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

Charge spectrum which minimally allows terms

i. Overview

We can say that a charge spectrum `x : ChargeSpectrum 𝓩` minimally allows a potential term `T : PotentialTerm` if it allows the term `T` and no strict subset of `x` allows `T`.

That is to say, you need all of the charges in `x` to allow the term `T`.

We show that any charge spectrum which allows `T` has a subset which minimally allows `T`.

We show that every charge spectrum which minimally allows `T` is of the form `allowsTermForm a b c T` for some `a b c : 𝓩`, and the reverse is true for `T` not equal to `W1` or `W2`.

ii. Key results

- `MinimallyAllowsTerm` : Predicate on charge spectra which is true if the charge spectrum minimally allows a potential term. - `allowsTerm_iff_subset_minimallyAllowsTerm` : A charge spectrum which allows a term has a subset which minimally allows the term, and vice versa. - `eq_allowsTermForm_of_minimallyAllowsTerm` : Any charge spectrum which minimally allows a term is of the form `allowsTermForm a b c T` for some `a b c : 𝓩`.

iii. Table of contents

- A. Charge spectra which minimally allow potential terms - A.1. Decidability of `MinimallyAllowsTerm` - A.2. A charge spectrum which minimally allows a term allows the term - A.3. Spectrum with a subset which minimally allows a term, allows the term - A.4. Minimally allows term iff only member of powerset allowing term - A.5. Minimally allows term iff powerset allowing term has cardinal one - A.6. A charge spectrum which allows a term has a subset which minimally allows the term - A.7. A charge spectrum allows a term iff it has a subset which minimally allows the term - A.8. Cardinality of spectrum which minimally allows term is at most degree of term - B. Relation between `MinimallyAllowsTerm` and `allowsTermForm` - B.1. A charge spectrum which minimally allows a term is of the form `allowsTermForm a b c T` - B.2. `allowsTermForm a b c T` minimally allows `T` if `T` is not `W1` or `W2`

iv. References

There are no known references for this material.

A. Charge spectra which minimally allow potential terms

We define the predicate `MinimallyAllowsTerm` on charge spectra which is true if the charge spectrum allows a given potential term and no strict subset of it allows the term.

We prove properties of charge spectra which minimally allow potential terms.

A.1. Decidability of `MinimallyAllowsTerm`

We show that `MinimallyAllowsTerm` is decidable.

A.2. A charge spectrum which minimally allows a term allows the term

Somewhat trivially a charge spectrum which minimally allows the term does indeed allow the term.

A.3. Spectrum with a subset which minimally allows a term, allows the term

If a charge spectrum `x` has a subset which minimally allows a term `T`, then `x` allows `T`.

A.4. Minimally allows term iff only member of powerset allowing term

A charge spectrum `x` minimally allows a term `T` if and only if the only member of its own powerset which allows `T` is itself.

A.5. Minimally allows term iff powerset allowing term has cardinal one

A charge spectrum `x` minimally allows a term `T` if and only the the number of members of its powerset which allow `T` is one.

A.6. A charge spectrum which allows a term has a subset which minimally allows the term

If a charge spectrum `x` allows a term `T`, then it has a subset which minimally allows `T`.

A.7. A charge spectrum allows a term iff it has a subset which minimally allows the term

We combine results above to show that a charge spectrum allows a term if and only if it has a subset which minimally allows the term.

A.8. Cardinality of spectrum which minimally allows term is at most degree of term

We show that the cardinality of a charge spectrum which minimally allows a term `T` is at most the degree of `T`.

B. Relation between `MinimallyAllowsTerm` and `allowsTermForm`

We now relate the predicate `MinimallyAllowsTerm` to charge spectra of the form `allowsTermForm a b c T`.

B.1. A charge spectrum which minimally allows a term is of the form `allowsTermForm a b c T`

We show that any charge spectrum which minimally allows a term `T` is of the form `allowsTermForm a b c T` for some `a b c : 𝓩`.

B.2. `allowsTermForm a b c T` minimally allows `T` if `T` is not `W1` or `W2`

We show that charge spectra of the form `allowsTermForm a b c T` minimally allow `T` provided that `T` is not one of `W1` or `W2`.

