Physlib.Particles.SuperSymmetry.SU5.FieldLabels

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.

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.

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.

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 $1$.