e^{i \alpha} & 0\\
0 & e^{-i \alpha}
\end{array}
\right)_{\alpha\neq \pi/2},
\\
CP1: &\hspace{1cm} &
X =
\left(
\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}
\right),
\\
CP2: &\hspace{1cm} &
X =
\left(
\begin{array}{cc}
0 & 1\\
-1 & 0
\end{array}
\right),
\\
CP3: &\hspace{1cm} &
X =
\left(
\begin{array}{cc}
\cos{\theta} & \sin{\theta}\\
-\sin{\theta} & \cos{\theta}
\end{array}
\right).\end{aligned}$$ Here $0<\alpha<\pi$ (but $\alpha\neq \pi/2$, since the case of $\alpha=\pi/2$ corresponds to the $Z_2$ symmetry), and $0<\theta<\pi/2$. Invariance under the full $U(2)$ global symmetry[^2] is obtained by requiring the invariance of the scalar potential under Eq. (\[S-transf-symmetry\]), for *all* unitary matrices $S$.

\[sec:MSSM\]Basis invariant probes of the MSSM
==============================================

\[sec:Higgs\_MSSM\]The Higgs sector of the MSSM
-----------------------------------------------

The Higgs potential of the MSSM (prior to including soft-supersymmetry-breaking dimension-two squared-mass terms) is a particular case of Eq. (\[VH1\]), with $$\begin{aligned}
m_{11}^2 &=& m_{22}^2,
\label{MSSM_m11_m22}
\\
m_{12}^2 &=& 0,
\label{MSSM_m12}
\\
\lambda_1 = \lambda_2 &=&
\tfrac{1}{4} (g^2 + {g^\prime}^2),
\label{MSSM_L1_L2}
\\
\lambda_3 &=&
\tfrac{1}{4} (g^2 - {g^\prime}^2),
\label{MSSM_L3}
\\
\lambda_4 &=&
- \tfrac{1}{2} g^2,
\label{MSSM_L4}
\\
\lambda_5=\lambda_6=\lambda_7 &=& 0,
\label{MSSM_L5_L6_L7}\end{aligned}$$ where $g$ and $g^\prime$ are the $SU(2)_L$ and $U(1)_Y$ gauge coupling constants, respectively. In this case, Eqs. (\[ynum\]) and (\[znum\]) become $$\begin{aligned}
Y_{11}=Y_{22}, &&