10 declarations

definition

Charge spectrum $x$ minimally allows potential term $T$

#MinimallyAllowsTerm

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 $T$ .

instance

Decidability of $x$ minimally allowing $T$

#instDecidableMinimallyAllowsTerm

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 $y = x$ .

theorem

If $x$ Minimally Allows $T$ , then $x$ Allows $T$

#allowsTerm_of_minimallyAllowsTerm

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

theorem

A Spectrum Allows a Term if it Contains a Subset that Minimally Allows it

#allowsTerm_of_has_minimallyAllowsTerm_subset

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 $T$ .

theorem

$x$ Minimally Allows $T$ iff $x$ is the Only Sub-Spectrum that Allows $T$

#minimallyAllowsTerm_iff_powerset_filter_eq

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 $\{x\}$ .

theorem

$x$ minimally allows $T \iff$ exactly one subset of $x$ allows $T$

#minimallyAllowsTerm_iff_powerset_countP_eq_one

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.

theorem

Every Charge Spectrum Allowing $T$ Contains a Subset Minimally Allowing $T$

#subset_minimallyAllowsTerm_of_allowsTerm

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 $T$ .

theorem

$x$ allows $T \iff \exists y \subseteq x$ such that $y$ minimally allows $T$

#allowsTerm_iff_subset_minimallyAllowsTerm

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 $T$ .

theorem

Cardinality of a Spectrum Minimally Allowing $T$ is at most $\text{deg}(T)$

#card_le_degree_of_minimallyAllowsTerm

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 $T$ .

theorem

$\text{allowsTermForm}(a, b, c, T)$ minimally allows $T$ for $T \neq W^1, W^2$

#allowsTermForm_minimallyAllowsTerm

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 $T$ .

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

Charge spectrum which minimally allows terms

i. Overview

ii. Key results

iii. Table of contents

iv. References

A. Charge spectra which minimally allow potential terms

A.1. Decidability of `MinimallyAllowsTerm`

A.2. A charge spectrum which minimally allows a term allows the term

A.3. Spectrum with a subset which minimally allows a term, allows the term

A.4. Minimally allows term iff only member of powerset allowing term

A.5. Minimally allows term iff powerset allowing term has cardinal one

A.6. A charge spectrum which allows a term has a subset which minimally allows the term

A.7. A charge spectrum allows a term iff it has a subset which minimally allows the term

A.8. Cardinality of spectrum which minimally allows term is at most degree of term

B. Relation between `MinimallyAllowsTerm` and `allowsTermForm`

B.1. A charge spectrum which minimally allows a term is of the form `allowsTermForm a b c T`

B.2. `allowsTermForm a b c T` minimally allows `T` if `T` is not `W1` or `W2`

Charge spectrum xxx minimally allows potential term TTT

Decidability of xxx minimally allowing TTT

If xxx Minimally Allows TTT, then xxx Allows TTT

A Spectrum Allows a Term if it Contains a Subset that Minimally Allows it

xxx Minimally Allows TTT iff xxx is the Only Sub-Spectrum that Allows TTT

xxx minimally allows T ⟺ T \iffT⟺ exactly one subset of xxx allows TTT

Every Charge Spectrum Allowing TTT Contains a Subset Minimally Allowing TTT

xxx allows T ⟺ ∃y⊆xT \iff \exists y \subseteq xT⟺∃y⊆x such that yyy minimally allows TTT

Cardinality of a Spectrum Minimally Allowing TTT is at most deg(T)\text{deg}(T)deg(T)

allowsTermForm(a,b,c,T)\text{allowsTermForm}(a, b, c, T)allowsTermForm(a,b,c,T) minimally allows TTT for T≠W1,W2T \neq W^1, W^2T=W1,W2

Charge spectrum $x$ minimally allows potential term $T$

Decidability of $x$ minimally allowing $T$

If $x$ Minimally Allows $T$ , then $x$ Allows $T$

$x$ Minimally Allows $T$ iff $x$ is the Only Sub-Spectrum that Allows $T$

$x$ minimally allows $T \iff$ exactly one subset of $x$ allows $T$

Every Charge Spectrum Allowing $T$ Contains a Subset Minimally Allowing $T$

$x$ allows $T \iff \exists y \subseteq x$ such that $y$ minimally allows $T$

Cardinality of a Spectrum Minimally Allowing $T$ is at most $\text{deg}(T)$

$\text{allowsTermForm}(a, b, c, T)$ minimally allows $T$ for $T \neq W^1, W^2$