Physlib.Particles.SuperSymmetry.SU5.FieldLabels

The field labels

i. Overview

To each field and it's conjugate in the `SU(5)` SUSY GUT we assign a label, namely an element of the inductive type `FieldLabel`.

This field label will be used to specify the representations of fields present in terms of the lagrangian.

ii. Key results

The key results are - `FieldLabel` : the inductive type of field labels. - `FieldLabel.RParity` : the R-parity of a field label, being `1` if it is in the non-trivial representation and `0` otherwise.

iii. Table of contents

A. The field labels
B. R-parity of a field label

iv. References

A. The field labels

The inductive type `FieldLabel` is defined, with one constructor for each field and it's conjugate present in the `SU(5)` SUSY GUT.

Note that there is only one `10` representation, as it is self-conjugate.

B. R-parity of a field label

We define the R-parity of a field label, being `1` if it is in the non-trivial representation and `0` otherwise.

4 declarations

inductive

Field labels in $SU(5)$ SUSY GUT

#FieldLabel

The type $\text{FieldLabel}$ is an inductive set consisting of labels for each field and its conjugate within an $SU(5)$ Supersymmetric Grand Unified Theory (SUSY GUT). These labels are used to specify the representations of the fields appearing in the Lagrangian of the theory.

instance

Decidability of equality for field labels in $SU(5)$ SUSY GUT

#instDecidableEqFieldLabel

The equality relation between any two field labels in the set $\text{FieldLabel}$ of the $SU(5)$ Supersymmetric Grand Unified Theory (SUSY GUT) is decidable. This implies that there is a computational procedure to determine whether two given labels represent the same field.

instance

The set of field labels $\text{FieldLabel}$ is finite

#instFintypeFieldLabel

The set of field labels in the $SU(5)$ Supersymmetric Grand Unified Theory (SUSY GUT), denoted as $\text{FieldLabel}$ , is finite. This ensures that there are a finite number of distinct labels representing the fields and their conjugates within the theory.

definition

R-parity of field labels in $SU(5)$ SUSY GUT

#RParity

The function assigns an R-parity value in $\text{Fin } 2$ to a given field label $\ell$ within an $SU(5)$ Supersymmetric Grand Unified Theory (SUSY GUT). Specifically, the Higgs field labels—representing the up-type Higgs ( $5_{H_u}, \bar{5}_{H_u}$ ) and down-type Higgs ( $5_{H_d}, \bar{5}_{H_d}$ )—are mapped to $0$ , while the matter field labels (in the $\mathbf{5}$ , $\mathbf{\bar{5}}$ , and $\mathbf{10}$ representations) are mapped to $1$ .