the THDM
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Symmetries leaving the scalar Lagrangian unchanged may be of two types. On the one hand, one may relate $\Phi_a$ with some unitary transformation of $\Phi_b$: $$\Phi_a \rightarrow \Phi_a^S = S_{ab} \Phi_b,
\label{S-transf-symmetry}$$ where $S$ is a unitary matrix. These are known as Higgs Family symmetries, or HF symmetries. As a result of this symmetry, $$\begin{aligned}
Y_{a b} & = &
S_{a \alpha}\, Y_{\alpha \beta}\, S_{b \beta}^\ast ,
\label{Y-S}
\\
Z_{ab,cd} & = &
S_{a \alpha}\, S_{c \gamma}\,
Z_{\alpha \beta, \gamma \delta}\, S_{b \beta}^\ast \, S_{d \delta}^\ast .
\label{Z-S}\end{aligned}$$ On the other hand, one may relate $\Phi_a$ with some unitary transformation of $\Phi_b^\ast$: $$\Phi_a \rightarrow \Phi^{\textrm{GCP}}_a
= X_{a \alpha} \Phi_\alpha^\ast,
\label{GCP}$$ where $X$ is an arbitrary unitary matrix.[^1] These are known as generalized CP symmetries, or GCP symmetries [@GCP1; @GCP2]. The potential is invariant under this symmetry if and only if $$\begin{aligned}
Y_{ab}^\ast
&=&
X_{\alpha a}^\ast Y_{\alpha \beta} X_{\beta b}
\nonumber\\
Z_{ab,cd}^\ast
&=&
X_{\alpha a}^\ast X_{\gamma c}^\ast
Z_{\alpha \beta, \gamma \delta} X_{\beta b} X_{\delta d}.
\label{YZ-CPtransf}\end{aligned}$$ Under the basis transformation of Eq. (\[basis-transf\]), the specific forms of the HF and GCP symmetries are altered, respectively, as follows: $$\begin{aligned}
S^\prime &=& U S U^\dagger ,
\label{S-prime}
\\
X^\prime &=& 