Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.Map

Given an additive monoid homomorphism $f: \mathcal{Z} \to \mathcal{Z}_1$ and an SU(5) charge spectrum $x$ defined over $\mathcal{Z}$, the function returns a new charge spectrum over $\mathcal{Z}_1$. The components of the new spectrum are obtained by mapping the Higgs charges $q_{H_d}$ and $q_{H_u}$ via $f$, and taking the images of the finite sets of charges $Q_5$ and $Q_{10}$ under $f$.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids. For any additive monoid homomorphism $f: \mathcal{Z} \to \mathcal{Z}_1$, the mapping of the empty $SU(5)$ charge spectrum $\emptyset$ over $\mathcal{Z}$ results in the empty $SU(5)$ charge spectrum over $\mathcal{Z}_1$: $$\text{map}(f, \emptyset) = \emptyset$$

Let $\mathcal{Z}$, $\mathcal{Z}_1$, and $\mathcal{Z}_2$ be additive monoids. For any additive monoid homomorphisms $f: \mathcal{Z} \to \mathcal{Z}_1$ and $g: \mathcal{Z}_1 \to \mathcal{Z}_2$, and any SU(5) charge spectrum $x$ defined over $\mathcal{Z}$, mapping the spectrum $x$ by $f$ and then mapping the result by $g$ is equivalent to mapping $x$ by the composite homomorphism $g \circ f$: $$\text{map}(g, \text{map}(f, x)) = \text{map}(g \circ f, x)$$

Let $x$ be an SU(5) charge spectrum defined over an additive monoid $\mathcal{Z}$. Mapping the charge spectrum $x$ using the identity homomorphism $\text{id}_{\mathcal{Z}}: \mathcal{Z} \to \mathcal{Z}$ returns the original spectrum $x$: $$\text{map}(\text{id}_{\mathcal{Z}}, x) = x$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids. For any additive monoid homomorphism $f: \mathcal{Z} \to \mathcal{Z}_1$, any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$, and any field label $F$, the set of charges associated with the field $F$ in the mapped spectrum $\text{map}(f, x)$ is equal to the image under $f$ of the set of charges associated with $F$ in the original spectrum $x$: $$\text{ofFieldLabel}(\text{map}(f, x), F) = f(\text{ofFieldLabel}(x, F))$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids. For any additive monoid homomorphism $f: \mathcal{Z} \to \mathcal{Z}_1$ and any two $SU(5)$ charge spectra $x, y$ defined over $\mathcal{Z}$, if $x \subseteq y$, then their mapped spectra satisfy $\text{map}(f, x) \subseteq \text{map}(f, y)$.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids. For any additive monoid homomorphism $f: \mathcal{Z} \to \mathcal{Z}_1$, any $SU(5)$ charge spectrum $x$ over $\mathcal{Z}$, and any potential term $T$ in the $SU(5)$ SUSY GUT, the finite set of total charges for $T$ in the mapped spectrum $\text{map}(f, x)$ is equal to the image under $f$ of the finite set of total charges for $T$ in the original spectrum $x$: $$\text{toFinset}(\text{ofPotentialTerm}(\text{map}(f, x), T)) = (\text{ofPotentialTerm}(x, T)).\text{toFinset.image}(f)$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ over $\mathcal{Z}$ and any potential term $T$ in the $SU(5)$ SUSY GUT, a charge value $i \in \mathcal{Z}_1$ is an element of the multiset of total charges for $T$ under the mapped spectrum $\text{map}(f, x)$ if and only if $i$ is an element of the multiset obtained by mapping the multiset of total charges for $T$ under the original spectrum $x$ via $f$. That is, $$i \in \text{ofPotentialTerm}(\text{map}(f, x), T) \iff i \in (\text{ofPotentialTerm}(x, T)).\text{map}(f)$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ over $\mathcal{Z}$ and any potential term $T$ in the $SU(5)$ SUSY GUT, a charge value $i \in \mathcal{Z}_1$ is an element of the multiset of total charges $\text{ofPotentialTerm}'(\text{map}(f, x), T)$ (the explicitly defined multiset of charges resulting from the combination of fields in $T$) if and only if $i$ is an element of the multiset obtained by mapping the multiset of total charges for $T$ under the original spectrum $x$ via $f$. That is, $$i \in \text{ofPotentialTerm}'(\text{map}(f, x), T) \iff i \in (\text{ofPotentialTerm}'(x, T)).\text{map}(f)$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids with decidable equality, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ over $\mathcal{Z}$ and any potential term $T$ in the $SU(5)$ SUSY GUT, the finite set of total charges for $T$ under the mapped spectrum $\text{map}(f, x)$ is the image under $f$ of the finite set of total charges for $T$ under the original spectrum $x$. That is, $$\text{set}(\text{ofPotentialTerm}'(\text{map}(f, x), T)) = f(\text{set}(\text{ofPotentialTerm}'(x, T)))$$ where $\text{ofPotentialTerm}'$ denotes the multiset of charges resulting from the specific combination of matter and Higgs fields in the term $T$, and $\text{set}(\cdot)$ denotes the conversion of that multiset to a finite set.

