Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.AllowsTerm

Let $\mathcal{Z}$ be an abelian group of charges and $x$ be a charge spectrum that assigns a multiset of charges to various field labels in an $SU(5)$ supersymmetric theory. A potential term $T$ is said to be **allowed** by the charge spectrum $x$ if the additive identity $0 \in \mathcal{Z}$ is an element of the multiset of all possible total charges associated with $T$ under the spectrum $x$. More explicitly, if $T$ is an interaction term composed of fields $F_1, F_2, \dots, F_n$, and $x$ assigns the multiset of charges $S_i \subset \mathcal{Z}$ to the field $F_i$, then $x$ allows $T$ if there exists a selection of charges $z_i \in S_i$ such that their sum vanishes: $$\sum_{i=1}^n z_i = 0$$ This condition signifies that there is a combination of field charges present in the spectrum that makes the potential term $T$ gauge-invariant (invariant under the symmetry group associated with $\mathcal{Z}$).

Let $x$ be a charge spectrum and $T$ be a potential term in an $SU(5)$ supersymmetric theory with charges in an abelian group $\mathcal{Z}$. The spectrum $x$ allows the potential term $T$ if and only if the additive identity $0 \in \mathcal{Z}$ is an element of the multiset of possible total charges $x.\text{ofPotentialTerm}' T$.

For a charge spectrum $x$ and a potential term $T$ within an $SU(5)$ theory, the property that $x$ allows $T$ (denoted $x.\text{AllowsTerm } T$) is decidable, provided that equality is decidable in the underlying charge group $\mathcal{Z}$. This means that it can be computationally determined whether there exists a selection of charges from $x$ assigned to the fields in $T$ such that their sum equals the additive identity $0 \in \mathcal{Z}$.

Let $\mathcal{Z}$ be an abelian group of charges. For any potential term $T$ and any two $SU(5)$ charge spectra $x, y \in \text{ChargeSpectrum } \mathcal{Z}$, if $y$ is a subset of $x$ (denoted $y \subseteq x$) and $y$ allows the potential term $T$, then $x$ also allows the potential term $T$.

For a potential term $T$ and three charges $a, b, c$ in the charge group $\mathcal{Z}$, the function `allowsTermForm` constructs a `ChargeSpectrum` representing the minimal set of charges assigned to the fields $H_u, H_d, M,$ and $10$ required to allow that term. This spectrum is defined such that at least one combination of these charges satisfies the gauge invariance condition (the total charge sums to zero) for the specific interaction $T$. The charge assignments for each potential term $T$ are as follows: - For $T = \mu$ (Higgs mass term $\mu H_u \bar{5}_H$): $H_d$ has charge $\{a\}$ and $H_u$ has charge $\{a\}$. - For $T = \beta$ (matter-Higgs mixing $\beta \bar{5}_M H_u$): $H_u$ has charge $\{a\}$ and $M$ has charge $\{a\}$. - For $T = \Lambda$ (trilinear matter interaction $\lambda \bar{5}_M \bar{5}_M 10$): $M$ has charges $\{a, b\}$ and $10$ has charge $\{-a-b\}$. - For $T = W^1$ ($10^3 \bar{5}_M$): $M$ has charge $\{-a-b-c\}$ and $10$ has charges $\{a, b, c\}$. - For $T = W^2$ ($10^3 \bar{5}_H$): $H_d$ has charge $\{-a-b-c\}$ and $10$ has charges $\{a, b, c\}$. - For $T = W^3$ ($\bar{5}_M^2 H_u^2$): $H_u$ has charge $\{-a\}$ and $M$ has charges $\{b, -b-2a\}$. - For $T = W^4$ ($\bar{5}_M \bar{5}_H H_u^2$): $H_d$ has charge $\{-c-2b\}$, $H_u$ has charge $\{-b\}$, and $M$ has charge $\{c\}$. - For $T = K^1$ ($10^2 5_M$): $M$ has charge $\{-a\}$ and $10$ has charges $\{b, -a-b\}$. - For $T = K^2$ ($\bar{5}_{H_u} \bar{5}_{H_d} 10$): $H_d$ has charge $\{a\}$, $H_u$ has charge $\{b\}$, and $10$ has charge $\{-a-b\}$. - For $T = \text{topYukawa}$ ($10^2 5_{H_u}$): $H_u$ has charge $\{-a\}$ and $10$ has charges $\{b, -a-b\}$. - For $T = \text{bottomYukawa}$ ($10 \bar{5}_M \bar{5}_H$): $H_d$ has charge $\{a\}$, $M$ has charge $\{b\}$, and $10$ has charge $\{-a-b\}$.

