Physlib.Particles.SuperSymmetry.SU5.Potential

The inductive type `PotentialTerm` serves as an index for the various interaction terms appearing in the superpotential $W$ and the Kähler potential $K$ of a supersymmetric $SU(5)$ Grand Unified Theory (GUT). These terms involve matter fields in the $\mathbf{\overline{5}}$ (denoted $\bar{5}M$) and $\mathbf{10}$ representations, as well as Higgs fields in the $\mathbf{5}$ and $\mathbf{\overline{5}}$ representations (denoted $5H_u$ and $\bar{5}H_d$). The specific terms indexed by this type include: - Superpotential terms: - $\mu$: The Higgs mass term $\mu (5H_u)(\bar{5}H_d)$. - $\beta_i$: The matter-Higgs mixing term $\beta_i (\bar{5}M^i)(5H_u)$. - $\lambda_{ijk}$: The trilinear matter interaction $\lambda_{ijk} (\bar{5}M^i)(\bar{5}M^j)(10^k)$. - $W^1_{ijkl}$: The dimension-5 operator $10^i 10^j 10^k \bar{5}M^l$. - $W^2_{ijk}$: The operator $10^i 10^j 10^k \bar{5}H_d$. - $W^3_{ij}$: The operator $\bar{5}M^i \bar{5}M^j 5H_u 5H_u$. - $W^4_i$: The operator $\bar{5}M^i \bar{5}H_d 5H_u 5H_u$. - Kähler potential terms: - $K^1_{ijk}$: The term $10^i 10^j 5M^k$. - $K^2_i$: The term $\bar{5}H_u \bar{5}H_d 10^i$.

The equality of interaction terms in the $SU(5)$ supersymmetric Grand Unified Theory (GUT), represented by the type `PotentialTerm`, is decidable. This implies that for any two potential terms—such as the Higgs mass term $\mu$, the matter-Higgs mixing terms $\beta_i$, or the various superpotential ($W^{1-4}$) and Kähler potential ($K^{1-2}$) operators—it can be algorithmically determined whether they are identical.

The set of potential terms in the $SU(5)$ supersymmetric Grand Unified Theory (GUT), represented by the type `PotentialTerm`, is finite. This type indexes the interaction terms present in the superpotential $W$ and the Kähler potential $K$, including the Higgs mass term $\mu$, the matter-Higgs mixing term $\beta$, the trilinear interaction $\lambda$, the higher-dimension operators $W^{1-4}$, the Kähler terms $K^{1-2}$, and the Yukawa couplings.

This function maps a potential term $T$ in the $SU(5)$ supersymmetric GUT model to the list of field labels that constitute that term. For each term $T \in \text{PotentialTerm}$, it returns the sequence of matter ($M$) or Higgs ($H_{u/d}$) field representations ($\mathbf{5}, \mathbf{\bar{5}}, \mathbf{10}$) involved: - $\mu \mapsto [\mathbf{\bar{5}}_{H_d}, \mathbf{5}_{H_u}]$ (Higgs mass term) - $\beta \mapsto [\mathbf{5}_{H_u}, \mathbf{\bar{5}}_M]$ (Matter-Higgs mixing) - $\Lambda \mapsto [\mathbf{\bar{5}}_M, \mathbf{\bar{5}}_M, \mathbf{10}_M]$ (Trilinear matter interaction) - $W^1 \mapsto [\mathbf{10}_M, \mathbf{10}_M, \mathbf{10}_M, \mathbf{\bar{5}}_M]$ (Dimension-5 operator) - $W^2 \mapsto [\mathbf{10}_M, \mathbf{10}_M, \mathbf{10}_M, \mathbf{\bar{5}}_{H_d}]$ (Higgs-mediated operator) - $W^3 \mapsto [\mathbf{\bar{5}}_M, \mathbf{\bar{5}}_M, \mathbf{5}_{H_u}, \mathbf{5}_{H_u}]$ - $W^4 \mapsto [\mathbf{\bar{5}}_M, \mathbf{\bar{5}}_{H_d}, \mathbf{5}_{H_u}, \mathbf{5}_{H_u}]$ - $K^1 \mapsto [\mathbf{10}_M, \mathbf{10}_M, \mathbf{5}_M]$ (Kähler term) - $K^2 \mapsto [\mathbf{\bar{5}}_{H_u}, \mathbf{\bar{5}}_{H_d}, \mathbf{10}_M]$ (Kähler term) - $\text{topYukawa} \mapsto [\mathbf{10}_M, \mathbf{10}_M, \mathbf{5}_{H_u}]$ (Top quark Yukawa coupling) - $\text{bottomYukawa} \mapsto [\mathbf{10}_M, \mathbf{\bar{5}}_M, \mathbf{\bar{5}}_{H_d}]$ (Bottom quark Yukawa coupling)