For any potential term $T$ of the $SU(5)$ SUSY GUT and any additive monoid homomorphism $f: \mathcal{Z} \to \mathcal{Z}_1$, the mapping under $f$ of the charge interaction form $\text{allowsTermForm}(a, b, c, T)$ for charges $a, b, c \in \mathcal{Z}$ is equal to the charge interaction form evaluated with the mapped charges $f(a), f(b)$, and $f(c)$. That is, $$f(\text{allowsTermForm}(a, b, c, T)) = \text{allowsTermForm}(f(a), f(b), f(c), T)$$ where $\text{allowsTermForm}$ represents the specific combination of charges required for the potential term $T$ to be gauge-invariant or otherwise "allowed."

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$ and any potential term $T$ of the $SU(5)$ SUSY GUT (such as Yukawa interactions or Higgs mass terms), if $x$ allows the term $T$, then the mapped charge spectrum $f(x)$ also allows the term $T$.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$, if $x$ is phenomenologically constrained, then the mapped charge spectrum $x.\text{map}(f)$ is also phenomenologically constrained. A charge spectrum is considered phenomenologically constrained if it allows any of the potential terms $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$ (associated with proton decay or R-parity violation in $SU(5)$ supersymmetry).

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$, if the mapped charge spectrum $\text{map}(f, x)$ is not phenomenologically constrained, then the original charge spectrum $x$ is also not phenomenologically constrained. A charge spectrum is considered phenomenologically constrained if it allows any of the potential terms (specifically $\mu, \beta, \Lambda, W_1, W_2, W_4, K_1$, or $K_2$) associated with proton decay or R-parity violation in $SU(5)$ supersymmetry.