Let $\mathcal{Z}$ be an abelian group of charges and $T$ be a potential term in an $SU(5)$ supersymmetric theory. For any charges $a, b, c \in \mathcal{Z}$, the charge spectrum constructed by $\text{allowsTermForm}(a, b, c, T)$ allows the potential term $T$. This means there exists a selection of charges from the spectrum for the fields constituting $T$ such that their sum is the additive identity $0 \in \mathcal{Z}$.

Let $\mathcal{Z}$ be an abelian group of charges. For any potential term $T$ and charge spectrum $x$, if there exist charges $a, b, c \in \mathcal{Z}$ such that $x = \text{allowsTermForm}(a, b, c, T)$, then the charge spectrum $x$ allows the potential term $T$. This means there exists a selection of charges from the spectrum $x$ for the fields constituting $T$ such that their sum is the additive identity $0 \in \mathcal{Z}$.

Let $\mathcal{Z}$ be an abelian group of charges. For any charges $a, b, c, a', b', c' \in \mathcal{Z}$ and any potential term $T$ of the $SU(5)$ SUSY GUT, excluding the dimension-5 operators $W^1$ ($10^3 \bar{5}_M$) and $W^2$ ($10^3 \bar{5}_{H_d}$), if the minimal charge spectrum $\text{allowsTermForm}(a, b, c, T)$ is a subset of $\text{allowsTermForm}(a', b', c', T)$ (denoted by the component-wise subset relation $\subseteq$), then the two charge spectra are equal: $\text{allowsTermForm}(a, b, c, T) = \text{allowsTermForm}(a', b', c', T)$.

For any charges $a, b, c$ in the charge group $\mathcal{Z}$ and any potential term $T$ in an $SU(5)$ supersymmetric grand unified theory, the cardinality of the minimal charge spectrum constructed by $\text{allowsTermForm}(a, b, c, T)$ is less than or equal to the degree of the potential term $T$. Mathematically, this is expressed as: $$\text{card}(\text{allowsTermForm}(a, b, c, T)) \leq \text{deg}(T)$$ where $\text{card}(x)$ denotes the total number of charges in the spectrum $x$, and $\text{deg}(T)$ is the number of field labels that constitute the interaction term $T$.

Let $\mathcal{Z}$ be an abelian group of charges. For any potential term $T$ and any $SU(5)$ charge spectrum $x$ over $\mathcal{Z}$, if $x$ allows the potential term $T$ (meaning there exists a selection of charges for the fields in $T$ from the spectrum $x$ that sums to zero), then there exist charges $a, b, c \in \mathcal{Z}$ such that the minimal charge spectrum $\text{allowsTermForm}(a, b, c, T)$ is a subset of $x$ and $\text{allowsTermForm}(a, b, c, T)$ itself allows the potential term $T$.

Let $\mathcal{Z}$ be an abelian group of charges. For any potential term $T$ in an $SU(5)$ supersymmetric theory and any charge spectrum $x$ over $\mathcal{Z}$, the spectrum $x$ allows the term $T$ if and only if there exist charges $a, b, c \in \mathcal{Z}$ such that the minimal charge spectrum $\text{allowsTermForm}(a, b, c, T)$ is a subset of $x$.

Let $\mathcal{Z}$ be an abelian group of charges. Suppose an $SU(5)$ charge spectrum $x$ allows a potential term $T$. Then there exists a subset charge spectrum $y \subseteq x$ such that $y$ also allows $T$ and the cardinality of $y$ is less than or equal to the degree of the term $T$: $$\text{card}(y) \le \text{deg}(T)$$ where $\text{card}(y)$ is the total number of charges in the spectrum $y$, and $\text{deg}(T)$ is the number of fields constituting the interaction term $T$.

