our notation. In section \[sec:MSSM\], we construct the basis invariant quantities that identify the supersymmetric limit of the scalar sector of the THDM. We draw our conclusions in section \[sec:conclusions\].

\[sec:notation\]The scalar sector of the THDM
=============================================

The scalar potential
--------------------

Let us consider a $SU(2) \otimes U(1)$ gauge theory with two hypercharge-one Higgs-doublets, denoted by $\Phi_a$, where $a=1,2$. The scalar potential may be written as $$\begin{aligned}
V_H
&=&
m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2
- \left[ m_{12}^2 \Phi_1^\dagger \Phi_2 + \textrm{h.c.} \right]
\nonumber\\[6pt]
&&
+ \tfrac{1}{2} \lambda_1 (\Phi_1^\dagger\Phi_1)^2
+ \tfrac{1}{2} \lambda_2 (\Phi_2^\dagger\Phi_2)^2
+ \lambda_3 (\Phi_1^\dagger\Phi_1) (\Phi_2^\dagger\Phi_2)
+ \lambda_4 (\Phi_1^\dagger\Phi_2) (\Phi_2^\dagger\Phi_1)
\nonumber\\[6pt]
&&
+ \left[
\tfrac{1}{2} \lambda_5 (\Phi_1^\dagger\Phi_2)^2
+ \lambda_6 (\Phi_1^\dagger\Phi_1) (\Phi_1^\dagger\Phi_2)
+ \lambda_7 (\Phi_2^\dagger\Phi_2) (\Phi_1^\dagger\Phi_2)
+ \textrm{h.c.}
\right],
\label{VH1}\end{aligned}$$ where $m_{11}^2$, $m_{22}^2$, and $\lambda_1,\lambda_2,\lambda_3,\lambda_4$ are real parameters, and $m_{12}^2$, $\lambda_5$, $\lambda_6$ and $\lambda_7$ are potentially complex. An alternative notation, useful for the construction of invariants and championed by Botella and Silva [@BS] is $$\begin{aligned}
V_H
&=&
Y_{ab} (\Phi_a^\dagger \Phi_b) +
\tfrac{1}{2}
Z_{ab,cd} (\Phi_a^\dagger \Phi_b) (\Phi_c^\dagger \Phi_d),
\label{VH2}\end{aligned}$$ where Hermiticity implies $$\begin{aligned}
Y_{ab} &=& Y_{ba}^\ast,
\nonumber\\
Z_{ab,cd} \equiv Z_{cd,ab} &=& Z_{ba,dc}^\ast.
\label{hermiticity_coefficients}\end{aligned}$$ One should be very careful when comparing Eqs. (\[VH1\]) and (\[VH2\]) among different authors, since the same 