Let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism between charge groups, and let $x$ be an $SU(5)$ charge spectrum defined over $\mathcal{Z}$. The mapped charge spectrum $x.\text{map}(f)$ is complete if and only if the original charge spectrum $x$ is complete. Here, a spectrum is considered complete if the up-type and down-type Higgs charges are present and the sets of $\bar{\mathbf{5}}$ and $\mathbf{10}$ charges are non-empty.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$, the finite set of charges associated with the Yukawa terms of the mapped spectrum $\text{map}(f, x)$ is equal to the image under $f$ of the finite set of charges associated with the Yukawa terms of the original spectrum $x$. The Yukawa terms are defined as the sum of the charges from the top and bottom Yukawa couplings. That is, $$(\text{map}(f, x)).\text{ofYukawaTerms}.\text{toFinset} = (\text{ofYukawaTerms}(x)).\text{toFinset}.\text{image}(f)$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$, a charge $i \in \mathcal{Z}_1$ is an element of the multiset of Yukawa charges of the mapped spectrum $\text{map}(f, x)$ if and only if $i$ is an element of the multiset obtained by mapping the original multiset of Yukawa charges of $x$ via $f$. That is, $$i \in (\text{map}(f, x)).\text{ofYukawaTerms} \iff i \in (\text{ofYukawaTerms}(x)).\text{map}(f)$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$ and any natural number $n$, let $S(x, n)$ denote the finite set of charges formed by summing up to $n$ charges from the multiset of Yukawa terms of $x$. The theorem states that the set of such sums for the mapped spectrum $\text{map}(f, x)$ is equal to the image under $f$ of the set of sums for the original spectrum $x$: $$\text{set}(\text{ofYukawaTermsNSum}(\text{map}(f, x), n)) = f(\text{set}(\text{ofYukawaTermsNSum}(x, n)))$$ where $\text{set}(\cdot)$ denotes the conversion of a multiset of charges to a finite set.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$, a natural number $n$, and a charge $i \in \mathcal{Z}_1$, $i$ is an element of the multiset of charges formed by summing up to $n$ Yukawa charges of the mapped spectrum $\text{map}(f, x)$ if and only if $i$ is an element of the multiset obtained by mapping the corresponding sums of the original spectrum $x$ via $f$. That is, $$i \in (\text{map}(f, x)).\text{ofYukawaTermsNSum}(n) \iff i \in (\text{ofYukawaTermsNSum}(x, n)).\text{map}(f)$$

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$, the finite set of charges associated with the phenomenologically constraining superpotential terms of the mapped spectrum $\text{map}(f, x)$ is equal to the image under $f$ of the finite set of charges for those same terms in the original spectrum $x$. That is, $$\text{set}(\text{phenoConstrainingChargesSP}(\text{map}(f, x))) = f(\text{set}(\text{phenoConstrainingChargesSP}(x)))$$ where $\text{phenoConstrainingChargesSP}$ denotes the multiset of charges resulting from the superpotential terms $\mu, \beta, \Lambda, W_1, W_2,$ and $W_4$ (which lead to proton decay or R-parity violation), and $\text{set}(\cdot)$ denotes the conversion of a multiset to a finite set.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$ and any natural number $n$, if the spectrum $x$ regenerates a phenomenologically constrained (dangerous) superpotential term through the insertion of up to $n$ Yukawa-related singlets, then the mapped charge spectrum $\text{map}(f, x)$ also regenerates a dangerous superpotential term at the same level $n$. Mathematically, if $Y_n(x) \cap P(x) \neq \emptyset$, where $Y_n(x)$ is the set of charges formed by summing up to $n$ Yukawa charges and $P(x)$ is the set of phenomenologically constraining charges, then $Y_n(\text{map}(f, x)) \cap P(\text{map}(f, x)) \neq \emptyset$.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge groups, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any $SU(5)$ charge spectrum $x$ defined over $\mathcal{Z}$ and any natural number $n$, if the mapped charge spectrum $\text{map}(f, x)$ does not regenerate a phenomenologically constrained (dangerous) superpotential term through the insertion of up to $n$ Yukawa-related singlets, then the original charge spectrum $x$ also does not regenerate a dangerous superpotential term at level $n$. Mathematically, this states that if $Y_n(\text{map}(f, x)) \cap P(\text{map}(f, x)) = \emptyset$, then $Y_n(x) \cap P(x) = \emptyset$, where $Y_n$ denotes the multiset of charges from $n$ Yukawa insertions and $P$ denotes the set of phenomenologically constrained charges.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge types, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. Given finite sets of charges $S_5, S_{10} \subset \mathcal{Z}$ and a target charge spectrum $x = (q_{H_d}, q_{H_u}, Q_{\bar{5}}, Q_{10})$ over $\mathcal{Z}_1$, this function returns the finite set of all charge spectra $y = (h_d, h_u, P_{\bar{5}}, P_{10})$ over $\mathcal{Z}$ such that: 1. The image of $y$ under $f$ is equal to $x$. That is, $\tilde{f}(h_d) = q_{H_d}$, $\tilde{f}(h_u) = q_{H_u}$, $f(P_{\bar{5}}) = Q_{\bar{5}}$, and $f(P_{10}) = Q_{10}$, where $\tilde{f}$ is the extension of $f$ to handle optional charges (mapping $\text{none}$ to $\text{none}$). 2. The components of $y$ are restricted such that $h_d, h_u \in S_5 \cup \{\text{none}\}$, $P_{\bar{5}} \subseteq S_5$, and $P_{10} \subseteq S_{10}$. This set corresponds to the preimage of $x$ under the mapping induced by $f$, restricted to the subset of spectra whose matter and Higgs charges are contained within $S_5$ and $S_{10}$.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge types, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. For any finite sets of charges $S_{\bar{5}}, S_{10} \subseteq \mathcal{Z}$ and any charge spectrum $x$ over $\mathcal{Z}_1$, the finite set $\text{preimageOfFinset}(S_{\bar{5}}, S_{10}, f, x)$ is equal to the set of all charge spectra $y$ over $\mathcal{Z}$ such that $y.\text{map}(f) = x$ and $y$ belongs to the set $\text{ofFinset}(S_{\bar{5}}, S_{10})$. (Note: $\text{ofFinset}(S_{\bar{5}}, S_{10})$ is the set of spectra whose components are restricted such that the Higgs charges are in $S_{\bar{5}} \cup \{\text{none}\}$, and the matter charge sets $Q_{\bar{5}}$ and $Q_{10}$ are subsets of $S_{\bar{5}}$ and $S_{10}$ respectively.)