The predicate $\text{InSuperPotential}$ determines whether a given potential term $T$ of type $\text{PotentialTerm}$ belongs to the superpotential $W$ of the $SU(5)$ supersymmetric GUT. It is defined to be true for the terms $\mu$, $\beta$, $\Lambda$, $W^1$, $W^2$, $W^3$, $W^4$, $\text{topYukawa}$, and $\text{bottomYukawa}$, and false for the Kähler potential terms $K^1$ and $K^2$.

The predicate $\text{InSuperPotential}(T)$, which determines whether a given potential term $T$ in the $SU(5)$ supersymmetric Grand Unified Theory (GUT) belongs to the superpotential $W$, is decidable. Specifically, for any term $T \in \text{PotentialTerm}$, the predicate is true for the superpotential terms $T \in \{\mu, \beta, \Lambda, W^1, W^2, W^3, W^4, \text{topYukawa}, \text{bottomYukawa}\}$ and false for the Kähler potential terms $T \in \{K^1, K^2\}$.

For any interaction term $T$ in the $SU(5)$ supersymmetric Grand Unified Theory (GUT), if $T$ is part of the superpotential $W$ (i.e., the predicate $\text{InSuperPotential}(T)$ holds), then the list of its constituent field labels $T.\text{toFieldLabel}$ does not contain the matter representation $\mathbf{5}_M$, the Higgs representation $\mathbf{5}_{H_d}$, or the conjugate Higgs representation $\mathbf{\bar{5}}_{H_u}$.

The degree of a potential term $T$ in the $SU(5)$ SUSY GUT model is defined as the number of fields (represented by their `FieldLabel`s) that constitute the term. Mathematically, for a term $T \in \text{PotentialTerm}$, its degree is the length of the list of field labels returned by the mapping $T.\text{toFieldLabel}$. For example, the Higgs mass term $\mu$ has a degree of 2, while the trilinear matter interaction $\lambda$ has a degree of 3.

In the $SU(5)$ supersymmetric Grand Unified Theory (GUT), for any interaction term $T$ in the superpotential or Kähler potential (including terms such as $\mu$, $\beta$, $\lambda$, $W^1$, $W^2$, $W^3$, $W^4$, $K^1$, and $K^2$), the degree of the term is at most 4. Here, the degree $\text{degree}(T)$ is defined as the number of constituent field labels in the term.

For a potential term $T$ in the $SU(5)$ SUSY GUT model, the R-parity is defined as the sum of the R-parities of its constituent fields. Specifically, if $T$ consists of a list of field labels $[f_1, f_2, \dots, f_n]$ (determined by `toFieldLabel`), its R-parity is calculated as $\sum_{i=1}^n R(f_i) \pmod 2$, where $R(f_i) \in \{0, 1\}$ is the R-parity of the field $f_i$. In this formulation, a total R-parity of $0$ corresponds to a term that preserves R-parity, while $1$ corresponds to a term that violates it.

In the $SU(5)$ supersymmetric Grand Unified Theory (GUT), a potential term $T$ (representing an interaction in the superpotential $W$ or the Kähler potential $K$) violates R-parity, denoted by $T.\text{RParity} = 1$, if and only if $T$ belongs to the set $\{\beta, \lambda, W^2, W^4, K^1, K^2\}$. These terms include the matter-Higgs mixing term $\beta$, the trilinear matter interaction $\lambda$, the higher-dimensional superpotential operators $W^2$ and $W^4$, and the Kähler potential terms $K^1$ and $K^2$.

The finite set of interaction terms in the $SU(5)$ supersymmetric Grand Unified Theory (GUT) potential that contribute to proton decay. This set consists of the following operators: - $W^1$: The dimension-5 superpotential operator $10^i 10^j 10^k \bar{5}M^l$. - $\lambda$: The trilinear matter superpotential interaction $\lambda_{ijk} \bar{5}M^i \bar{5}M^j 10^k$. - $W^2$: The superpotential operator $10^i 10^j 10^k \bar{5}H_d$. - $K^1$: The Kähler potential term $10^i 10^j 5M^k$.