Given a charge spectrum $x$ with Higgs charges $q_{H_d}, q_{H_u} \in \mathcal{Z} \cup \{\bot\}$ and sets of matter charges $Q_5, Q_{10} \subset \mathcal{Z}$, the proposition `AllowsTermQ5 x q5 T` defines whether adding a charge $q_5 \in \mathcal{Z}$ to the $\mathbf{\overline{5}}$ representation allows the potential term $T$ due to the inclusion of $q_5$. The conditions for each term $T$ are: - For $\mu$: Always false. - For $\beta_i$: $q_{H_u} \neq \bot$ and $q_5 = q_{H_u}$. - For $\lambda_{ijk}$: There exist $q_1 \in Q_5 \cup \{q_5\}$ and $q_2 \in Q_{10}$ such that $q_1 + q_5 + q_2 = 0$. - For $W^4_i$: $q_{H_d} \neq \bot$, $q_{H_u} \neq \bot$, and $q_5 + q_{H_d} - 2q_{H_u} = 0$. - For $K^1_{ijk}$: There exist $q_1, q_2 \in Q_{10}$ such that $-q_5 + q_1 + q_2 = 0$. - For $W^1_{ijkl}$: There exist $q_1, q_2, q_3 \in Q_{10}$ such that $q_5 + q_1 + q_2 + q_3 = 0$. - For $W^2_{ijk}$: Always false. - For the bottom Yukawa term: $q_{H_d} \neq \bot$ and there exists $y \in Q_{10}$ such that $y + q_5 + q_{H_d} = 0$. - For the top Yukawa term: Always false. - For $K^2_i$: Always false. - For $W^3_{ij}$: $q_{H_u} \neq \bot$ and there exists $y \in Q_5 \cup \{q_5\}$ such that $y + q_5 - 2q_{H_u} = 0$.

Given a charge group $\mathcal{Z}$ with decidable equality, for any charge spectrum $x$, charge $q_5 \in \mathcal{Z}$, and potential term $T$, it is decidable whether the term $T$ is allowed in the theory's potential due to the addition of $q_5$ to the $\mathbf{\overline{5}}$ representation matter charges. This decidability is established by evaluating the specific charge constraints associated with the term $T$ (such as the Higgs mass term $\mu$, Yukawa couplings, or dimension-5 operators) against the charges present in $x$ and the additional charge $q_5$.

Let $\mathcal{Z}$ be an abelian group of charges. Consider an $SU(5)$ SUSY GUT charge spectrum $x$ characterized by Higgs charges $q_{H_d}, q_{H_u} \in \mathcal{Z} \cup \{\bot\}$ and finite sets of matter charges $Q_5, Q_{10} \subset \mathcal{Z}$. For any potential term $T$ and any additional charge $q_5 \in \mathcal{Z}$, if $T$ is allowed by the charge spectrum formed by inserting $q_5$ into the set $Q_5$ (i.e., the spectrum $\langle q_{H_d}, q_{H_u}, Q_5 \cup \{q_5\}, Q_{10} \rangle$), then either the original spectrum $\langle q_{H_d}, q_{H_u}, Q_5, Q_{10} \rangle$ already allows $T$, or $T$ is allowed specifically due to the inclusion of $q_5$ as defined by the proposition `AllowsTermQ5`.

Consider a charge spectrum with Higgs charges $q_{H_d}, q_{H_u} \in \mathcal{Z} \cup \{\bot\}$ and sets of matter charges $Q_5, Q_{10} \subseteq \mathcal{Z}$. For any potential term $T$ and charge $q_5 \in \mathcal{Z}$, if $T$ is allowed due to the addition of $q_5$ to the $\mathbf{\overline{5}}$ representation (i.e., `AllowsTermQ5` holds for the spectrum $\langle q_{H_d}, q_{H_u}, Q_5, Q_{10} \rangle$ and charge $q_5$), then the charge spectrum obtained by inserting $q_5$ into $Q_5$ allows the potential term $T$.

Let $\mathcal{Z}$ be an abelian group of charges. For any $SU(5)$ charge spectrum consisting of Higgs charges $q_{H_d}, q_{H_u} \in \mathcal{Z} \cup \{\bot\}$ and finite sets of matter charges $Q_5, Q_{10} \subset \mathcal{Z}$, and for any potential term $T$ and additional charge $q_5 \in \mathcal{Z}$, the potential term $T$ is allowed by the augmented spectrum $\langle q_{H_d}, q_{H_u}, Q_5 \cup \{q_5\}, Q_{10} \rangle$ if and only if $T$ is allowed specifically due to the addition of $q_5$ to the $\mathbf{\overline{5}}$ representation (denoted by `AllowsTermQ5`) or $T$ is already allowed by the original spectrum $\langle q_{H_d}, q_{H_u}, Q_5, Q_{10} \rangle$.

