Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.MinimallyAllowsTerm.Basic

A charge spectrum $x$ is said to **minimally allow** a potential term $T$ if for every sub-spectrum $y \subseteq x$, $y$ allows $T$ if and only if $y = x$. This condition implies that $x$ allows $T$ and no proper subset of $x$ allows $T$.

For any charge spectrum $x$ and potential term $T$, the property that $x$ minimally allows $T$ is decidable. This property, denoted by `MinimallyAllowsTerm`, holds if for every sub-spectrum $y \in \mathcal{P}(x)$, the spectrum $y$ allows the term $T$ if and only if $y = x$.

If a charge spectrum $x$ minimally allows a potential term $T$, then it holds that $x$ allows $T$.

For an $SU(5)$ charge spectrum $x$ and a potential term $T$ in an $SU(5)$ supersymmetric theory, if there exists a sub-spectrum $y$ in the powerset of $x$ (that is, $y \subseteq x$) such that $y$ minimally allows the term $T$, then $x$ allows the term $T$.

For a charge spectrum $x$ and a potential term $T$, $x$ minimally allows $T$ if and only if the set of all sub-spectra $y \subseteq x$ that allow $T$ is equal to the singleton set $\{x\}$.

For a charge spectrum $x$ and a potential term $T$ in an $SU(5)$ supersymmetric theory, $x$ minimally allows $T$ if and only if the number of sub-spectra $y \subseteq x$ that allow $T$ is exactly one. Here, $x$ minimally allows $T$ means that $x$ allows the term (there exist charges in the spectrum that sum to zero for the fields in $T$) and no proper subset of $x$ does.

If a charge spectrum $x$ allows a potential term $T$, then there exists a sub-spectrum $y \subseteq x$ that minimally allows $T$. Here, $y$ minimally allows $T$ means that $y$ allows $T$ and no proper subset of $y$ allows $T$.

For an $SU(5)$ charge spectrum $x$ and a potential term $T$, $x$ allows the term $T$ if and only if there exists a sub-spectrum $y \subseteq x$ that minimally allows $T$. A sub-spectrum $y$ minimally allows $T$ if it allows $T$ and no proper subset of $y$ allows $T$.

If an $SU(5)$ charge spectrum $x$ minimally allows a potential term $T$, then the cardinality of the spectrum $x$ is less than or equal to the degree of the term $T$: \[ \text{card}(x) \le \text{deg}(T) \] where $\text{card}(x)$ is the total number of charges in the spectrum $x$ and $\text{deg}(T)$ is the number of fields that constitute the interaction term $T$.

Let $\mathcal{Z}$ be an abelian group of charges. For any charges $a, b, c \in \mathcal{Z}$ and any potential term $T$ of the $SU(5)$ SUSY GUT, provided $T$ is not the dimension-5 operator $W^1$ ($10^3 \bar{5}_M$) or $W^2$ ($10^3 \bar{5}_{H_d}$), the charge spectrum $\text{allowsTermForm}(a, b, c, T)$ minimally allows the term $T$. This means that the spectrum $\text{allowsTermForm}(a, b, c, T)$ allows the term $T$, and no proper subset of this spectrum allows $T$.