y \implies x.\text{phenoConstrainingChargesSP} \subseteq y.\text{phenoConstrainingChargesSP}.$$

For a given charge spectrum $x$ and a charge $q_5$, the predicate `IsPhenoConstrainedQ5` is defined as the condition that the addition of $q_5$ to the spectrum $x$ allows at least one of the following phenomenologically constrained terms: $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1,$ or $K_2$. These terms typically correspond to superpotential or Kähler potential operators that lead to physical effects such as proton decay or R-parity violation.

For a charge spectrum $x$ and a charge $q_5 \in \mathcal{Z}$, the condition that the addition of $q_5$ to the $Q_5$ set of $x$ makes the spectrum phenomenologically constrained (denoted by `IsPhenoConstrainedQ5`) is equivalent to the spectrum allowing at least one of the following physical terms: $\beta, \Lambda, W_4, K_1,$ or $W_1$. That is, $$x.\text{IsPhenoConstrainedQ5}(q_5) \iff \text{AllowsTermQ5}(x, q_5, \beta) \lor \dots \lor \text{AllowsTermQ5}(x, q_5, W_1).$$

For a given charge spectrum $x$ and a charge $q_5 \in \mathcal{Z}$, the predicate `IsPhenoConstrainedQ5` is decidable. This means there is an algorithmic procedure to determine whether the addition of the charge $q_5$ to the spectrum $x$ allows terms in the superpotential or Kähler potential—specifically $\beta, \Lambda, W_4, K_1$, or $W_1$—that lead to phenomenological issues such as proton decay or R-parity violation.

Let $x = \langle q_{H_d}, q_{H_u}, Q_5, Q_{10} \rangle$ be a charge spectrum over a group $\mathcal{Z}$. For any charge $q_5 \in \mathcal{Z}$, the charge spectrum $\langle q_{H_d}, q_{H_u}, Q_5 \cup \{q_5\}, Q_{10} \rangle$ is phenomenologically constrained if and only if either the addition of $q_5$ to $x$ satisfies the predicate `IsPhenoConstrainedQ5` or the original spectrum $x$ is already phenomenologically constrained.

Given a charge spectrum $x$ and a specific charge $q_{10} \in \mathcal{Z}$ (representing a charge in the $\mathbf{10}$ representation of $SU(5)$), the proposition `IsPhenoConstrainedQ10` is true if the inclusion of $q_{10}$ allows at least one of the following terms in the superpotential or Kähler potential: $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$. These terms typically correspond to operators that lead to proton decay or R-parity violation in the phenomenological analysis of the model.

For a charge spectrum $x$ and a charge $q_{10} \in \mathcal{Z}$, the spectrum is phenomenologically constrained by the addition of $q_{10}$ (denoted by `IsPhenoConstrainedQ10`) if and only if the inclusion of $q_{10}$ allows at least one of the following terms: $\Lambda, W_1, W_2, K_1$, or $K_2$. These terms typically correspond to operators in the superpotential or Kähler potential that lead to proton decay or R-parity violation.

For a given charge spectrum $x$ and a specific charge $q_{10} \in \mathcal{Z}$ (representing a charge in the $\mathbf{10}$ representation of $SU(5)$), the property that the spectrum is phenomenologically constrained by $q_{10}$ (denoted as $x.\text{IsPhenoConstrainedQ10}(q_{10})$) is decidable. This means there is a computational procedure to determine if the inclusion of $q_{10}$ allows terms in the superpotential or Kähler potential—specifically $\Lambda, W_1, W_2, K_1$, or $K_2$—which correspond to operators leading to proton decay or R-parity violation.

For a charge spectrum $x = \langle q_{H_d}, q_{H_u}, Q_5, Q_{10} \rangle$ over a group $\mathcal{Z}$ and a charge $q_{10} \in \mathcal{Z}$, the spectrum $\langle q_{H_d}, q_{H_u}, Q_5, Q_{10} \cup \{q_{10}\} \rangle$ is phenomenologically constrained if and only if $x$ is phenomenologically constrained by the addition of $q_{10}$ (denoted as $x.\text{IsPhenoConstrainedQ10}(q_{10})$) or the original spectrum $x$ is already phenomenologically constrained. A charge spectrum is phenomenologically constrained if it allows terms in the superpotential or Kähler potential (such as $\mu, \beta, \Lambda, W_i, K_i$) that lead to proton decay or $R$-parity violation.