of the invariant Higgs fields $H_U$, $H_D$, $H_U^\prime$ and $H_D^\prime$ \[defined by \], can be rewritten as: \[gensusygauginoinv\] [L]{}\_= \^\^\_[ij]{}(|f\_U|\^j\_[H\_U]{}H\_U\^[i]{} +|f\_D|\^j\_[H\_D]{}H\_D\^[i]{}) +\^(|f\^\_U|\^i\_[H\_U]{} H\_U\^[i]{} -|f\_D\^|\^i\_[H\_D]{}H\_D\^[i]{})+[h.c.]{} However, this form is not particularly useful outside of the supersymmetric limit, since $\{H_U\,,\,\widetilde H_D\}$ and $\{H^\prime_U\,,\,\widetilde H^\prime_D\}$ are *not* orthogonal pairs of hypercharge-one Higgs doublet fields in the general case. Of course, one can always rewrite in terms of one of the four basis choices of hyper-charge one doublet Higgs fields listed in by inserting the appropriate form for $\Phi_a$ given in into .

\[sec:ESL\]Basis-invariant probes of the supersymmetric Higgs interactions
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Supersymmetry imposes strong constraints on the scalar Higgs potential and the gaugino-higgsino-Higgs interactions. These constraints must involve basis-independent combinations of the scalar potential parameters $Y_{ab}$, $Z_{abcd}$, and the gaugino-higgsino-Higgs couplings $f^a_U$, $f_D^a$, $f^{\prime\,a}_U$, $f^{\prime\,a}_D$. It is straightforward to find the necessary relations. First we exhibit the basis invariant relations that enforce supersymmetric gaugino-higgsino-Higgs couplings:

$$\begin{aligned}
f_U^a f_D^{a \ast} = 0,
&\hspace{1cm}&
f_U^a f_D^{\prime a \ast} = 0,
\nonumber\\
f_U^{\prime a} f_D^{a \ast} = 0,
&\hspace{1cm}&
f_U^{\prime a} f_D^{\prime a \ast} = 0,
\nonumber\\
f_U^a f_U^{a \ast} = f_D^a f_D^{a \ast}= g^2,
&\hspace{1cm}&
f_U^{\prime a} f_U^{\prime a \ast}
= f_D^{\prime a} f_D^{\prime a \ast} 