Given a charge spectrum $x$ consisting of Higgs charges $q_{H_d}$ and $q_{H_u}$, a set of charges for fields in the $\mathbf{\overline{5}}$ representation $Q_5$, and a set of charges for fields in the $\mathbf{10}$ representation $Q_{10}$, the proposition `AllowsTermQ10 x q10 T` is true if the potential term $T$ is allowed by adding a specific charge $q_{10}$ to the $10$-representation sector. The conditions for each potential term $T$ are as follows: - $\Lambda_{ijk}$ (trilinear matter interaction $\lambda_{ijk} (\bar{5}M^i)(\bar{5}M^j)(10^k)$): $\exists q_5, q_5' \in Q_5$ such that $q_5 + q_5' + q_{10} = 0$. - $K^1_{ijk}$ (Kähler term $10^i 10^j 5M^k$): $\exists q_5 \in Q_5$ and $q_{10}' \in Q_{10} \cup \{q_{10}\}$ such that $-q_5 + q_{10}' + q_{10} = 0$. - $W^1_{ijkl}$ (dimension-5 operator $10^i 10^j 10^k \bar{5}M^l$): $\exists q_5 \in Q_5$ and $q_{10}', q_{10}'' \in Q_{10} \cup \{q_{10}\}$ such that $q_5 + q_{10}' + q_{10}'' + q_{10} = 0$. - $W^2_{ijk}$ (operator $10^i 10^j 10^k \bar{5}H_d$): $q_{H_d}$ is present and $\exists q_{10}', q_{10}'' \in Q_{10} \cup \{q_{10}\}$ such that $q_{H_d} + q_{10}' + q_{10}'' + q_{10} = 0$. - $\text{bottomYukawa}$: $q_{H_d}$ is present and $\exists q_5 \in Q_5$ such that $q_{10} + q_5 + q_{H_d} = 0$. - $\text{topYukawa}$: $q_{H_u}$ is present and $\exists q_{10}' \in Q_{10} \cup \{q_{10}\}$ such that $q_{10} + q_{10}' - q_{H_u} = 0$. - $K^2_i$ (Kähler term $\bar{5}H_u \bar{5}H_d 10^i$): $q_{H_d}$ and $q_{H_u}$ are both present and $q_{H_d} + q_{H_u} + q_{10} = 0$. - For $T \in \{\mu, \beta, W^4, W^3\}$, the proposition is always false.

Given a charge spectrum $x$ (consisting of Higgs charges $q_{H_d}, q_{H_u}$ and finite sets of representation charges $Q_5, Q_{10} \subset \mathcal{Z}$), an $SU(5)$ potential term $T$, and a specific charge $q_{10} \in \mathcal{Z}$, the proposition `AllowsTermQ10 x q10 T` is decidable, provided that equality in the charge group $\mathcal{Z}$ is decidable. This proposition determines whether the interaction term $T$ can be formed with total charge zero by selecting charges from the spectrum $x$ and the newly provided charge $q_{10}$ for the fields in the $\mathbf{10}$ representation.

Let $\mathcal{Z}$ be an abelian group of charges. Consider a potential term $T$, a charge $q_{10} \in \mathcal{Z}$, and a charge spectrum $x$ consisting of Higgs charges $q_{H_d}, q_{H_u}$, and sets of matter charges $Q_5$ and $Q_{10}$. If the potential term $T$ is allowed by the charge spectrum obtained by adding $q_{10}$ to the $\mathbf{10}$-representation sector (i.e., using the set $Q_{10} \cup \{q_{10}\}$), then either the original spectrum $x$ already allowed $T$, or $T$ is allowed specifically due to the inclusion of the new charge $q_{10}$ (the condition `AllowsTermQ10 x q10 T` holds).

Let $\mathcal{Z}$ be an abelian group of charges. Consider a charge spectrum $x = (q_{H_d}, q_{H_u}, Q_5, Q_{10})$, where $q_{H_d}, q_{H_u}$ are optional Higgs charges and $Q_5, Q_{10} \subset \mathcal{Z}$ are finite sets of matter charges. For any potential term $T$ and any charge $q_{10} \in \mathcal{Z}$, if the proposition `AllowsTermQ10 x q10 T` holds (meaning $T$ is allowed specifically by using $q_{10}$ as a charge for a field in the $\mathbf{10}$ representation), then the potential term $T$ is allowed by the augmented charge spectrum $(q_{H_d}, q_{H_u}, Q_5, Q_{10} \cup \{q_{10}\})$.

Let $\mathcal{Z}$ be an abelian group of charges. Consider a potential term $T$, a charge $q_{10} \in \mathcal{Z}$, and an $SU(5)$ charge spectrum $x = (q_{H_d}, q_{H_u}, Q_5, Q_{10})$. The augmented charge spectrum obtained by adding $q_{10}$ to the $\mathbf{10}$-representation sector, $(q_{H_d}, q_{H_u}, Q_5, Q_{10} \cup \{q_{10}\})$, allows the term $T$ if and only if either the original spectrum $x$ already allowed $T$ or the term $T$ is allowed specifically due to the inclusion of $q_{10}$ (i.e., the condition `AllowsTermQ10 x q10 T` holds).