Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.Yukawa

Given a charge spectrum $x$ over a set of charges $\mathcal{Z}$, the function `ofYukawaTerms` returns the multiset of charges in $\mathcal{Z}$ associated with the Yukawa terms in the superpotential. It is calculated as the sum of the multisets of charges obtained from the top Yukawa term and the bottom Yukawa term: \[ x.\text{ofPotentialTerm}'(\text{topYukawa}) + x.\text{ofPotentialTerm}'(\text{bottomYukawa}) \] These charges correspond to the negatives of the charges of the singlets required to regenerate the Yukawa terms in the potential.

Let $\mathcal{Z}$ be a set of charges. For any two $SU(5)$ charge spectra $x, y \in \text{ChargeSpectrum } \mathcal{Z}$, if $x$ is a subset of $y$ ($x \subseteq y$), then the multiset of charges associated with the Yukawa terms of $x$ is contained within the multiset of charges associated with the Yukawa terms of $y$.

For a charge spectrum $x$ over a set of charges $\mathcal{Z}$ and a natural number $n$, this function returns the multiset of charges in $\mathcal{Z}$ formed by summing up to $n$ charges associated with the Yukawa terms of the spectrum. If $Y$ is the multiset of charges of the Yukawa terms (defined by `ofYukawaTerms`), the function `ofYukawaTermsNSum` at level $n$ produces a multiset containing elements of the form $\sum_{i=1}^k s_i$, where $0 \le k \le n$ and each $s_i \in Y$. Physically, these represent the charges of terms that can be regenerated through the insertion of up to $n$ singlets associated with the Yukawa couplings in the superpotential.

Let $\mathcal{Z}$ be a set of charges. For any two $SU(5)$ charge spectra $x, y \in \text{ChargeSpectrum } \mathcal{Z}$ and any natural number $n$, if $x$ is a subset of $y$ ($x \subseteq y$), then the multiset of charges formed by summing up to $n$ Yukawa term charges of $x$ is contained within the multiset of charges formed by summing up to $n$ Yukawa term charges of $y$. That is, $x.\text{ofYukawaTermsNSum}(n) \subseteq y.\text{ofYukawaTermsNSum}(n)$.

Given a charge spectrum $x$ and a natural number $n$, this proposition asserts that the multiset of charges formed by summing up to $n$ Yukawa-associated charges, denoted here as $Y_n(x)$, has a non-empty intersection with the multiset of phenomenologically constraining (or "dangerous") superpotential charges, denoted $P(x)$. Mathematically, the condition is: \[ Y_n(x) \cap P(x) \neq \emptyset \] In a physical context, this means that the insertions of up to $n$ Yukawa-related singlets are capable of regenerating a superpotential term that is phenomenologically constrained (such as those leading to proton decay or R-parity violation).

For a given $SU(5)$ charge spectrum $x$ and a natural number $n$, the property $\text{YukawaGeneratesDangerousAtLevel}(x, n)$ is decidable. This property holds if the intersection of the multiset of charges formed by summing up to $n$ Yukawa-associated charges, denoted $Y_n(x)$, and the multiset of phenomenologically constraining superpotential charges, $P(x)$, is non-empty. Mathematically, the decidability applies to the condition: \[ Y_n(x) \cap P(x) \neq \emptyset \] In a physical context, this means there is a computational procedure to determine whether the insertions of up to $n$ Yukawa-related singlets can regenerate a phenomenologically constrained (dangerous) superpotential term.

For a charge spectrum $x$ and a natural number $n$, the proposition that $x$ regenerates a dangerous term at level $n$ holds if and only if the intersection of the multiset of charges formed by summing up to $n$ Yukawa-associated charges, denoted as $Y_n(x)$, and the multiset of phenomenologically constraining superpotential charges, denoted as $P(x)$, is non-empty. That is: \[ \text{YukawaGeneratesDangerousAtLevel}(x, n) \iff Y_n(x) \cap P(x) \neq \emptyset \]

For a charge spectrum $x$ and a natural number $n$, the property that the spectrum regenerates a phenomenologically constrained (dangerous) superpotential term with up to $n$ Yukawa singlet insertions holds if and only if the intersection of the set of charges formed by summing up to $n$ Yukawa charges and the set of phenomenologically constraining superpotential charges is non-empty. Mathematically, this is expressed as: \[ x.\text{YukawaGeneratesDangerousAtLevel } n \iff \text{Set}(Y_n(x)) \cap \text{Set}(P(x)) \neq \emptyset \] where $\text{Set}(Y_n(x))$ is the set of charges obtained from summing up to $n$ Yukawa terms, and $\text{Set}(P(x))$ is the set of phenomenologically constraining charges.

For any natural number $n$, the empty charge spectrum $\emptyset$ does not regenerate any phenomenologically constrained superpotential terms through the insertion of up to $n$ Yukawa-related singlets.

For any two $SU(5)$ charge spectra $x$ and $y$ such that $x \subseteq y$, and for any natural number $n$, if $x$ generates a phenomenologically constrained (dangerous) superpotential term through the insertion of up to $n$ Yukawa-related singlets (i.e., $\text{YukawaGeneratesDangerousAtLevel } x \ n$ holds), then $y$ also generates a dangerous superpotential term at the same level $n$.

Let $x$ be a charge spectrum and $n$ be a natural number. If $x$ generates a phenomenologically constrained (dangerous) superpotential term through the insertion of up to $n$ Yukawa-related singlets, then it also generates such a term through the insertion of up to $n + 1$ singlets. Mathematically, $x.\text{YukawaGeneratesDangerousAtLevel } n \implies x.\text{YukawaGeneratesDangerousAtLevel } (n + 1)$.

Let $x$ be a charge spectrum and $n, k$ be natural numbers. If $x$ generates a phenomenologically constrained (dangerous) superpotential term through the insertion of up to $n$ Yukawa-related singlets, then it also generates such a term through the insertion of up to $n + k$ singlets. Mathematically, this is expressed as: $x.\text{YukawaGeneratesDangerousAtLevel } n \implies x.\text{YukawaGeneratesDangerousAtLevel } (n + k)$.

Let $x$ be a charge spectrum and $n, m$ be natural numbers. If $n \le m$ and $x$ generates a phenomenologically constrained (dangerous) superpotential term through the insertion of up to $n$ Yukawa-related singlets, then it also generates such a term through the insertion of up to $m$ singlets. Mathematically, if $n \le m$, then $x.\text{YukawaGeneratesDangerousAtLevel } n \implies x.\text{YukawaGeneratesDangerousAtLevel } m$.