Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.OfPotentialTerm

Given a charge spectrum $x$, which assigns a multiset of charges in an abelian group $\mathcal{Z}$ to each field label, and a potential term $T$ in the $SU(5)$ SUSY GUT, this function calculates the multiset of all possible total charges for that term. Let the potential term $T$ be composed of fields with labels $F_1, F_2, \dots, F_n$ (obtained via `toFieldLabel`). If $S_i$ is the multiset of charges associated with the field label $F_i$ in the spectrum $x$, the function returns the multiset $\{ z_1 + z_2 + \dots + z_n \mid z_i \in S_i \}$. This result is computed by performing a left-fold over the list of multisets $S_i$ using a multiset addition operation (the Minkowski sum), starting with the identity multiset $\{0\}$.

For any two $SU(5)$ charge spectra $x, y \in \text{ChargeSpectrum } \mathcal{Z}$ such that $x \subseteq y$, and for any potential term $T$, the multiset of total charges associated with $T$ under spectrum $x$ is a sub-multiset of the charges associated with $T$ under spectrum $y$, i.e., $\text{ofPotentialTerm}(x, T) \subseteq \text{ofPotentialTerm}(y, T)$.

For any potential term $T$ in the $SU(5)$ Grand Unified Theory, the multiset of total charges `ofPotentialTerm` associated with $T$ given the empty charge spectrum $\emptyset$ is the empty multiset $\emptyset$.

Given a charge spectrum $y$ (characterized by the sets of charges $Q_5$ and $Q_{10}$ for matter fields and optional charges $q_{H_u}$ and $q_{H_d}$ for Higgs fields) and a potential term $T$ in an $SU(5)$ supersymmetric theory, this function computes the multiset of charges resulting from the combination of fields in $T$. The computation is performed via an explicit case-by-case mapping: * For the Higgs mass term $\mu$: the multiset $\{q_{H_d} - q_{H_u}\}$ (provided both Higgs charges are defined). * For the matter-Higgs mixing term $\beta$: the multiset $\{-q_{H_u} + q \mid q \in Q_5\}$. * For the trilinear matter interaction $\Lambda$: the multiset $\{q_1 + q_2 + q_3 \mid q_1, q_2 \in Q_5, q_3 \in Q_{10}\}$. * For the dimension-5 operators $W^1, W^2, W^3, W^4$: * $W^1$: $\{q_1 + q_2 + q_3 + q_4 \mid q_1 \in Q_5, q_2, q_3, q_4 \in Q_{10}\}$. * $W^2$: $\{q_{H_d} + q_1 + q_2 + q_3 \mid q_1, q_2, q_3 \in Q_{10}\}$. * $W^3$: $\{-2q_{H_u} + q_1 + q_2 \mid q_1, q_2 \in Q_5\}$. * $W^4$: $\{q_{H_d} - 2q_{H_u} + q \mid q \in Q_5\}$. * For the Kähler potential terms $K^1, K^2$: * $K^1$: $\{-q_1 + q_2 + q_3 \mid q_1 \in Q_5, q_2, q_3 \in Q_{10}\}$. * $K^2$: $\{q_{H_d} + q_{H_u} + q \mid q \in Q_{10}\}$. * For the Yukawa couplings: * `topYukawa`: $\{-q_{H_u} + q_1 + q_2 \mid q_1, q_2 \in Q_{10}\}$. * `bottomYukawa`: $\{q_{H_d} + q_1 + q_2 \mid q_1 \in Q_5, q_2 \in Q_{10}\}$. If a required Higgs charge is undefined (`none`), the resulting multiset is empty ($\emptyset$). This version is optimized for computational decidability.

For a given $SU(5)$ charge spectrum $x$, the multiset of charges associated with the Higgs mass potential term $\mu$ is the collection containing the single charge $q_{H_d} - q_{H_u}$, where $q_{H_d}$ and $q_{H_u}$ are the charges of the Higgs fields $H_d$ and $H_u$ respectively. If either $q_{H_d}$ or $q_{H_u}$ is undefined in the charge spectrum $x$, the resulting multiset is empty.

For a given $SU(5)$ charge spectrum $x$, the multiset of charges associated with the matter-Higgs mixing potential term $\beta$ is the collection of all charges $-q_{H_u} + q$, where $q_{H_u}$ is the charge of the $H_u$ Higgs field and $q \in Q_5$ is a charge from the set of charges associated with the $\mathbf{\bar{5}}$ representation. If the Higgs charge $q_{H_u}$ is undefined in the charge spectrum $x$, the resulting multiset is empty.

For a given charge spectrum $x$ in an $SU(5)$ supersymmetric theory, the multiset of charges associated with the dimension-5 potential term $W^2$ is the collection of all sums $q_{H_d} + q_1 + q_2 + q_3$, where $q_{H_d}$ is the charge of the $H_d$ Higgs field and $q_1, q_2, q_3 \in Q_{10}$ are charges from the set of charges associated with the $\mathbf{10}$ representation. If the Higgs charge $q_{H_d}$ is undefined, the resulting multiset is empty.