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids, and let $f : \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. Given finite sets of charges $S_5, S_{10} \subset \mathcal{Z}$ and a target charge spectrum $x = (q_{H_d}, q_{H_u}, Q_5, Q_{10})$ defined over $\mathcal{Z}_1$, this function computes the cardinality of the preimage of $x$ under the mapping $f$ within a restricted set of spectra. Specifically, it calculates the number of charge spectra $y$ over $\mathcal{Z}$ such that $y.\text{map}(f) = x$, where the Higgs charges of $y$ are restricted to $S_5 \cup \{\text{none}\}$ and the sets of charges for the matter fields are subsets of $S_5$ and $S_{10}$ respectively. The value is computed as the product of: - The number of elements $h_d \in S_5 \cup \{\text{none}\}$ such that $f(h_d) = q_{H_d}$. - The number of elements $h_u \in S_5 \cup \{\text{none}\}$ such that $f(h_u) = q_{H_u}$. - The number of subsets $P_5 \subseteq S_5$ such that the image $f(P_5) = Q_5$. - The number of subsets $P_{10} \subseteq S_{10}$ such that the image $f(P_{10}) = Q_{10}$.

Let $\mathcal{Z}$ and $\mathcal{Z}_1$ be additive monoids representing charge types, and let $f: \mathcal{Z} \to \mathcal{Z}_1$ be an additive monoid homomorphism. Given finite sets of charges $S_5, S_{10} \subset \mathcal{Z}$ and a target charge spectrum $x$ over $\mathcal{Z}_1$, the value computed by the function `preimageOfFinsetCard` is equal to the cardinality of the finite set `preimageOfFinset`. Specifically, this set consists of all charge spectra $y = (h_d, h_u, P_{\bar{5}}, P_{10})$ over $\mathcal{Z}$ such that the image of $y$ under $f$ is equal to $x$, and $y$ is restricted such that the Higgs charges $h_d, h_u \in S_5 \cup \{\text{none}\}$, and the matter charge sets satisfy $P_{\bar{5}} \subseteq S_5$ and $P_{10} \subseteq S_{10}$.