For a given charge spectrum $x$ in an $SU(5)$ supersymmetric theory, the multiset of charges associated with the dimension-5 potential term $W^3$ is the collection of all sums $-2q_{H_u} + q_1 + q_2$, where $q_{H_u}$ is the charge of the $H_u$ Higgs field and $q_1, q_2 \in Q_5$ are charges from the set of charges associated with the $\mathbf{\bar{5}}$ representation. If the Higgs charge $q_{H_u}$ is undefined, the resulting multiset is empty.

For a given charge spectrum $x$ in an $SU(5)$ supersymmetric theory, the multiset of charges associated with the dimension-5 potential term $W^4$ is the collection of all sums $q_{H_d} - 2q_{H_u} + q$, where $q_{H_d}$ and $q_{H_u}$ are the charges of the Higgs fields $H_d$ and $H_u$ respectively, and $q \in Q_5$ is a charge from the set of charges associated with the $\mathbf{\bar{5}}$ representation. If either Higgs charge $q_{H_d}$ or $q_{H_u}$ is undefined, the resulting multiset is empty.

For a given charge spectrum $x$ in an $SU(5)$ supersymmetric theory, let $Q_{10}$ be the set of charges associated with the $\mathbf{10}$ representation, and let $q_{H_d}$ and $q_{H_u}$ be the optional charges of the Higgs fields $H_d$ and $H_u$. The multiset of charges associated with the Kähler potential term $K^2$ is the collection of all sums $q_{H_d} + q_{H_u} + q$, where $q \in Q_{10}$. If either $q_{H_d}$ or $q_{H_u}$ is undefined, the resulting multiset is empty.

For a charge spectrum $x$ in an $SU(5)$ supersymmetric theory, let $Q_{10}$ be the set of charges for the matter fields in the $\mathbf{10}$ representation and $q_{H_u}$ be the optional charge for the $H_u$ Higgs field. The multiset of charges associated with the top Yukawa coupling term, denoted by $\text{ofPotentialTerm}'(x, \text{topYukawa})$, is given by the multiset of values $\{-q_{H_u} + q_1 + q_2\}$ for all $q_1, q_2 \in Q_{10}$. If $q_{H_u}$ is undefined, the resulting multiset is empty.

For a charge spectrum $x$ in an $SU(5)$ supersymmetric theory, let $Q_5$ and $Q_{10}$ be the sets of charges for the matter fields and $q_{H_d}$ be the optional charge for the $H_d$ Higgs field. The multiset of charges associated with the bottom Yukawa coupling term, denoted by $\text{ofPotentialTerm}'(x, \text{bottomYukawa})$, is equal to the multiset of sums $\{q_{H_d} + q_5 + q_{10} \mid q_5 \in Q_5, q_{10} \in Q_{10}\}$. If $q_{H_d}$ is undefined, the resulting multiset is empty.

In an $SU(5)$ supersymmetric Grand Unified Theory (GUT), for any potential term $T$, the explicit multiset of charges $\text{ofPotentialTerm}'(\emptyset, T)$ associated with the empty charge spectrum $\emptyset$ is empty. The empty charge spectrum refers to a configuration where no charges are assigned to the Higgs fields $H_u$ and $H_d$, and the sets of charges $Q_5$ and $Q_{10}$ for matter fields are empty.

For any charge spectrum $x$ (mapping field labels to multisets of charges in an abelian group $\mathcal{Z}$) and any potential term $T$ in an $SU(5)$ supersymmetric theory, the multiset of total charges computed from the constituent field labels of $T$, denoted by $\text{ofPotentialTerm}(x, T)$, is a sub-multiset of the explicitly defined multiset of charges for that term, denoted by $\text{ofPotentialTerm}'(x, T)$.

For any charge spectrum $x$ in an $SU(5)$ supersymmetric theory and any potential term $T$, the explicitly defined multiset of charges $\text{ofPotentialTerm}'(x, T)$ is a sub-multiset of the multiset $\text{ofPotentialTerm}(x, T)$, where $\text{ofPotentialTerm}(x, T)$ is the multiset of total charges calculated by summing the charges of the constituent fields of $T$.

For any potential term $T$ in an $SU(5)$ supersymmetric theory, any charge spectrum $y$ (assigning multisets of charges to field labels), and any charge $n \in \mathcal{Z}$, $n$ is an element of the multiset $\text{ofPotentialTerm}(y, T)$ (the multiset of total charges calculated by summing the charges of the constituent fields of $T$) if and only if it is an element of the explicitly defined multiset $\text{ofPotentialTerm}'(y, T)$.

For any two $SU(5)$ charge spectra $x, y \in \text{ChargeSpectrum } \mathcal{Z}$ such that $x \subseteq y$, and for any potential term $T$, the explicitly defined multiset of charges associated with $T$ under spectrum $x$ is a sub-multiset of those associated with $T$ under spectrum $y$, i.e., $\text{ofPotentialTerm}'(x, T) \subseteq \text{ofPotentialTerm}'(y, T)$.